3654:
2929:
3649:{\displaystyle {\begin{aligned}ds^{2}&={\begin{bmatrix}du&dv\end{bmatrix}}{\begin{bmatrix}E&F\\F&G\end{bmatrix}}{\begin{bmatrix}du\\dv\end{bmatrix}}\\&={\begin{bmatrix}du'&dv'\end{bmatrix}}{\begin{bmatrix}{\dfrac {\partial u}{\partial u'}}&{\dfrac {\partial u}{\partial v'}}\\{\dfrac {\partial v}{\partial u'}}&{\dfrac {\partial v}{\partial v'}}\end{bmatrix}}^{\mathsf {T}}{\begin{bmatrix}E&F\\F&G\end{bmatrix}}{\begin{bmatrix}{\dfrac {\partial u}{\partial u'}}&{\dfrac {\partial u}{\partial v'}}\\{\dfrac {\partial v}{\partial u'}}&{\dfrac {\partial v}{\partial v'}}\end{bmatrix}}{\begin{bmatrix}du'\\dv'\end{bmatrix}}\\&={\begin{bmatrix}du'&dv'\end{bmatrix}}{\begin{bmatrix}E'&F'\\F'&G'\end{bmatrix}}{\begin{bmatrix}du'\\dv'\end{bmatrix}}\\&=(ds')^{2}\,.\end{aligned}}}
14777:
2382:
14374:
4512:
14772:{\displaystyle D\varphi ={\begin{bmatrix}{\frac {\partial \varphi ^{1}}{\partial x^{1}}}&{\frac {\partial \varphi ^{1}}{\partial x^{2}}}&\dots &{\frac {\partial \varphi ^{1}}{\partial x^{n}}}\\{\frac {\partial \varphi ^{2}}{\partial x^{1}}}&{\frac {\partial \varphi ^{2}}{\partial x^{2}}}&\dots &{\frac {\partial \varphi ^{2}}{\partial x^{n}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial \varphi ^{m}}{\partial x^{1}}}&{\frac {\partial \varphi ^{m}}{\partial x^{2}}}&\dots &{\frac {\partial \varphi ^{m}}{\partial x^{n}}}\end{bmatrix}}.}
2018:
5101:
3977:
5850:
2377:{\displaystyle {\begin{bmatrix}E'&F'\\F'&G'\end{bmatrix}}={\begin{bmatrix}{\frac {\partial u}{\partial u'}}&{\frac {\partial u}{\partial v'}}\\{\frac {\partial v}{\partial u'}}&{\frac {\partial v}{\partial v'}}\end{bmatrix}}^{\mathsf {T}}{\begin{bmatrix}E&F\\F&G\end{bmatrix}}{\begin{bmatrix}{\frac {\partial u}{\partial u'}}&{\frac {\partial u}{\partial v'}}\\{\frac {\partial v}{\partial u'}}&{\frac {\partial v}{\partial v'}}\end{bmatrix}}}
4507:{\displaystyle {\begin{aligned}\mathbf {a} \cdot \mathbf {b} &=a_{1}b_{1}{\vec {r}}_{u}\cdot {\vec {r}}_{u}+a_{1}b_{2}{\vec {r}}_{u}\cdot {\vec {r}}_{v}+a_{2}b_{1}{\vec {r}}_{v}\cdot {\vec {r}}_{u}+a_{2}b_{2}{\vec {r}}_{v}\cdot {\vec {r}}_{v}\\&=a_{1}b_{1}E+a_{1}b_{2}F+a_{2}b_{1}F+a_{2}b_{2}G.\\&={\begin{bmatrix}a_{1}&a_{2}\end{bmatrix}}{\begin{bmatrix}E&F\\F&G\end{bmatrix}}{\begin{bmatrix}b_{1}\\b_{2}\end{bmatrix}}\,.\end{aligned}}}
982:
18926:
2918:
4840:
5496:
19827:
11221:
20432:
12514:
18634:
5096:{\displaystyle {\begin{aligned}g\left(\lambda \mathbf {a} +\mu \mathbf {a} ',\mathbf {b} \right)&=\lambda g(\mathbf {a} ,\mathbf {b} )+\mu g\left(\mathbf {a} ',\mathbf {b} \right),\quad {\text{and}}\\g\left(\mathbf {a} ,\lambda \mathbf {b} +\mu \mathbf {b} '\right)&=\lambda g(\mathbf {a} ,\mathbf {b} )+\mu g\left(\mathbf {a} ,\mathbf {b} '\right)\end{aligned}}}
603:
5845:{\displaystyle {\begin{aligned}&\iint _{D}{\sqrt {\left({\vec {r}}_{u}\cdot {\vec {r}}_{u}\right)\left({\vec {r}}_{v}\cdot {\vec {r}}_{v}\right)-\left({\vec {r}}_{u}\cdot {\vec {r}}_{v}\right)^{2}}}\,du\,dv\\={}&\iint _{D}{\sqrt {EG-F^{2}}}\,du\,dv\\={}&\iint _{D}{\sqrt {\det {\begin{bmatrix}E&F\\F&G\end{bmatrix}}}}\,du\,dv\end{aligned}}}
19190:
2572:
6445:
459:-plane. One of the chief aims of Gauss's investigations was to deduce those features of the surface which could be described by a function which would remain unchanged if the surface underwent a transformation in space (such as bending the surface without stretching it), or a change in the particular parametric form of the same geometrical surface.
9596:
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2697:
17557:
18623:
1951:
10895:
17792:
17180:
20165:
12190:
18921:{\displaystyle g=J^{\mathsf {T}}J={\begin{bmatrix}\cos ^{2}\theta +\sin ^{2}\theta &-r\sin \theta \cos \theta +r\sin \theta \cos \theta \\-r\cos \theta \sin \theta +r\cos \theta \sin \theta &r^{2}\sin ^{2}\theta +r^{2}\cos ^{2}\theta \end{bmatrix}}={\begin{bmatrix}1&0\\0&r^{2}\end{bmatrix}}}
3962:
6103:
977:{\displaystyle {\begin{aligned}s&=\int _{a}^{b}\left\|{\frac {d}{dt}}{\vec {r}}(u(t),v(t))\right\|\,dt\\&=\int _{a}^{b}{\sqrt {u'(t)^{2}\,{\vec {r}}_{u}\cdot {\vec {r}}_{u}+2u'(t)v'(t)\,{\vec {r}}_{u}\cdot {\vec {r}}_{v}+v'(t)^{2}\,{\vec {r}}_{v}\cdot {\vec {r}}_{v}}}\,dt\,,\end{aligned}}}
1696:
18984:
10630:
11569:
8759:
17334:
For a pseudo-Riemannian metric, the length formula above is not always defined, because the term under the square root may become negative. We generally only define the length of a curve when the quantity under the square root is always of one sign or the other. In this case, define
1156:
1475:
8940:
8250:
14172:
19822:{\displaystyle g={\begin{bmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{bmatrix}}\quad {\text{or}}\quad g={\begin{bmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}}\,.}
9376:
7828:
5330:
2913:{\displaystyle {\begin{bmatrix}du\\dv\end{bmatrix}}={\begin{bmatrix}{\dfrac {\partial u}{\partial u'}}&{\dfrac {\partial u}{\partial v'}}\\{\dfrac {\partial v}{\partial u'}}&{\dfrac {\partial v}{\partial v'}}\end{bmatrix}}{\begin{bmatrix}du'\\dv'\end{bmatrix}}}
18069:
11216:{\displaystyle {\begin{aligned}&\alpha G^{-1}\beta ^{\mathsf {T}}\\={}&\left(\alpha A\right)\left(A^{-1}G^{-1}\left(A^{-1}\right)^{\mathsf {T}}\right)\left(A^{\mathsf {T}}\beta ^{\mathsf {T}}\right)\\={}&\alpha G^{-1}\beta ^{\mathsf {T}}.\end{aligned}}}
2426:
17341:
18468:
10338:
11907:
13223:
8370:
8615:
17580:
7454:
20027:
16974:
15476:
20427:{\displaystyle g_{\mu \nu }={\begin{bmatrix}\left(1-{\frac {2GM}{rc^{2}}}\right)&0&0&0\\0&-\left(1-{\frac {2GM}{rc^{2}}}\right)^{-1}&0&0\\0&0&-r^{2}&0\\0&0&0&-r^{2}\sin ^{2}\theta \end{bmatrix}}\,,}
12509:{\displaystyle {\begin{aligned}\alpha &=a_{1}\theta ^{1}+a_{2}\theta ^{2}+\cdots +a_{n}\theta ^{n}\\&={\big \lbrack }{\begin{array}{cccc}a_{1}&a_{2}&\dots &a_{n}\end{array}}{\big \rbrack }\theta \\&=a\theta \end{aligned}}}
7656:
12072:
4719:
15368:
15133:
17820:
In many cases, whenever a calculation calls for the length to be used, a similar calculation using the energy may be done as well. This often leads to simpler formulas by avoiding the need for the square-root. Thus, for example, the
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19185:{\displaystyle g_{ij}=\sum _{kl}\delta _{kl}{\frac {\partial x^{k}}{\partial q^{i}}}{\frac {\partial x^{l}}{\partial q^{j}}}=\sum _{k}{\frac {\partial x^{k}}{\partial q^{i}}}{\frac {\partial x^{k}}{\partial q^{j}}}.}
9249:
14859:
6440:{\displaystyle {\begin{aligned}g_{p}(aU_{p}+bV_{p},Y_{p})&=ag_{p}(U_{p},Y_{p})+bg_{p}(V_{p},Y_{p})\,,\quad {\text{and}}\\g_{p}(Y_{p},aU_{p}+bV_{p})&=ag_{p}(Y_{p},U_{p})+bg_{p}(Y_{p},V_{p})\,.\end{aligned}}}
2934:
1570:
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9591:{\displaystyle q_{m}\left(\sum _{i}\xi ^{i}X_{i}\right)=\left(\xi ^{1}\right)^{2}+\left(\xi ^{2}\right)^{2}+\cdots +\left(\xi ^{p}\right)^{2}-\left(\xi ^{p+1}\right)^{2}-\cdots -\left(\xi ^{n}\right)^{2}}
8067:
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1285:
608:
528:
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15289:
7699:
2567:{\displaystyle J={\begin{bmatrix}{\frac {\partial u}{\partial u'}}&{\frac {\partial u}{\partial v'}}\\{\frac {\partial v}{\partial u'}}&{\frac {\partial v}{\partial v'}}\end{bmatrix}}\,.}
18473:
17833:: they describe the motion of a "free particle" (a particle feeling no forces) that is confined to move on the manifold, but otherwise moves freely, with constant momentum, within the manifold.
17552:{\displaystyle L=\int _{a}^{b}{\sqrt {\left|\sum _{i,j=1}^{n}g_{ij}(\gamma (t))\left({\frac {d}{dt}}x^{i}\circ \gamma (t)\right)\left({\frac {d}{dt}}x^{j}\circ \gamma (t)\right)\right|}}\,dt\,.}
16873:
10900:
9943:
6108:
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18618:{\displaystyle {\begin{aligned}x&=r\cos \theta \\y&=r\sin \theta \\J&={\begin{bmatrix}\cos \theta &-r\sin \theta \\\sin \theta &r\cos \theta \end{bmatrix}}\,.\end{aligned}}}
13773:
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7058:
13391:
17562:
While these formulas use coordinate expressions, they are in fact independent of the coordinates chosen; they depend only on the metric, and the curve along which the formula is integrated.
15602:
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10205:
11783:
6916:
13040:
8265:
20534:, the metric tensor was allowed to be non-symmetric; however, the antisymmetric part of such a tensor plays no role in the contexts described here, so it will not be further considered.
17787:{\displaystyle E={\frac {1}{2}}\int _{a}^{b}\sum _{i,j=1}^{n}g_{ij}(\gamma (t))\left({\frac {d}{dt}}x^{i}\circ \gamma (t)\right)\left({\frac {d}{dt}}x^{j}\circ \gamma (t)\right)\,dt\,.}
12675:
10127:. Whereas the metric itself provides a way to measure the length of (or angle between) vector fields, the inverse metric supplies a means of measuring the length of (or angle between)
8506:
13520:
17175:{\displaystyle L=\int _{a}^{b}{\sqrt {\sum _{i,j=1}^{n}g_{ij}(\gamma (t))\left({\frac {d}{dt}}x^{i}\circ \gamma (t)\right)\left({\frac {d}{dt}}x^{j}\circ \gamma (t)\right)}}\,dt\,.}
16427:
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1013:
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15393:
18149:
20543:
The notation of using square brackets to denote the basis in terms of which the components are calculated is not universal. The notation employed here is modeled on that of
7510:
11946:
16748:
3672:(non-euclidean geometry) of tangent vectors in a manner independent of the parametric description of the surface. Any tangent vector at a point of the parametric surface
18247:. The volume form also gives a way to integrate functions on the manifold, and this geometric integral agrees with the integral obtained by the canonical Borel measure.
13606:
20472:
18232:
4577:
15300:
15060:
9817:
3957:{\displaystyle {\begin{aligned}\mathbf {a} &=a_{1}{\vec {r}}_{u}+a_{2}{\vec {r}}_{v}\\\mathbf {b} &=b_{1}{\vec {r}}_{u}+b_{2}{\vec {r}}_{v}\end{aligned}}}
16260:
19444:
19254:
5386:
5159:
474:
of a piece of the surface. The study of these invariants of a surface led Gauss to introduce the predecessor of the modern notion of the metric tensor.
305:
14905:
20046:
10744:
10439:
7118:
1946:{\displaystyle E'={\vec {r}}_{u'}\cdot {\vec {r}}_{u'},\quad F'={\vec {r}}_{u'}\cdot {\vec {r}}_{v'},\quad G'={\vec {r}}_{v'}\cdot {\vec {r}}_{v'}.}
4746:
22711:
16771:
8983:
8381:
1691:{\displaystyle ds^{2}={\begin{bmatrix}du&dv\end{bmatrix}}{\begin{bmatrix}E&F\\F&G\end{bmatrix}}{\begin{bmatrix}du\\dv\end{bmatrix}}}
21902:
10625:{\displaystyle \alpha ={\big \lbrack }{\begin{array}{cccc}\alpha _{1}&\alpha _{2}&\dots &\alpha _{n}\end{array}}{\big \rbrack }\,.}
11564:{\displaystyle X=v^{1}X_{1}+v^{2}X_{2}+\dots +v^{n}X_{n}=\mathbf {f} {\begin{bmatrix}v^{1}\\v^{2}\\\vdots \\v^{n}\end{bmatrix}}=\mathbf {f} v}
9135:
8754:{\displaystyle {\frac {\partial }{\partial y^{i}}}=\sum _{k=1}^{n}{\frac {\partial x^{k}}{\partial y^{i}}}{\frac {\partial }{\partial x^{k}}}}
6488:
20950:
14807:
19336:
1151:{\displaystyle {\vec {r}}_{u}={\frac {\partial {\vec {r}}}{\partial u}}\,,\quad {\vec {r}}_{v}={\frac {\partial {\vec {r}}}{\partial v}}\,.}
22706:
20816:
9987:
1470:{\displaystyle E={\vec {r}}_{u}\cdot {\vec {r}}_{u},\quad F={\vec {r}}_{u}\cdot {\vec {r}}_{v},\quad G={\vec {r}}_{v}\cdot {\vec {r}}_{v}.}
15766:
14297:
13921:
3682:
18289:
8935:{\displaystyle g_{ij}\left=\sum _{k,l=1}^{n}{\frac {\partial x^{k}}{\partial y^{i}}}g_{kl}\left{\frac {\partial x^{l}}{\partial y^{j}}}.}
21993:
21113:
20963:
8245:{\displaystyle \mathbf {f} =\left(X_{1}={\frac {\partial }{\partial x^{1}}},\dots ,X_{n}={\frac {\partial }{\partial x^{n}}}\right)\,.}
19832:
For a curve with—for example—constant time coordinate, the length formula with this metric reduces to the usual length formula. For a
22017:
14167:{\displaystyle \varphi _{*}(v)=\sum _{i=1}^{n}\sum _{a=1}^{m}v^{i}{\frac {\partial \varphi ^{a}}{\partial x^{i}}}\mathbf {e} _{a}\,.}
6618:
3668:
to the surface, as well as the angle between two tangent vectors. In contemporary terms, the metric tensor allows one to compute the
12755:
2669:, that transformed in this way on passing from one system of coordinates to another. The upshot is that the first fundamental form (
2587:
22212:
21590:
18358:
17197:
16589:
10119:
7944:
3664:
Another interpretation of the metric tensor, also considered by Gauss, is that it provides a way in which to compute the length of
226:
14203:
6835:
1176:
11247:
whatsoever. Consequently, the equation may be assigned a meaning independently of the choice of basis. The entries of the matrix
5344:
is another numerical quantity which should depend only on the surface itself, and not on how it is parameterized. If the surface
480:
21143:
21118:
5325:{\displaystyle \cos(\theta )={\frac {g(\mathbf {a} ,\mathbf {b} )}{\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|}}\,.}
21739:
15627:
22082:
21005:
20898:
20751:
20719:
20660:
20630:
14285:
It follows from a straightforward calculation that the matrix of the induced metric in the basis of coordinate vector fields
7823:{\displaystyle \mathbf {f} \mapsto \mathbf {f} '=\left(\sum _{k}X_{k}a_{k1},\dots ,\sum _{k}X_{k}a_{kn}\right)=\mathbf {f} A}
10361:
22308:
18064:{\displaystyle \Lambda f=\int _{U}f\,d\mu _{g}=\int _{\varphi (U)}f\circ \varphi ^{-1}(x){\sqrt {\left|\det g\right|}}\,dx}
15248:
22361:
21889:
16839:
9864:
22645:
21774:
21453:
19425:
13668:
11625:
6980:
530:
in the description below; E, F, and G in the matrix can contain any number as long as the matrix is positive definite.
20:
13289:
10333:{\displaystyle \alpha _{p}\left(aX_{p}+bY_{p}\right)=a\alpha _{p}\left(X_{p}\right)+b\alpha _{p}\left(Y_{p}\right)\,.}
21233:
20978:
20943:
20867:
20799:
19206:
comes equipped with a natural metric induced from the ambient
Euclidean metric, through the process explained in the
15560:
15164:
11902:{\displaystyle \theta ^{i}(X_{j})={\begin{cases}1&\mathrm {if} \ i=j\\0&\mathrm {if} \ i\not =j.\end{cases}}}
13218:{\displaystyle v^{\mathsf {T}}G=v^{\mathsf {T}}\left(A^{-1}\right)^{\mathsf {T}}A^{\mathsf {T}}GA=v^{\mathsf {T}}GA}
8365:{\displaystyle g_{ij}\left=g\left({\frac {\partial }{\partial x^{i}}},{\frac {\partial }{\partial x^{j}}}\right)\,.}
22410:
21655:
20531:
20518:
17324:
8610:{\displaystyle g_{ij}\left=g\left({\frac {\partial }{\partial y^{i}}},{\frac {\partial }{\partial y^{j}}}\right).}
22393:
22002:
20766:
16158:
5884:
20682:
2679:
under changes in the coordinate system, and that this follows exclusively from the transformation properties of
22769:
18132:
12592:
22605:
22012:
21506:
21438:
13444:
9356:
9314:
7449:{\displaystyle g(v,w)=\sum _{i,j=1}^{n}v^{i}w^{j}g\left(X_{i},X_{j}\right)=\sum _{i,j=1}^{n}v^{i}w^{j}g_{ij}}
470:
between a pair of curves drawn along the surface and meeting at a common point. A third such quantity is the
20022:{\displaystyle ds^{2}=c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}=dr^{\mu }dr_{\mu }=g_{\mu \nu }dr^{\mu }dr^{\nu }\,.}
16337:
12108:
19:
This article is about metric structures on manifolds. For the specific case of spacetime of relativity, see
22764:
22754:
22590:
22313:
22087:
21531:
21040:
20936:
17869:
11735:
A frame also allows covectors to be expressed in terms of their components. For the basis of vector fields
11282:
1166:
990:
20674:
18972:
in this coordinate system. The metric tensor with respect to arbitrary (possibly curvilinear) coordinates
17829:
to either the length or the energy. In the latter case, the geodesic equations are seen to arise from the
15374:
22635:
21769:
20450:
18939:
18269:
18095:
15948:
15471:{\displaystyle (\mathrm {T} M\otimes \mathrm {T} M)^{*}\cong \mathrm {T} ^{*}M\otimes \mathrm {T} ^{*}M,}
14990:
13802:
13245:
transforms covariantly. The operation of associating to the (contravariant) components of a vector field
12947:
10657:
7872:
5904:
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22274:
22255:
22022:
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17873:
17813:
of a point particle moving on the surface of a manifold. Thus, for example, in Jacobi's formulation of
14178:
9744:
7651:{\displaystyle g(v,w)=\mathbf {v} ^{\mathsf {T}}G\mathbf {w} =\mathbf {w} ^{\mathsf {T}}G\mathbf {v} }
6055:. A function of two vector arguments is bilinear if it is linear separately in each argument. Thus if
1016:
22177:
22042:
21252:
21190:
20487:
18124:
17830:
12067:{\displaystyle \theta ={\begin{bmatrix}\theta ^{1}\\\theta ^{2}\\\vdots \\\theta ^{n}\end{bmatrix}}.}
21734:
16436:
is symmetric. There is thus a natural one-to-one correspondence between symmetric bilinear forms on
11832:
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22427:
22119:
21961:
21836:
21754:
21708:
21415:
21123:
20983:
20919:
20808:
17861:
allows one to develop a theory of integrating functions on the manifold by means of the associated
17814:
16128:
9330:
203:
20617:, Graduate Texts in Mathematics, vol. 130 (2nd ed.), Berlin, New York: Springer-Verlag,
15014:
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16512:
15692:
15537:
15028:
15024:
13866:
9277:
5487:
4714:{\displaystyle g(\mathbf {a} ,\mathbf {b} )=a_{1}b_{1}E+a_{1}b_{2}F+a_{2}b_{1}F+a_{2}b_{2}G\,.}
1514:
551:
297:
43:
21195:
16564:
which, analogously, gives an abstract formulation of "raising the index" on a covector field.
15363:{\displaystyle \tau :\mathrm {T} M\otimes \mathrm {T} M{\stackrel {\cong }{\to }}TM\otimes TM}
11716:(that is, inversely or in the opposite way) under a change of basis by the nonsingular matrix
198:
from the early 19th century, it was not until the early 20th century that its properties as a
22249:
22244:
21846:
21801:
21281:
21226:
20915:
17826:
16762:
16080:
15128:{\displaystyle g_{\otimes }\in \Gamma \left((\mathrm {T} M\otimes \mathrm {T} M)^{*}\right).}
15044:
10166:
274:
54:
31:
15238:
by extending linearly to linear combinations of simple elements. The original bilinear form
22580:
22518:
22366:
22070:
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22032:
22007:
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21749:
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21501:
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13849:
12738:
Now, the metric tensor gives a means to identify vectors and covectors as follows. Holding
7460:
6034:
2007:
256:
164:
18090:
of the matrix formed by the components of the metric tensor in the coordinate chart. That
13407:, one applies the same construction but with the inverse metric instead of the metric. If
8:
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22192:
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15385:
14986:
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144:
22691:
18445:
The
Euclidean metric in some other common coordinate systems can be written as follows.
18218:{\displaystyle \omega ={\sqrt {\left|\det g\right|}}\,dx^{1}\wedge \cdots \wedge dx^{n}}
16542:
gives an abstract formulation of "lowering the index" on a vector field. The inverse of
13900:, gives a means for taking the dot product of these tangent vectors. This is called the
65:
allows defining distances and angles there. More precisely, a metric tensor at a point
22660:
22615:
22512:
22383:
22187:
21875:
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21185:
21082:
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20844:
20812:
19844:
19833:
19435:
18265:
18261:
18113:
17817:, the metric tensor can be seen to correspond to the mass tensor of a moving particle.
16515:
of the tangent bundle to the cotangent bundle. This section has the same smoothness as
16167:
16143:
6460:
4725:
1024:
270:
238:
207:
22197:
21092:
19560:{\displaystyle r^{\mu }\rightarrow \left(x^{0},x^{1},x^{2},x^{3}\right)=(ct,x,y,z)\,,}
9370:
can be chosen locally so that the quadratic form diagonalizes in the following manner
4553:. It is more profitably viewed, however, as a function that takes a pair of arguments
194:
While the notion of a metric tensor was known in some sense to mathematicians such as
22595:
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22570:
22477:
22388:
22202:
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21976:
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10132:
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7834:
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20036:
describes the spacetime around a spherically symmetric body, such as a planet, or a
22759:
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22527:
22482:
22405:
22376:
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22162:
22157:
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18240:
18094:
is well-defined on functions supported in coordinate neighborhoods is justified by
17927:
16943:
15733:
15525:
15502:
9120:
7674:
7103:
6707:
2401:
583:
230:
222:
218:
210:, who first codified the notion of a tensor. The metric tensor is an example of a
19320:{\displaystyle g={\begin{bmatrix}1&0\\0&\sin ^{2}\theta \end{bmatrix}}\,.}
5468:{\displaystyle \iint _{D}\left|{\vec {r}}_{u}\times {\vec {r}}_{v}\right|\,du\,dv}
5208:{\displaystyle \left\|\mathbf {a} \right\|={\sqrt {g(\mathbf {a} ,\mathbf {a} )}}}
422:{\displaystyle {\vec {r}}(u,\,v)={\bigl (}x(u,\,v),\,y(u,\,v),\,z(u,\,v){\bigr )}}
22676:
22585:
22415:
22371:
22137:
21811:
21759:
21703:
21683:
21585:
21473:
21340:
21311:
20993:
20928:
20904:
20697:
20648:
20636:
19431:
18963:
18948:
17894:
17570:
Given a segment of a curve, another frequently defined quantity is the (kinetic)
14994:
13810:
11937:
10352:
9338:
9108:
7690:
6799:
5899:
5876:
2417:
245:
234:
62:
14969:{\displaystyle g_{p}:\mathrm {T} _{p}M\times \mathrm {T} _{p}M\to \mathbf {R} .}
10736:
of covector fields, define the inverse metric applied to these two covectors by
6463:. A function of two vector arguments is symmetric provided that for all vectors
22542:
22467:
22437:
22335:
22328:
22268:
22239:
22109:
22104:
22065:
21851:
21816:
21713:
21546:
21536:
21526:
21448:
21420:
21405:
21390:
21306:
20492:
20149:{\displaystyle \left(x^{0},x^{1},x^{2},x^{3}\right)=(ct,r,\theta ,\varphi )\,,}
17810:
17186:
15729:
15696:
14882:
10845:{\displaystyle {\tilde {g}}(\alpha ,\beta )=\alpha G^{-1}\beta ^{\mathsf {T}}.}
10516:{\displaystyle \alpha _{i}=\alpha \left(X_{i}\right)\,,\quad i=1,2,\dots ,n\,.}
9949:
9126:
7224:{\displaystyle v=\sum _{i=1}^{n}v^{i}X_{i}\,,\quad w=\sum _{i=1}^{n}w^{i}X_{i}}
6603:. A bilinear function is nondegenerate provided that, for every tangent vector
5486:, and the absolute value denotes the length of a vector in Euclidean space. By
3665:
1522:
1162:
90:
21796:
20622:
9659:
negative signs in any such expression. Equivalently, the metric has signature
22748:
22728:
22552:
22547:
22532:
22522:
22472:
22449:
22323:
22283:
22224:
22172:
21971:
21788:
21605:
21478:
20840:
17913:
17858:
16908:
15608:
15543:
14998:
14878:
14792:
7490:
6600:
6052:
5962:
5483:
4815:
3968:
229:
under changes to the coordinate system. Thus a metric tensor is a covariant
78:
74:
58:
4804:{\displaystyle g(\mathbf {a} ,\mathbf {b} )=g(\mathbf {b} ,\mathbf {a} )\,.}
143:. A manifold equipped with a positive-definite metric tensor is known as a
22655:
22650:
22492:
22459:
22432:
22340:
21981:
21856:
21660:
21645:
21610:
21458:
21443:
20729:
17314:
14788:
14392:
6946:
6942:
6818:
5947:
5895:
5341:
2649:
1504:
433:
211:
184:
20890:
20547:. Typically, such explicit dependence on the basis is entirely suppressed.
17331:, it represents the square of the differential with respect to arclength.
16819:{\displaystyle \mathrm {T} ^{*}M\otimes \mathrm {T} ^{*}M\to \mathbf {R} }
16519:: it is continuous, differentiable, smooth, or real-analytic according as
14787:
The notion of a metric can be defined intrinsically using the language of
9093:{\displaystyle G\left=\left((Dy)^{-1}\right)^{\mathsf {T}}G\left(Dy)^{-1}}
8490:{\displaystyle y^{i}=y^{i}(x^{1},x^{2},\dots ,x^{n}),\quad i=1,2,\dots ,n}
22498:
22487:
22444:
22345:
21946:
21744:
21718:
21640:
21329:
21268:
19837:
18959:
18128:
18087:
16830:
13887:
8114:
6030:
5860:
3669:
1161:
The integrand is the restriction to the curve of the square root of the (
94:
27:
9244:{\displaystyle q_{m}(X_{m})=g_{m}(X_{m},X_{m})\,,\quad X_{m}\in T_{m}M.}
22723:
22681:
22507:
22420:
22052:
21956:
21867:
21625:
21138:
20832:
20783:
20037:
19229:
15923:
15040:
14854:{\displaystyle g:\mathrm {T} M\times _{M}\mathrm {T} M\to \mathbf {R} }
12539:
11759:
10527:
9701:
8637:
1970:
1706:
Suppose now that a different parameterization is selected, by allowing
1020:
591:
188:
19409:{\displaystyle ds^{2}=d\theta ^{2}+\sin ^{2}\theta \,d\varphi ^{2}\,.}
9707:
Certain metric signatures which arise frequently in applications are:
8083:
is said to transform covariantly with respect to changes in the frame
2577:
A matrix which transforms in this way is one kind of what is called a
22537:
22502:
22207:
22094:
21600:
21551:
16965:
13837:
10094:{\displaystyle G^{-1}=A^{-1}G^{-1}\left(A^{-1}\right)^{\mathsf {T}}.}
8500:
the metric tensor will determine a different matrix of coefficients,
463:
20687:
Commentationes
Societatis Regiae Scientiarum Gottingesis Recentiores
18962:
with respect to the
Euclidean metric. Thus the metric tensor is the
15839:{\displaystyle S_{g}X_{p}\,{\stackrel {\text{def}}{=}}\,g(X_{p},-),}
14354:{\displaystyle G(\mathbf {e} )=(D\varphi )^{\mathsf {T}}(D\varphi )}
13987:{\displaystyle v=v^{1}\mathbf {e} _{1}+\dots +v^{n}\mathbf {e} _{n}}
9969:). The inverse metric satisfies a transformation law when the frame
3758:{\displaystyle \mathbf {p} =p_{1}{\vec {r}}_{u}+p_{2}{\vec {r}}_{v}}
22701:
22696:
22686:
22077:
21898:
21630:
21615:
21010:
18342:{\displaystyle g={\begin{bmatrix}1&0\\0&1\end{bmatrix}}\,.}
17565:
15740:. This isomorphism is obtained by setting, for each tangent vector
13788:
10868:
The resulting definition, although it involves the choice of basis
10128:
8118:
6807:
595:
449:
47:
21324:
21286:
17798:
5965:, consisting of all tangent vectors to the manifold at the point
176:
22293:
21650:
21242:
17854:
16928:
of the metric tensor relative to the coordinate vector fields.
10123:, or with respect to the inverse of the change of basis matrix
2578:
199:
20817:"Méthodes de calcul différentiel absolu et leurs applications"
12849:{\displaystyle g_{p}(X_{p},-):Y_{p}\mapsto g_{p}(X_{p},Y_{p})}
12718:
rather than its inverse). The covariance of the components of
12179:
on tangent vectors can be expanded in terms of the dual basis
2630:{\displaystyle {\begin{bmatrix}E&F\\F&G\end{bmatrix}}}
288:
of points on the surface depending on two auxiliary variables
18435:{\displaystyle L=\int _{a}^{b}{\sqrt {(dx)^{2}+(dy)^{2}}}\,.}
17295:{\displaystyle ds^{2}=\sum _{i,j=1}^{n}g_{ij}(p)dx^{i}dx^{j}}
16643:{\displaystyle S_{g}^{-1}:\mathrm {T} ^{*}M\to \mathrm {T} M}
12941:
in the dual basis are given by the entries of the row vector
8062:{\displaystyle g_{ij}=\sum _{k,l=1}^{n}a_{ki}g_{kl}a_{lj}\,.}
467:
195:
17841:
In analogy with the case of surfaces, a metric tensor on an
16112:. It follows from the definition of non-degeneracy that the
15691:
Using duality as above, a metric is often identified with a
14275:{\displaystyle g(v,w)=\varphi _{*}(v)\cdot \varphi _{*}(w).}
3287:
3111:
2657:
first observed the significance of a system of coefficients
1280:{\displaystyle (ds)^{2}=E\,(du)^{2}+2F\,du\,dv+G\,(dv)^{2},}
20734:
Mathematical thought from ancient to modern times, Volume 3
17857:
of subsets of the manifold. The resulting natural positive
11895:
523:{\textstyle {\begin{bmatrix}E&F\\F&G\end{bmatrix}}}
471:
21211:
20807:
20712:
Schaum's
Outline of Theory and Problems of Tensor Calculus
16761:. Such a nonsingular symmetric mapping gives rise (by the
13640:) associating to the (covariant) components of a covector
2654:
1564:
Using matrix notation, the first fundamental form becomes
20769:, vol. 93, Providence: American Mathematical Society
20473:
Basic introduction to the mathematics of curved spacetime
13896:
is a metric which, when restricted to vectors tangent to
57:) that allows defining distances and angles, just as the
15681:{\displaystyle g_{p}:E_{p}\times E_{p}\to \mathbf {R} .}
13630:
in an essential way, and thus defines a vector field on
8620:
This new system of functions is related to the original
6580:{\displaystyle g_{p}(X_{p},Y_{p})=g_{p}(Y_{p},X_{p})\,.}
2650:
Invariance of arclength under coordinate transformations
237:
point of view, a metric tensor field is defined to be a
147:. Such a metric tensor can be thought of as specifying
18098:. It extends to a unique positive linear functional on
17908:
is a manifold with a (pseudo-)Riemannian metric tensor
11712:. In other words, the components of a vector transform
191:
of the distance function (taken in a suitable manner).
20190:
19717:
19595:
19269:
18880:
18667:
18543:
18304:
13034:, the right-hand side of this equation transforms via
11972:
11450:
10402:{\displaystyle p\mapsto \alpha _{p}\left(X_{p}\right)}
5791:
4461:
4425:
4387:
3567:
3511:
3471:
3419:
3251:
3070:
3028:
2992:
2962:
2873:
2741:
2706:
2596:
2441:
2255:
2219:
2087:
2027:
1661:
1625:
1595:
489:
483:
20646:
20581:
20168:
20049:
19856:
19583:
19447:
19419:
19339:
19257:
18987:
18637:
18471:
18361:
18292:
18152:
17947:
17583:
17344:
17200:
17185:
In connection with this geometrical application, the
16977:
16842:
16774:
16662:
16653:
which is nonsingular and symmetric in the sense that
16592:
16340:
16170:
15951:
15769:
15630:
15563:
15396:
15303:
15251:
15167:
15063:
14908:
14810:
14377:
14300:
14206:
14039:
13924:
13671:
13547:
13447:
13292:
13043:
12950:
12758:
12724:
is notationally designated by placing the indices of
12595:
12193:
12111:
11949:
11786:
11726:
is notationally designated by placing the indices of
11628:
11325:
11267:
have been raised to indicate the transformation law (
10898:
10747:
10660:
10539:
10442:
10364:
10208:
9990:
9867:
9379:
9138:
8986:
8773:
8648:
8509:
8384:
8268:
8140:
7947:
7875:
7702:
7513:
7267:
7121:
6983:
6838:
6621:
6491:
6106:
5907:
5499:
5389:
5243:
5162:
4843:
4749:
4580:
3980:
3795:
3685:
3380:
3351:
3320:
3291:
3204:
3175:
3144:
3115:
2932:
2834:
2805:
2774:
2745:
2700:
2590:
2429:
2021:
1745:
1573:
1314:
1179:
1036:
993:
606:
308:
20681:
translated by A. M. Hiltebeitel and J. C. Morehead;
17836:
15515:
is symmetric as a bilinear mapping, it follows that
15284:{\displaystyle g_{\otimes }\circ \tau =g_{\otimes }}
14782:
7862:, the matrix of components of the metric changes by
296:. Thus a parametric surface is (in today's terms) a
151:
distance on the manifold. On a
Riemannian manifold
20778:
20683:"Disquisitiones generales circa superficies curvas"
20655:(3rd ed.), Berlin, New York: Springer-Verlag,
16868:{\displaystyle \mathrm {T} M\otimes \mathrm {T} M.}
13430:are the components of a covector in the dual basis
9938:{\displaystyle g_{ij}=g\left(X_{i},X_{j}\right)\,.}
8375:Relative to a new system of local coordinates, say
6037:), so that the following conditions are satisfied:
5490:for the cross product, the integral can be written
20958:
20862:(2nd ed.), New York: Chelsea Publishing Co.,
20426:
20148:
20021:
19821:
19559:
19408:
19319:
19184:
18920:
18617:
18434:
18341:
18217:
18063:
17786:
17551:
17294:
17174:
16867:
16818:
16742:
16642:
16421:
16254:
16035:
15838:
15680:
15596:
15470:
15362:
15283:
15216:
15127:
15008:
14968:
14899:to each fiber is a nondegenerate bilinear mapping
14853:
14771:
14353:
14274:
14166:
13986:
13859:, which means that the ambient Euclidean space is
13767:
13600:
13514:
13385:
13217:
13010:
12848:
12669:
12508:
12164:
12066:
11901:
11688:
11563:
11215:
10844:
10704:
10624:
10515:
10401:
10332:
10093:
9937:
9590:
9243:
9092:
8934:
8753:
8609:
8489:
8364:
8244:
8061:
7927:
7822:
7650:
7448:
7223:
7052:
6910:
6676:
6579:
6439:
5928:
5844:
5467:
5324:
5207:
5095:
4803:
4713:
4506:
3956:
3757:
3648:
2912:
2629:
2566:
2376:
1945:
1690:
1469:
1279:
1150:
1007:
976:
522:
421:
155:, the length of a smooth curve between two points
16878:
15001:, depending on the case of interest, and whether
13768:{\displaystyle v^{i}=\sum _{k=1}^{n}g^{ik}a_{k}.}
13260:the (covariant) components of the covector field
11689:{\displaystyle X=\mathbf {fA} v=\mathbf {f} v\,.}
7053:{\displaystyle g_{ij}=g\left(X_{i},X_{j}\right).}
4517:This is plainly a function of the four variables
2416:arranged in this way therefore transforms by the
262:Disquisitiones generales circa superficies curvas
22746:
18260:The most familiar example is that of elementary
18166:
18041:
17566:The energy, variational principles and geodesics
15713:
13538:has components which transform contravariantly:
13386:{\displaystyle a_{i}=\sum _{k=1}^{n}v^{k}g_{ki}}
6003:which takes as inputs a pair of tangent vectors
5783:
2646:) is known as the metric tensor of the surface.
20580:For the terminology "musical isomorphism", see
19195:
16432:This bilinear form is symmetric if and only if
15597:{\displaystyle g:E\times _{M}E\to \mathbf {R} }
15542:More generally, one may speak of a metric in a
15217:{\displaystyle g_{\otimes }(v\otimes w)=g(v,w)}
11276:
10880:in an essential way. Indeed, changing basis to
9766:. More generally, a metric tensor in dimension
8259:has components relative to this frame given by
6677:{\displaystyle Y_{p}\mapsto g_{p}(X_{p},Y_{p})}
1525:of the square of the displacement undergone by
187:. Conversely, the metric tensor itself is the
18352:The length of a curve reduces to the formula:
11587:for some uniquely determined smooth functions
9852:be a basis of vector fields, and as above let
9284:. If the metric is positive-definite at every
5147:In particular, the length of a tangent vector
179:of the lengths of all such curves; this makes
21883:
21227:
20944:
20885:, Weinheim: Wiley-VCH Verlag GmbH & Co.,
12445:
12358:
10613:
10559:
10427:has components in the basis of vector fields
1701:
414:
342:
20696:
20612:
20556:
20500:, a technique to visualize the metric tensor
16534:, which associates to every vector field on
15910:. That is, in terms of the pairing between
15531:
15490:is regarded also as a section of the bundle
12930:, then the components of the covector field
10142:is a covector field. To wit, for each point
7938:or, in terms of the entries of this matrix,
7481:and arranging the components of the vectors
6911:{\displaystyle g(X,Y)(p)=g_{p}(X_{p},Y_{p})}
1023:has been applied, and the subscripts denote
17849:gives rise to a natural way to measure the
13646:the (contravariant) components of a vector
6930:This section assumes some familiarity with
6925:
462:One natural such invariant quantity is the
21890:
21876:
21234:
21220:
21114:Fundamental theorem of Riemannian geometry
20951:
20937:
20921:Differential Analysis on Complex Manifolds
18127:, then it is possible to define a natural
17809:can be seen to directly correspond to the
16319:defines a non-degenerate bilinear form on
13865:, the induced metric tensor is called the
11720:. The contravariance of the components of
8072:For this reason, the system of quantities
7244:, then the value of the metric applied to
101:consists of a metric tensor at each point
20854:
20676:General Investigations of Curved Surfaces
20597:
20420:
20142:
20015:
19815:
19553:
19402:
19388:
19313:
18607:
18428:
18335:
18179:
18054:
17970:
17780:
17773:
17545:
17538:
17168:
17161:
16415:
15807:
15790:
14160:
12670:{\displaystyle \alpha =a\theta =a\theta }
11682:
11226:So that the right-hand side of equation (
10698:
10618:
10509:
10477:
10326:
9931:
9302:. More generally, if the quadratic forms
9207:
8358:
8238:
8128:, determines a basis of vector fields on
8055:
7169:
6573:
6429:
6263:
5910:
5831:
5824:
5751:
5744:
5691:
5684:
5458:
5451:
5318:
4797:
4707:
4496:
3638:
2560:
1254:
1241:
1234:
1205:
1144:
1088:
966:
959:
915:
846:
764:
706:
546:are taken to depend on a third variable,
405:
392:
382:
369:
359:
330:
267:General investigations of curved surfaces
16:Structure defining distance on a manifold
21897:
21591:Covariance and contravariance of vectors
20679:, New York: Raven Press (published 1965)
15227:and is defined on arbitrary elements of
15015:universal property of the tensor product
10169:condition holds for all tangent vectors
9747:(the term semi-Riemannian is also used).
9114:
8092:
2655:Ricci-Curbastro & Levi-Civita (1900)
466:drawn along the surface. Another is the
20702:The large scale structure of space-time
14008:are the standard coordinate vectors in
13624:does not depend on the choice of basis
13515:{\displaystyle v=G^{-1}a^{\mathsf {T}}}
9125:Associated to any metric tensor is the
1714:to depend on another pair of variables
217:The components of a metric tensor in a
163:can be defined by integration, and the
22747:
20875:
20760:
20568:
19570:the metric is, depending on choice of
18650:
16422:{\displaystyle g_{S}(X_{p},Y_{p})=\,.}
14333:
13506:
13192:
13149:
13137:
13107:
13064:
12985:
12165:{\displaystyle \theta =A^{-1}\theta .}
11232:) is unaffected by changing the basis
11200:
11123:
11100:
11078:
10968:
10833:
10715:That is, the row vector of components
10082:
9046:
7902:
7612:
7554:
3240:
2208:
21871:
21215:
20932:
20914:
20728:
20669:
20613:Dodson, C. T. J.; Poston, T. (1991),
20544:
20517:More precisely, the integrand is the
18133:positively oriented coordinate system
16448:and symmetric linear isomorphisms of
13876:is an immersion onto the submanifold
1008:{\displaystyle \left\|\cdot \right\|}
261:
20879:Introduction to mathematical physics
20582:Gallot, Hulin & Lafontaine (2004
19836:curve, the length formula gives the
19330:This is usually written in the form
19210:. In standard spherical coordinates
14801:
13438:
12184:
11613:change in such a way that equation (
11316:
10738:
9981:
9786:is sometimes also called Lorentzian.
9625:of positive signs. The signature of
6974:
6937:The components of the metric in any
3786:. If two tangent vectors are given:
2400:where the superscript T denotes the
2012:
1736:
1305:
1170:
20924:, Berlin, New York: Springer-Verlag
20741:
20709:
20585:
19207:
18255:
16917:, whose entries are the components
16833:to a section of the tensor product
16290:Conversely, any linear isomorphism
16036:{\displaystyle =g_{p}(X_{p},Y_{p})}
15550:is a vector bundle over a manifold
14197:, the induced metric is defined by
14026:goes over to the vector tangent to
13011:{\displaystyle a=v^{\mathsf {T}}G.}
10705:{\displaystyle \alpha =\alpha A\,.}
9754:is four-dimensional with signature
9739:is a pseudo-Riemannian metric, and
7928:{\displaystyle G=A^{\mathsf {T}}GA}
7252:is determined by the coefficients (
3659:
2404:. The matrix with the coefficients
202:were understood by, in particular,
13:
21454:Tensors in curvilinear coordinates
20521:of this differential to the curve.
19426:Metric tensor (general relativity)
19420:Lorentzian metrics from relativity
19248:-plane, the metric takes the form
19163:
19148:
19129:
19114:
19082:
19067:
19048:
19033:
18146:the volume form is represented as
17948:
17912:, then there is a unique positive
17845:-dimensional paracompact manifold
16855:
16844:
16795:
16777:
16633:
16616:
16127:is reduced to zero, and so by the
15452:
15434:
15412:
15401:
15384:is finite-dimensional, there is a
15322:
15311:
15100:
15089:
15077:
14942:
14924:
14836:
14818:
14742:
14727:
14701:
14686:
14665:
14650:
14605:
14590:
14564:
14549:
14528:
14513:
14490:
14475:
14449:
14434:
14413:
14398:
14132:
14117:
13911:is a tangent vector at a point of
11873:
11870:
11844:
11841:
11310:, any smooth tangent vector field
8913:
8898:
8853:
8838:
8735:
8731:
8713:
8698:
8655:
8651:
8583:
8579:
8558:
8554:
8337:
8333:
8312:
8308:
8217:
8213:
8173:
8169:
5929:{\displaystyle \mathbb {R} ^{n+1}}
3391:
3383:
3362:
3354:
3331:
3323:
3302:
3294:
3215:
3207:
3186:
3178:
3155:
3147:
3126:
3118:
2845:
2837:
2816:
2808:
2785:
2777:
2756:
2748:
2538:
2530:
2511:
2503:
2482:
2474:
2455:
2447:
2352:
2344:
2325:
2317:
2296:
2288:
2269:
2261:
2184:
2176:
2157:
2149:
2128:
2120:
2101:
2093:
1135:
1118:
1079:
1062:
244:on each tangent space that varies
21:Metric tensor (general relativity)
14:
22786:
20860:Lectures on Differential Geometry
17868:A measure can be defined, by the
17837:Canonical measure and volume form
16907:, the metric tensor appears as a
15621:which is bilinear in each fiber:
15147:is defined on simple elements of
14783:Intrinsic definitions of a metric
13778:
9824:
9129:defined in each tangent space by
5376:-plane, then the surface area of
5348:is parameterized by the function
1521:. Intuitively, it represents the
20532:classical unified field theories
17309:associated to the metric, while
16812:
16499:defines a section of the bundle
15671:
15590:
14959:
14847:
14308:
14150:
13974:
13943:
13755:
13734:
13686:
13588:
13555:
13496:
13482:
13455:
13376:
13352:
13307:
13205:
13182:
13162:
13097:
13077:
13051:
12998:
12975:
12958:
12878:. This operation takes a vector
12660:
12646:
12626:
12609:
12495:
12481:
12457:
12432:
12404:
12381:
12339:
12318:
12288:
12267:
12243:
12222:
12152:
12119:
12046:
12014:
11989:
11957:
11801:
11675:
11664:
11653:
11650:
11639:
11636:
11554:
11543:
11524:
11492:
11467:
11441:
11420:
11380:
11346:
11190:
11166:
11152:
11113:
11037:
10997:
10955:
10928:
10911:
10823:
10799:
10785:
10688:
10668:
10547:
10041:
9998:
9885:
9060:
8996:
8887:
8793:
8529:
8287:
8142:
8035:
7965:
7915:
7883:
7813:
7713:
7704:
7641:
7633:
7625:
7602:
7594:
7583:
7575:
7567:
7544:
7536:
7439:
7001:
5307:
5294:
5280:
5272:
5196:
5188:
5168:
5076:
5067:
5045:
5037:
5007:
4995:
4984:
4953:
4941:
4918:
4910:
4884:
4872:
4860:
4790:
4782:
4765:
4757:
4596:
4588:
3994:
3986:
3878:
3801:
3687:
1959:
20767:Graduate Studies in Mathematics
20763:Topics in Differential Geometry
20453:content of the central object.
19705:
19699:
16891:. In a local coordinate system
16159:symmetric linear transformation
15009:Metric as a section of a bundle
10481:
9957:, which is identified with the
9211:
8459:
7173:
6267:
4965:
1877:
1811:
1416:
1365:
1092:
251:
21930:Differentiable/Smooth manifold
20651:; Lafontaine, Jacques (2004),
20591:
20574:
20562:
20550:
20537:
20524:
20511:
20139:
20112:
19550:
19523:
19458:
19232:, the angle measured from the
18417:
18407:
18395:
18385:
18031:
18025:
18001:
17995:
17765:
17759:
17715:
17709:
17670:
17667:
17661:
17655:
17523:
17517:
17473:
17467:
17428:
17425:
17419:
17413:
17263:
17257:
17151:
17145:
17101:
17095:
17056:
17053:
17047:
17041:
16942:be a piecewise-differentiable
16879:Arclength and the line element
16808:
16629:
16412:
16383:
16377:
16351:
16249:
16213:
16207:
16171:
16030:
16004:
15988:
15952:
15830:
15811:
15667:
15586:
15420:
15397:
15333:
15211:
15199:
15190:
15178:
15108:
15085:
15005:can support such a structure.
14955:
14843:
14348:
14339:
14328:
14318:
14312:
14304:
14266:
14260:
14244:
14238:
14222:
14210:
14056:
14050:
13759:
13751:
13738:
13730:
13690:
13682:
13592:
13584:
13562:
13551:
13501:
13492:
13486:
13478:
13459:
13451:
13380:
13372:
13356:
13348:
13311:
13303:
13209:
13201:
13187:
13178:
13166:
13158:
13102:
13093:
13084:
13073:
13059:
13047:
13002:
12994:
12980:
12971:
12962:
12954:
12914:. In a basis of vector fields
12843:
12817:
12804:
12788:
12769:
12664:
12656:
12650:
12642:
12633:
12622:
12616:
12605:
12586:) continues to hold. That is,
12499:
12491:
12485:
12477:
12461:
12453:
12436:
12428:
12408:
12400:
12385:
12377:
12343:
12335:
12322:
12314:
12292:
12284:
12271:
12263:
12247:
12239:
12226:
12218:
12156:
12148:
12126:
12115:
12050:
12042:
12018:
12010:
11993:
11985:
11961:
11953:
11821:
11808:
11805:
11797:
11679:
11671:
11657:
11646:
11558:
11550:
11528:
11520:
11496:
11488:
11471:
11463:
11424:
11416:
11384:
11376:
11350:
11342:
11195:
11186:
11171:
11162:
11156:
11148:
11118:
11109:
11042:
11033:
11001:
10993:
10963:
10951:
10936:
10924:
10918:
10907:
10874:, does not actually depend on
10828:
10819:
10804:
10795:
10789:
10781:
10772:
10760:
10754:
10692:
10684:
10675:
10664:
10551:
10543:
10368:
10161:defined on tangent vectors at
10117:The inverse metric transforms
10046:
10037:
10006:
9994:
9889:
9881:
9858:be the matrix of coefficients
9609:. Any two such expressions of
9204:
9178:
9162:
9149:
9078:
9068:
9027:
9017:
8453:
8408:
8039:
8031:
7972:
7961:
7919:
7911:
7890:
7879:
7708:
7645:
7637:
7629:
7621:
7607:
7598:
7587:
7579:
7571:
7563:
7549:
7540:
7529:
7517:
7443:
7435:
7283:
7271:
7005:
6997:
6905:
6879:
6863:
6857:
6854:
6842:
6671:
6645:
6632:
6570:
6544:
6528:
6502:
6426:
6400:
6381:
6355:
6332:
6287:
6260:
6234:
6215:
6189:
6166:
6121:
6029:, and produces as an output a
5659:
5637:
5604:
5582:
5553:
5531:
5434:
5412:
5311:
5303:
5298:
5290:
5284:
5268:
5256:
5250:
5200:
5184:
5172:
5164:
5049:
5033:
4922:
4906:
4794:
4778:
4769:
4753:
4600:
4584:
4246:
4224:
4182:
4160:
4118:
4096:
4054:
4032:
3938:
3906:
3861:
3829:
3743:
3711:
3629:
3614:
1923:
1896:
1857:
1830:
1791:
1764:
1452:
1430:
1401:
1379:
1350:
1328:
1265:
1255:
1216:
1206:
1190:
1180:
1127:
1100:
1071:
1044:
1001:
995:
945:
923:
906:
899:
876:
854:
843:
837:
826:
820:
794:
772:
755:
748:
702:
698:
695:
689:
680:
674:
668:
662:
637:
594:of that curve is given by the
409:
396:
386:
373:
363:
350:
334:
321:
315:
1:
21507:Exterior covariant derivative
21439:Tensor (intrinsic definition)
20606:
18131:from the metric tensor. In a
15865:which sends a tangent vector
15714:Tangent–cotangent isomorphism
15554:, then a metric is a mapping
14895:such that the restriction of
12580:in such a way that equation (
9359:, a basis of tangent vectors
9265:is positive for all non-zero
8945:Or, in terms of the matrices
6088:are three tangent vectors at
5866:
2691:. Indeed, by the chain rule,
2640:with the transformation law (
533:
21532:Raising and lowering indices
21041:Raising and lowering indices
20704:, Cambridge University Press
19196:The round metric on a sphere
18268:metric tensor. In the usual
18096:Jacobian change of variables
17870:Riesz representation theorem
17825:may be obtained by applying
15242:is symmetric if and only if
15019:
14981:
14867:
11287:In a basis of vector fields
11283:Raising and lowering indices
11277:Raising and lowering indices
10416:for any smooth vector field
9816:, then the metric is called
9762:, then the metric is called
9727:is a Riemannian metric, and
9645:, signifying that there are
9621:) will have the same number
6806:. More precisely, given any
5136:plane, and any real numbers
7:
22636:Classification of manifolds
21770:Gluon field strength tensor
21241:
20876:Vaughn, Michael T. (2007),
20530:In several formulations of
20456:
20441:(inside the matrix) is the
20159:we can write the metric as
18940:Cartesian coordinate system
18250:
17327:to the image of a curve in
16968:of the curve is defined by
16743:{\displaystyle \left=\left}
14991:continuously differentiable
14189:.) Given two such vectors,
13803:continuously differentiable
13658:
13636:
13611:Consequently, the quantity
13528:
12582:
12522:
11615:
11577:
11314:can be written in the form
11269:
11228:
10858:
10131:fields; that is, fields of
10107:
7254:
7066:
3676:can be written in the form
2671:
2642:
2390:
1734:) for the new variables is
1730:
1499:
1483:
1293:
10:
22791:
21581:Cartan formalism (physics)
21401:Penrose graphical notation
21062:Pseudo-Riemannian manifold
20468:Pseudo-Riemannian manifold
19423:
18951:, the partial derivatives
18283:coordinates, we can write
17874:positive linear functional
16887:is a Riemannian metric on
16513:vector bundle isomorphisms
15724:The metric tensor gives a
15717:
15535:
13601:{\displaystyle v=A^{-1}v.}
12706:. That is, the components
11597:. Upon changing the basis
11280:
10433:. These are determined by
10138:To see this, suppose that
9770:other than 4 of signature
9745:pseudo-Riemannian manifold
9357:Sylvester's law of inertia
9118:
9111:of the coordinate change.
6929:
3768:for suitable real numbers
1702:Coordinate transformations
109:that varies smoothly with
18:
22712:over commutative algebras
22669:
22628:
22561:
22458:
22354:
22301:
22292:
22128:
22051:
21990:
21910:
21787:
21727:
21676:
21669:
21561:
21492:
21429:
21373:
21320:
21267:
21260:
21253:Glossary of tensor theory
21249:
21191:Geometrization conjecture
21178:
21152:
21106:
21075:
20971:
20809:Ricci-Curbastro, Gregorio
20761:Michor, Peter W. (2008),
20736:, Oxford University Press
20623:10.1007/978-3-642-10514-2
20488:List of coordinate charts
17831:principle of least action
16583:defines a linear mapping
15532:Metric in a vector bundle
13852:. More specifically, for
13436:, then the column vector
12565:transform when the basis
12090:for a nonsingular matrix
11619:) remains true. That is,
5380:is given by the integral
4567:which are vectors in the
2420:of the coordinate change
139:for every nonzero vector
97:), and a metric field on
22428:Riemann curvature tensor
21837:Gregorio Ricci-Curbastro
21709:Riemann curvature tensor
21416:Van der Waerden notation
20700:; Ellis, G.F.R. (1973),
20693:(1827), pp. 99–146.
20584:, p. 75). See also
20557:Dodson & Poston 1991
20504:
18933:trigonometric identities
18233:coordinate differentials
16265:for all tangent vectors
16046:for all tangent vectors
15017:, any bilinear mapping (
14368:is the Jacobian matrix:
13021:Under a change of basis
12893:and produces a covector
12874:on the tangent space at
12077:Under a change of basis
11603:by a nonsingular matrix
10412:is a smooth function of
9629:is the pair of integers
9341:, then the signature of
9331:pseudo-Riemannian metric
9321:, then the signature of
7689:, respectively. Under a
6926:Components of the metric
6918:is a smooth function of
6794:in the tangent space at
225:whose entries transform
204:Gregorio Ricci-Curbastro
21807:Elwin Bruno Christoffel
21740:Angular momentum tensor
21411:Tetrad (index notation)
21381:Abstract index notation
16831:double dual isomorphism
15736:, sometimes called the
13836:if its differential is
12735:in the lower position.
11732:in the upper position.
10530:of these components by
10191:, and all real numbers
9975:is changed by a matrix
9325:is this signature, and
7093:form the entries of an
6100:are real numbers, then
5335:
269:) considered a surface
242:symmetric bilinear form
22220:Manifold with boundary
21935:Differential structure
21621:Levi-Civita connection
21201:Uniformization theorem
21134:Nash embedding theorem
21067:Riemannian volume form
21026:Levi-Civita connection
20443:gravitational constant
20428:
20150:
20023:
19823:
19561:
19410:
19321:
19208:induced metric section
19186:
18922:
18619:
18436:
18343:
18264:: the two-dimensional
18219:
18065:
17827:variational principles
17797:This usage comes from
17788:
17641:
17553:
17399:
17307:first fundamental form
17296:
17243:
17176:
17027:
16869:
16820:
16744:
16644:
16423:
16256:
16037:
15840:
15682:
15598:
15538:metric (vector bundle)
15472:
15364:
15285:
15218:
15129:
14970:
14855:
14773:
14355:
14276:
14168:
14103:
14082:
13988:
13886:. The usual Euclidean
13867:first fundamental form
13769:
13716:
13602:
13516:
13387:
13337:
13219:
13012:
12850:
12671:
12510:
12175:Any linear functional
12166:
12068:
11903:
11690:
11565:
11217:
10846:
10706:
10626:
10517:
10403:
10334:
10165:so that the following
10150:determines a function
10095:
9939:
9592:
9245:
9094:
8936:
8834:
8755:
8694:
8611:
8491:
8366:
8246:
8101:real-valued functions
8063:
8004:
7929:
7824:
7652:
7450:
7401:
7315:
7225:
7200:
7148:
7054:
6912:
6775:assigns to each point
6767:A metric tensor field
6695:constant and allowing
6678:
6581:
6441:
5930:
5846:
5469:
5326:
5209:
5097:
4805:
4715:
4508:
3958:
3759:
3650:
2914:
2631:
2568:
2378:
1947:
1728:. Then the analog of (
1692:
1515:first fundamental form
1471:
1281:
1152:
1009:
978:
586:in parametric surface
550:, taking values in an
524:
423:
298:vector-valued function
235:coordinate-independent
221:take on the form of a
175:can be defined as the
22770:Differential geometry
21847:Jan Arnoldus Schouten
21802:Augustin-Louis Cauchy
21282:Differential geometry
20891:10.1002/9783527618859
20821:Mathematische Annalen
20671:Gauss, Carl Friedrich
20449:represents the total
20429:
20151:
20024:
19824:
19562:
19411:
19322:
19187:
18923:
18620:
18437:
18344:
18220:
18066:
17904:. More precisely, if
17815:Maupertuis' principle
17805:, where the integral
17789:
17615:
17554:
17373:
17297:
17217:
17177:
17001:
16870:
16821:
16763:tensor-hom adjunction
16745:
16645:
16424:
16257:
16081:linear transformation
16038:
15841:
15683:
15599:
15473:
15365:
15286:
15219:
15130:
15045:tensor product bundle
14971:
14856:
14774:
14356:
14277:
14169:
14083:
14062:
13989:
13770:
13696:
13603:
13517:
13388:
13317:
13220:
13013:
12851:
12672:
12511:
12167:
12069:
11904:
11691:
11566:
11218:
10847:
10707:
10627:
10518:
10404:
10335:
10096:
9948:One can consider the
9940:
9593:
9276:, then the metric is
9246:
9115:Signature of a metric
9095:
8937:
8808:
8756:
8674:
8612:
8492:
8367:
8247:
8093:Metric in coordinates
8064:
7978:
7930:
7825:
7653:
7451:
7375:
7289:
7226:
7180:
7128:
7055:
6913:
6798:in a way that varies
6710:. That is, for every
6679:
6582:
6442:
5969:. A metric tensor at
5931:
5847:
5470:
5327:
5210:
5098:
4834:separately. That is,
4818:, meaning that it is
4806:
4716:
4571:-plane. That is, put
4509:
3959:
3760:
3651:
2915:
2632:
2569:
2379:
1948:
1693:
1472:
1282:
1153:
1010:
979:
525:
477:The metric tensor is
424:
275:Cartesian coordinates
248:from point to point.
32:differential geometry
22367:Covariant derivative
21918:Topological manifold
21822:Carl Friedrich Gauss
21755:stress–energy tensor
21750:Cauchy stress tensor
21502:Covariant derivative
21464:Antisymmetric tensor
21396:Multi-index notation
21124:Gauss–Bonnet theorem
21031:Covariant derivative
20744:Riemannian manifolds
20166:
20047:
20034:Schwarzschild metric
19854:
19581:
19445:
19438:), with coordinates
19337:
19255:
18985:
18635:
18469:
18359:
18290:
18150:
17945:
17898:continuous functions
17581:
17342:
17198:
16975:
16840:
16772:
16660:
16590:
16538:a covector field on
16338:
16168:
16129:rank–nullity theorem
15949:
15767:
15628:
15561:
15394:
15301:
15249:
15165:
15061:
14985:) is required to be
14906:
14808:
14795:. In these terms, a
14375:
14298:
14204:
14177:(This is called the
14037:
13922:
13850:immersed submanifold
13669:
13545:
13445:
13290:
13041:
12948:
12920:, if a vector field
12756:
12749:fixed, the function
12593:
12191:
12109:
11947:
11784:
11626:
11323:
11259:, where the indices
10896:
10745:
10658:
10651:changes by the rule
10537:
10440:
10362:
10206:
9988:
9865:
9377:
9136:
8984:
8771:
8646:
8507:
8382:
8266:
8138:
7945:
7873:
7700:
7511:
7265:
7119:
6981:
6836:
6832:, the real function
6684:obtained by holding
6619:
6489:
6104:
5905:
5497:
5387:
5241:
5222:between two vectors
5160:
4841:
4747:
4578:
3978:
3971:of the dot product,
3793:
3683:
2930:
2698:
2588:
2427:
2019:
1743:
1571:
1312:
1177:
1034:
991:
604:
481:
448:, and defined in an
306:
257:Carl Friedrich Gauss
22765:Concepts in physics
22755:Riemannian geometry
22401:Exterior derivative
22003:Atiyah–Singer index
21952:Riemannian manifold
21699:Nonmetricity tensor
21554:(2nd-order tensors)
21522:Hodge star operator
21512:Exterior derivative
21361:Transport phenomena
21346:Continuum mechanics
21302:Multilinear algebra
21196:Poincaré conjecture
21057:Riemannian manifold
21045:Musical isomorphism
20960:Riemannian geometry
20813:Levi-Civita, Tullio
20746:, Springer Verlag,
20710:Kay, David (1988),
20653:Riemannian Geometry
20647:Gallot, Sylvestre;
20559:, Chapter VII §3.04
20498:Tissot's indicatrix
20463:Riemannian manifold
20040:. With coordinates
19240:the angle from the
19200:The unit sphere in
18382:
17895:compactly supported
17803:classical mechanics
17614:
17365:
16998:
16725:
16685:
16610:
15738:musical isomorphism
15726:natural isomorphism
15720:Musical isomorphism
15511:with itself. Since
15386:natural isomorphism
11607:, the coefficients
11238:to any other basis
10423:Any covector field
10351:is assumed to be a
9733:Riemannian manifold
9649:positive signs and
9613:(at the same point
9348:does not depend on
7234:are two vectors at
5488:Lagrange's identity
1025:partial derivatives
737:
635:
145:Riemannian manifold
89:that maps pairs of
42:) is an additional
22707:Secondary calculus
22661:Singularity theory
22616:Parallel transport
22384:De Rham cohomology
22023:Generalized Stokes
21832:Tullio Levi-Civita
21775:Metric tensor (GR)
21689:Levi-Civita symbol
21542:Tensor contraction
21356:General relativity
21292:Euclidean geometry
21186:General relativity
21129:Hopf–Rinow theorem
21076:Types of manifolds
21052:Parallel transport
20833:10.1007/BF01454201
20780:Misner, Charles W.
20742:Lee, John (1997),
20424:
20414:
20146:
20019:
19845:spacetime interval
19843:In this case, the
19819:
19809:
19693:
19557:
19436:special relativity
19406:
19317:
19307:
19182:
19110:
19016:
18918:
18912:
18866:
18615:
18613:
18601:
18432:
18368:
18339:
18329:
18262:Euclidean geometry
18245:differential forms
18243:in the algebra of
18215:
18114:partition of unity
18061:
17926:such that for any
17823:geodesic equations
17784:
17600:
17549:
17351:
17292:
17172:
16984:
16911:, denoted here by
16865:
16816:
16753:for all covectors
16740:
16708:
16668:
16640:
16593:
16419:
16252:
16161:in the sense that
16144:linear isomorphism
16033:
15836:
15678:
15594:
15468:
15360:
15281:
15214:
15125:
14966:
14851:
14769:
14760:
14351:
14272:
14164:
13984:
13840:at every point of
13765:
13656:. In components, (
13598:
13512:
13398:lowering the index
13383:
13215:
13008:
12859:of tangent vector
12846:
12693:, it follows that
12667:
12506:
12504:
12441:
12162:
12064:
12055:
11899:
11894:
11764:linear functionals
11686:
11561:
11533:
11213:
11211:
10842:
10702:
10635:Under a change of
10622:
10609:
10513:
10399:
10355:in the sense that
10330:
10133:linear functionals
10091:
9935:
9686:of the metric has
9588:
9404:
9241:
9090:
8932:
8751:
8607:
8487:
8362:
8242:
8059:
7925:
7866:as well. That is,
7820:
7780:
7738:
7648:
7446:
7258:) by bilinearity:
7221:
7050:
6932:coordinate vectors
6908:
6674:
6577:
6437:
6435:
5926:
5842:
5840:
5816:
5465:
5322:
5205:
5093:
5091:
4801:
4726:symmetric function
4711:
4504:
4502:
4490:
4450:
4414:
3954:
3952:
3755:
3646:
3644:
3598:
3556:
3500:
3450:
3408:
3404:
3375:
3344:
3315:
3276:
3232:
3228:
3199:
3168:
3139:
3099:
3049:
3017:
2981:
2910:
2904:
2862:
2858:
2829:
2798:
2769:
2727:
2627:
2621:
2564:
2554:
2374:
2368:
2244:
2200:
2072:
1943:
1688:
1682:
1650:
1614:
1467:
1277:
1148:
1005:
974:
972:
723:
621:
520:
514:
436:of real variables
419:
208:Tullio Levi-Civita
22742:
22741:
22624:
22623:
22389:Differential form
22043:Whitney embedding
21977:Differential form
21865:
21864:
21827:Hermann Grassmann
21783:
21782:
21735:Moment of inertia
21596:Differential form
21571:Affine connection
21386:Einstein notation
21369:
21368:
21297:Exterior calculus
21277:Coordinate system
21209:
21208:
20900:978-3-527-40627-2
20794:, W. H. Freeman,
20753:978-0-387-98322-6
20721:978-0-07-033484-7
20662:978-3-540-20493-0
20632:978-3-540-52018-4
20588:, pp. 27–29)
20306:
20232:
19840:along the curve.
19703:
19177:
19143:
19101:
19096:
19062:
19004:
18938:In general, in a
18449:Polar coordinates
18426:
18177:
18052:
17863:Lebesgue integral
17741:
17691:
17598:
17536:
17499:
17449:
17190:differential form
17159:
17127:
17077:
16255:{\displaystyle =}
15851:linear functional
15804:
15802:
15342:
14875:
14874:
14756:
14715:
14679:
14619:
14578:
14542:
14504:
14463:
14427:
14146:
13654:raising the index
13634:. The operation (
13536:
13535:
12872:linear functional
12554:. The components
12530:
12529:
11879:
11850:
11585:
11584:
10866:
10865:
10757:
10115:
10114:
9800:-dimensional and
9395:
9300:Riemannian metric
9278:positive-definite
8927:
8867:
8749:
8727:
8669:
8597:
8572:
8351:
8326:
8231:
8187:
8115:coordinate system
8113:, giving a local
7771:
7729:
7074:
7073:
6817:and any (smooth)
6271:
5883:; for instance a
5822:
5742:
5682:
5662:
5640:
5607:
5585:
5556:
5534:
5437:
5415:
5356:
5316:
5234:is calculated by
5203:
4969:
4822:in each variable
4249:
4227:
4185:
4163:
4121:
4099:
4057:
4035:
3941:
3909:
3864:
3832:
3746:
3714:
3403:
3374:
3343:
3314:
3227:
3198:
3167:
3138:
2857:
2828:
2797:
2768:
2550:
2523:
2494:
2467:
2398:
2397:
2364:
2337:
2308:
2281:
2196:
2169:
2140:
2113:
1967:
1966:
1926:
1899:
1860:
1833:
1794:
1767:
1533:
1491:
1490:
1455:
1433:
1404:
1382:
1353:
1331:
1301:
1300:
1142:
1130:
1103:
1086:
1074:
1047:
957:
948:
926:
879:
857:
797:
775:
665:
654:
582:will trace out a
562:
538:If the variables
464:length of a curve
318:
122:positive-definite
87:bilinear function
22782:
22734:Stratified space
22692:Fréchet manifold
22406:Interior product
22299:
22298:
21996:
21892:
21885:
21878:
21869:
21868:
21842:Bernhard Riemann
21674:
21673:
21517:Exterior product
21484:Two-point tensor
21469:Symmetric tensor
21351:Electromagnetism
21265:
21264:
21236:
21229:
21222:
21213:
21212:
20953:
20946:
20939:
20930:
20929:
20925:
20911:
20884:
20872:
20851:
20804:
20788:Wheeler, John A.
20770:
20756:
20737:
20724:
20705:
20680:
20665:
20649:Hulin, Dominique
20643:
20600:
20595:
20589:
20578:
20572:
20566:
20560:
20554:
20548:
20541:
20535:
20528:
20522:
20515:
20483:Finsler manifold
20478:Clifford algebra
20448:
20440:
20433:
20431:
20430:
20425:
20419:
20418:
20405:
20404:
20395:
20394:
20358:
20357:
20321:
20320:
20312:
20308:
20307:
20305:
20304:
20303:
20290:
20279:
20238:
20234:
20233:
20231:
20230:
20229:
20216:
20205:
20181:
20180:
20155:
20153:
20152:
20147:
20108:
20104:
20103:
20102:
20090:
20089:
20077:
20076:
20064:
20063:
20028:
20026:
20025:
20020:
20014:
20013:
20001:
20000:
19988:
19987:
19972:
19971:
19959:
19958:
19943:
19942:
19927:
19926:
19911:
19910:
19895:
19894:
19882:
19881:
19869:
19868:
19828:
19826:
19825:
19820:
19814:
19813:
19704:
19701:
19698:
19697:
19572:metric signature
19566:
19564:
19563:
19558:
19519:
19515:
19514:
19513:
19501:
19500:
19488:
19487:
19475:
19474:
19457:
19456:
19415:
19413:
19412:
19407:
19401:
19400:
19381:
19380:
19368:
19367:
19352:
19351:
19326:
19324:
19323:
19318:
19312:
19311:
19298:
19297:
19247:
19243:
19239:
19235:
19227:
19221:
19205:
19191:
19189:
19188:
19183:
19178:
19176:
19175:
19174:
19161:
19160:
19159:
19146:
19144:
19142:
19141:
19140:
19127:
19126:
19125:
19112:
19109:
19097:
19095:
19094:
19093:
19080:
19079:
19078:
19065:
19063:
19061:
19060:
19059:
19046:
19045:
19044:
19031:
19029:
19028:
19015:
19000:
18999:
18977:
18957:
18946:
18927:
18925:
18924:
18919:
18917:
18916:
18909:
18908:
18871:
18870:
18857:
18856:
18847:
18846:
18828:
18827:
18818:
18817:
18698:
18697:
18679:
18678:
18655:
18654:
18653:
18624:
18622:
18621:
18616:
18614:
18606:
18605:
18461:
18441:
18439:
18438:
18433:
18427:
18425:
18424:
18403:
18402:
18384:
18381:
18376:
18348:
18346:
18345:
18340:
18334:
18333:
18282:
18256:Euclidean metric
18241:exterior product
18238:
18230:
18224:
18222:
18221:
18216:
18214:
18213:
18192:
18191:
18178:
18176:
18172:
18160:
18145:
18122:
18111:
18093:
18085:
18078:
18074:
18070:
18068:
18067:
18062:
18053:
18051:
18047:
18035:
18024:
18023:
18005:
18004:
17983:
17982:
17966:
17965:
17940:
17928:coordinate chart
17925:
17911:
17907:
17903:
17892:
17878:
17852:
17848:
17844:
17808:
17801:, specifically,
17793:
17791:
17790:
17785:
17772:
17768:
17752:
17751:
17742:
17740:
17729:
17722:
17718:
17702:
17701:
17692:
17690:
17679:
17654:
17653:
17640:
17635:
17613:
17608:
17599:
17591:
17558:
17556:
17555:
17550:
17537:
17535:
17531:
17530:
17526:
17510:
17509:
17500:
17498:
17487:
17480:
17476:
17460:
17459:
17450:
17448:
17437:
17412:
17411:
17398:
17393:
17367:
17364:
17359:
17330:
17322:
17312:
17301:
17299:
17298:
17293:
17291:
17290:
17278:
17277:
17256:
17255:
17242:
17237:
17213:
17212:
17181:
17179:
17178:
17173:
17160:
17158:
17154:
17138:
17137:
17128:
17126:
17115:
17108:
17104:
17088:
17087:
17078:
17076:
17065:
17040:
17039:
17026:
17021:
17000:
16997:
16992:
16963:
16949:
16944:parametric curve
16941:
16927:
16916:
16906:
16896:
16890:
16886:
16874:
16872:
16871:
16866:
16858:
16847:
16825:
16823:
16822:
16817:
16815:
16804:
16803:
16798:
16786:
16785:
16780:
16760:
16756:
16749:
16747:
16746:
16741:
16739:
16735:
16724:
16716:
16699:
16695:
16684:
16676:
16649:
16647:
16646:
16641:
16636:
16625:
16624:
16619:
16609:
16601:
16582:
16581:
16580:
16563:
16552:
16541:
16537:
16533:
16522:
16518:
16510:
16498:
16487:
16483:
16476:
16472:
16471:
16459:
16447:
16435:
16428:
16426:
16425:
16420:
16411:
16410:
16398:
16397:
16376:
16375:
16363:
16362:
16350:
16349:
16330:
16318:
16314:
16313:
16286:
16275:
16261:
16259:
16258:
16253:
16248:
16247:
16235:
16234:
16225:
16224:
16206:
16205:
16193:
16192:
16183:
16182:
16156:
16141:
16126:
16111:
16107:
16106:
16094:
16078:
16067:
16056:
16042:
16040:
16039:
16034:
16029:
16028:
16016:
16015:
16003:
16002:
15987:
15986:
15974:
15973:
15964:
15963:
15941:
15937:
15936:
15921:
15909:
15879:
15875:
15864:
15845:
15843:
15842:
15837:
15823:
15822:
15806:
15805:
15803:
15800:
15798:
15793:
15789:
15788:
15779:
15778:
15759:
15734:cotangent bundle
15709:
15687:
15685:
15684:
15679:
15674:
15666:
15665:
15653:
15652:
15640:
15639:
15620:
15614:
15603:
15601:
15600:
15595:
15593:
15582:
15581:
15553:
15549:
15526:symmetric tensor
15523:
15514:
15510:
15503:cotangent bundle
15500:
15489:
15477:
15475:
15474:
15469:
15461:
15460:
15455:
15443:
15442:
15437:
15428:
15427:
15415:
15404:
15383:
15369:
15367:
15366:
15361:
15344:
15343:
15341:
15336:
15331:
15325:
15314:
15290:
15288:
15287:
15282:
15280:
15279:
15261:
15260:
15241:
15237:
15223:
15221:
15220:
15215:
15177:
15176:
15157:
15146:
15134:
15132:
15131:
15126:
15121:
15117:
15116:
15115:
15103:
15092:
15073:
15072:
15053:
15038:
15004:
14975:
14973:
14972:
14967:
14962:
14951:
14950:
14945:
14933:
14932:
14927:
14918:
14917:
14898:
14894:
14888:
14869:
14860:
14858:
14857:
14852:
14850:
14839:
14834:
14833:
14821:
14802:
14778:
14776:
14775:
14770:
14765:
14764:
14757:
14755:
14754:
14753:
14740:
14739:
14738:
14725:
14716:
14714:
14713:
14712:
14699:
14698:
14697:
14684:
14680:
14678:
14677:
14676:
14663:
14662:
14661:
14648:
14620:
14618:
14617:
14616:
14603:
14602:
14601:
14588:
14579:
14577:
14576:
14575:
14562:
14561:
14560:
14547:
14543:
14541:
14540:
14539:
14526:
14525:
14524:
14511:
14505:
14503:
14502:
14501:
14488:
14487:
14486:
14473:
14464:
14462:
14461:
14460:
14447:
14446:
14445:
14432:
14428:
14426:
14425:
14424:
14411:
14410:
14409:
14396:
14367:
14360:
14358:
14357:
14352:
14338:
14337:
14336:
14311:
14290:
14281:
14279:
14278:
14273:
14259:
14258:
14237:
14236:
14196:
14192:
14188:
14184:
14173:
14171:
14170:
14165:
14159:
14158:
14153:
14147:
14145:
14144:
14143:
14130:
14129:
14128:
14115:
14113:
14112:
14102:
14097:
14081:
14076:
14049:
14048:
14029:
14025:
14021:
14017:
14013:
14007:
13993:
13991:
13990:
13985:
13983:
13982:
13977:
13971:
13970:
13952:
13951:
13946:
13940:
13939:
13914:
13910:
13899:
13895:
13885:
13875:
13864:
13858:
13847:
13843:
13831:
13827:
13817:
13808:
13800:
13796:
13786:
13774:
13772:
13771:
13766:
13758:
13750:
13749:
13737:
13729:
13728:
13715:
13710:
13689:
13681:
13680:
13652:given is called
13651:
13645:
13633:
13629:
13623:
13607:
13605:
13604:
13599:
13591:
13580:
13579:
13558:
13530:
13521:
13519:
13518:
13513:
13511:
13510:
13509:
13499:
13485:
13477:
13476:
13458:
13439:
13435:
13429:
13392:
13390:
13389:
13384:
13379:
13371:
13370:
13355:
13347:
13346:
13336:
13331:
13310:
13302:
13301:
13282:
13259:
13244:
13240:
13224:
13222:
13221:
13216:
13208:
13197:
13196:
13195:
13185:
13165:
13154:
13153:
13152:
13142:
13141:
13140:
13134:
13130:
13129:
13112:
13111:
13110:
13100:
13080:
13069:
13068:
13067:
13054:
13033:
13017:
13015:
13014:
13009:
13001:
12990:
12989:
12988:
12978:
12961:
12940:
12929:
12923:
12919:
12913:
12892:
12888:
12877:
12869:
12855:
12853:
12852:
12847:
12842:
12841:
12829:
12828:
12816:
12815:
12803:
12802:
12781:
12780:
12768:
12767:
12748:
12734:
12723:
12717:
12709:
12705:
12692:
12680:whence, because
12676:
12674:
12673:
12668:
12663:
12649:
12629:
12612:
12579:
12570:
12564:
12553:
12537:
12524:
12515:
12513:
12512:
12507:
12505:
12498:
12484:
12467:
12460:
12449:
12448:
12442:
12435:
12427:
12426:
12407:
12399:
12398:
12384:
12376:
12375:
12362:
12361:
12349:
12342:
12334:
12333:
12321:
12313:
12312:
12291:
12283:
12282:
12270:
12262:
12261:
12246:
12238:
12237:
12225:
12217:
12216:
12185:
12182:
12178:
12171:
12169:
12168:
12163:
12155:
12144:
12143:
12122:
12101:
12095:
12089:
12073:
12071:
12070:
12065:
12060:
12059:
12049:
12041:
12040:
12017:
12009:
12008:
11992:
11984:
11983:
11960:
11935:
11908:
11906:
11905:
11900:
11898:
11897:
11877:
11876:
11848:
11847:
11820:
11819:
11804:
11796:
11795:
11776:
11757:
11731:
11725:
11719:
11711:
11695:
11693:
11692:
11687:
11678:
11667:
11656:
11642:
11612:
11606:
11602:
11596:
11579:
11570:
11568:
11567:
11562:
11557:
11546:
11538:
11537:
11527:
11519:
11518:
11495:
11487:
11486:
11470:
11462:
11461:
11444:
11436:
11435:
11423:
11415:
11414:
11396:
11395:
11383:
11375:
11374:
11362:
11361:
11349:
11341:
11340:
11317:
11313:
11309:
11266:
11262:
11258:
11252:
11246:
11237:
11222:
11220:
11219:
11214:
11212:
11205:
11204:
11203:
11193:
11182:
11181:
11169:
11155:
11142:
11133:
11129:
11128:
11127:
11126:
11116:
11105:
11104:
11103:
11088:
11084:
11083:
11082:
11081:
11075:
11071:
11070:
11053:
11052:
11040:
11029:
11028:
11011:
11007:
11000:
10982:
10973:
10972:
10971:
10958:
10947:
10946:
10931:
10914:
10902:
10888:
10879:
10873:
10860:
10851:
10849:
10848:
10843:
10838:
10837:
10836:
10826:
10815:
10814:
10802:
10788:
10759:
10758:
10750:
10739:
10735:
10731:
10721:transforms as a
10720:
10711:
10709:
10708:
10703:
10691:
10671:
10650:
10644:
10640:
10631:
10629:
10628:
10623:
10617:
10616:
10610:
10606:
10605:
10589:
10588:
10577:
10576:
10563:
10562:
10550:
10522:
10520:
10519:
10514:
10476:
10472:
10471:
10452:
10451:
10432:
10426:
10419:
10415:
10408:
10406:
10405:
10400:
10398:
10394:
10393:
10380:
10379:
10350:
10346:
10339:
10337:
10336:
10331:
10325:
10321:
10320:
10307:
10306:
10291:
10287:
10286:
10273:
10272:
10257:
10253:
10252:
10251:
10236:
10235:
10218:
10217:
10198:
10194:
10190:
10179:
10164:
10160:
10149:
10145:
10141:
10126:
10109:
10100:
10098:
10097:
10092:
10087:
10086:
10085:
10079:
10075:
10074:
10057:
10056:
10044:
10033:
10032:
10017:
10016:
10001:
9982:
9978:
9974:
9956:
9944:
9942:
9941:
9936:
9930:
9926:
9925:
9924:
9912:
9911:
9888:
9880:
9879:
9857:
9851:
9815:
9803:
9799:
9792:
9785:
9777:
9769:
9761:
9757:
9753:
9742:
9738:
9730:
9726:
9722:
9714:
9699:
9689:
9685:
9674:
9658:
9648:
9644:
9628:
9624:
9620:
9616:
9612:
9608:
9604:
9597:
9595:
9594:
9589:
9587:
9586:
9581:
9577:
9576:
9553:
9552:
9547:
9543:
9542:
9519:
9518:
9513:
9509:
9508:
9485:
9484:
9479:
9475:
9474:
9457:
9456:
9451:
9447:
9446:
9429:
9425:
9424:
9423:
9414:
9413:
9403:
9389:
9388:
9369:
9351:
9347:
9336:
9328:
9324:
9320:
9312:
9297:
9293:
9283:
9275:
9264:
9250:
9248:
9247:
9242:
9234:
9233:
9221:
9220:
9203:
9202:
9190:
9189:
9177:
9176:
9161:
9160:
9148:
9147:
9121:Metric signature
9106:
9099:
9097:
9096:
9091:
9089:
9088:
9067:
9063:
9051:
9050:
9049:
9043:
9039:
9038:
9037:
9007:
9003:
8999:
8976:
8960:
8941:
8939:
8938:
8933:
8928:
8926:
8925:
8924:
8911:
8910:
8909:
8896:
8894:
8890:
8881:
8880:
8868:
8866:
8865:
8864:
8851:
8850:
8849:
8836:
8833:
8828:
8804:
8800:
8796:
8786:
8785:
8760:
8758:
8757:
8752:
8750:
8748:
8747:
8746:
8730:
8728:
8726:
8725:
8724:
8711:
8710:
8709:
8696:
8693:
8688:
8670:
8668:
8667:
8666:
8650:
8636:by means of the
8635:
8616:
8614:
8613:
8608:
8603:
8599:
8598:
8596:
8595:
8594:
8578:
8573:
8571:
8570:
8569:
8553:
8540:
8536:
8532:
8522:
8521:
8496:
8494:
8493:
8488:
8452:
8451:
8433:
8432:
8420:
8419:
8407:
8406:
8394:
8393:
8371:
8369:
8368:
8363:
8357:
8353:
8352:
8350:
8349:
8348:
8332:
8327:
8325:
8324:
8323:
8307:
8294:
8290:
8281:
8280:
8258:
8251:
8249:
8248:
8243:
8237:
8233:
8232:
8230:
8229:
8228:
8212:
8207:
8206:
8188:
8186:
8185:
8184:
8168:
8163:
8162:
8145:
8131:
8127:
8123:
8112:
8100:
8088:
8082:
8068:
8066:
8065:
8060:
8054:
8053:
8038:
8030:
8029:
8017:
8016:
8003:
7998:
7968:
7960:
7959:
7934:
7932:
7931:
7926:
7918:
7907:
7906:
7905:
7886:
7865:
7861:
7845:
7829:
7827:
7826:
7821:
7816:
7808:
7804:
7803:
7802:
7790:
7789:
7779:
7761:
7760:
7748:
7747:
7737:
7720:
7716:
7707:
7688:
7682:
7672:
7666:
7657:
7655:
7654:
7649:
7644:
7636:
7628:
7617:
7616:
7615:
7605:
7597:
7586:
7578:
7570:
7559:
7558:
7557:
7547:
7539:
7503:
7497:
7488:
7484:
7480:
7474:
7455:
7453:
7452:
7447:
7442:
7434:
7433:
7421:
7420:
7411:
7410:
7400:
7395:
7371:
7367:
7366:
7365:
7353:
7352:
7335:
7334:
7325:
7324:
7314:
7309:
7251:
7247:
7243:
7230:
7228:
7227:
7222:
7220:
7219:
7210:
7209:
7199:
7194:
7168:
7167:
7158:
7157:
7147:
7142:
7111:
7104:symmetric matrix
7102:
7092:
7081:
7068:
7059:
7057:
7056:
7051:
7046:
7042:
7041:
7040:
7028:
7027:
7004:
6996:
6995:
6975:
6971:
6921:
6917:
6915:
6914:
6909:
6904:
6903:
6891:
6890:
6878:
6877:
6831:
6827:
6823:
6816:
6812:
6805:
6797:
6793:
6783:a metric tensor
6782:
6778:
6774:
6770:
6762:
6732:
6721:
6708:identically zero
6705:
6694:
6683:
6681:
6680:
6675:
6670:
6669:
6657:
6656:
6644:
6643:
6631:
6630:
6614:
6598:
6586:
6584:
6583:
6578:
6569:
6568:
6556:
6555:
6543:
6542:
6527:
6526:
6514:
6513:
6501:
6500:
6484:
6473:
6458:
6446:
6444:
6443:
6438:
6436:
6425:
6424:
6412:
6411:
6399:
6398:
6380:
6379:
6367:
6366:
6354:
6353:
6331:
6330:
6315:
6314:
6299:
6298:
6286:
6285:
6272:
6269:
6259:
6258:
6246:
6245:
6233:
6232:
6214:
6213:
6201:
6200:
6188:
6187:
6165:
6164:
6152:
6151:
6136:
6135:
6120:
6119:
6099:
6095:
6091:
6087:
6076:
6065:
6050:
6028:
6024:
6013:
6002:
5972:
5968:
5960:
5945:
5936:. At each point
5935:
5933:
5932:
5927:
5925:
5924:
5913:
5893:
5882:
5874:
5858:
5851:
5849:
5848:
5843:
5841:
5823:
5821:
5820:
5782:
5780:
5779:
5766:
5743:
5741:
5740:
5722:
5720:
5719:
5706:
5683:
5681:
5680:
5675:
5671:
5670:
5669:
5664:
5663:
5655:
5648:
5647:
5642:
5641:
5633:
5620:
5616:
5615:
5614:
5609:
5608:
5600:
5593:
5592:
5587:
5586:
5578:
5569:
5565:
5564:
5563:
5558:
5557:
5549:
5542:
5541:
5536:
5535:
5527:
5518:
5516:
5515:
5503:
5481:
5474:
5472:
5471:
5466:
5450:
5446:
5445:
5444:
5439:
5438:
5430:
5423:
5422:
5417:
5416:
5408:
5399:
5398:
5379:
5375:
5371:
5368:over the domain
5367:
5357:
5354:
5347:
5331:
5329:
5328:
5323:
5317:
5315:
5314:
5310:
5301:
5297:
5287:
5283:
5275:
5263:
5233:
5227:
5221:
5214:
5212:
5211:
5206:
5204:
5199:
5191:
5180:
5175:
5171:
5152:
5143:
5139:
5135:
5131:
5124:
5118:
5111:
5106:for any vectors
5102:
5100:
5099:
5094:
5092:
5088:
5084:
5083:
5079:
5070:
5048:
5040:
5019:
5015:
5014:
5010:
4998:
4987:
4970:
4967:
4961:
4957:
4956:
4948:
4944:
4921:
4913:
4892:
4888:
4887:
4879:
4875:
4863:
4833:
4827:
4810:
4808:
4807:
4802:
4793:
4785:
4768:
4760:
4739:
4733:
4720:
4718:
4717:
4712:
4703:
4702:
4693:
4692:
4677:
4676:
4667:
4666:
4651:
4650:
4641:
4640:
4625:
4624:
4615:
4614:
4599:
4591:
4570:
4566:
4559:
4552:
4543:
4534:
4525:
4513:
4511:
4510:
4505:
4503:
4495:
4494:
4487:
4486:
4473:
4472:
4455:
4454:
4419:
4418:
4411:
4410:
4399:
4398:
4375:
4365:
4364:
4355:
4354:
4339:
4338:
4329:
4328:
4313:
4312:
4303:
4302:
4287:
4286:
4277:
4276:
4261:
4257:
4256:
4251:
4250:
4242:
4235:
4234:
4229:
4228:
4220:
4216:
4215:
4206:
4205:
4193:
4192:
4187:
4186:
4178:
4171:
4170:
4165:
4164:
4156:
4152:
4151:
4142:
4141:
4129:
4128:
4123:
4122:
4114:
4107:
4106:
4101:
4100:
4092:
4088:
4087:
4078:
4077:
4065:
4064:
4059:
4058:
4050:
4043:
4042:
4037:
4036:
4028:
4024:
4023:
4014:
4013:
3997:
3989:
3963:
3961:
3960:
3955:
3953:
3949:
3948:
3943:
3942:
3934:
3930:
3929:
3917:
3916:
3911:
3910:
3902:
3898:
3897:
3881:
3872:
3871:
3866:
3865:
3857:
3853:
3852:
3840:
3839:
3834:
3833:
3825:
3821:
3820:
3804:
3785:
3776:
3764:
3762:
3761:
3756:
3754:
3753:
3748:
3747:
3739:
3735:
3734:
3722:
3721:
3716:
3715:
3707:
3703:
3702:
3690:
3675:
3660:Length and angle
3655:
3653:
3652:
3647:
3645:
3637:
3636:
3627:
3607:
3603:
3602:
3595:
3580:
3561:
3560:
3553:
3543:
3531:
3521:
3505:
3504:
3497:
3484:
3459:
3455:
3454:
3447:
3432:
3413:
3412:
3405:
3402:
3401:
3389:
3381:
3376:
3373:
3372:
3360:
3352:
3345:
3342:
3341:
3329:
3321:
3316:
3313:
3312:
3300:
3292:
3281:
3280:
3245:
3244:
3243:
3237:
3236:
3229:
3226:
3225:
3213:
3205:
3200:
3197:
3196:
3184:
3176:
3169:
3166:
3165:
3153:
3145:
3140:
3137:
3136:
3124:
3116:
3104:
3103:
3096:
3083:
3058:
3054:
3053:
3022:
3021:
2986:
2985:
2949:
2948:
2919:
2917:
2916:
2911:
2909:
2908:
2901:
2886:
2867:
2866:
2859:
2856:
2855:
2843:
2835:
2830:
2827:
2826:
2814:
2806:
2799:
2796:
2795:
2783:
2775:
2770:
2767:
2766:
2754:
2746:
2732:
2731:
2690:
2686:
2682:
2668:
2664:
2660:
2636:
2634:
2633:
2628:
2626:
2625:
2573:
2571:
2570:
2565:
2559:
2558:
2551:
2549:
2548:
2536:
2528:
2524:
2522:
2521:
2509:
2501:
2495:
2493:
2492:
2480:
2472:
2468:
2466:
2465:
2453:
2445:
2415:
2411:
2407:
2402:matrix transpose
2392:
2383:
2381:
2380:
2375:
2373:
2372:
2365:
2363:
2362:
2350:
2342:
2338:
2336:
2335:
2323:
2315:
2309:
2307:
2306:
2294:
2286:
2282:
2280:
2279:
2267:
2259:
2249:
2248:
2213:
2212:
2211:
2205:
2204:
2197:
2195:
2194:
2182:
2174:
2170:
2168:
2167:
2155:
2147:
2141:
2139:
2138:
2126:
2118:
2114:
2112:
2111:
2099:
2091:
2077:
2076:
2069:
2059:
2047:
2037:
2013:
2005:
2001:
1997:
1993:
1986:
1979:
1961:
1952:
1950:
1949:
1944:
1939:
1938:
1937:
1928:
1927:
1919:
1912:
1911:
1910:
1901:
1900:
1892:
1885:
1873:
1872:
1871:
1862:
1861:
1853:
1846:
1845:
1844:
1835:
1834:
1826:
1819:
1807:
1806:
1805:
1796:
1795:
1787:
1780:
1779:
1778:
1769:
1768:
1760:
1753:
1737:
1727:
1720:
1713:
1709:
1697:
1695:
1694:
1689:
1687:
1686:
1655:
1654:
1619:
1618:
1586:
1585:
1560:
1557:is increased by
1556:
1552:
1549:is increased by
1548:
1544:
1534:
1531:
1520:
1512:
1503:) is called the
1496:
1485:
1476:
1474:
1473:
1468:
1463:
1462:
1457:
1456:
1448:
1441:
1440:
1435:
1434:
1426:
1412:
1411:
1406:
1405:
1397:
1390:
1389:
1384:
1383:
1375:
1361:
1360:
1355:
1354:
1346:
1339:
1338:
1333:
1332:
1324:
1306:
1295:
1286:
1284:
1283:
1278:
1273:
1272:
1224:
1223:
1198:
1197:
1171:
1157:
1155:
1154:
1149:
1143:
1141:
1133:
1132:
1131:
1123:
1116:
1111:
1110:
1105:
1104:
1096:
1087:
1085:
1077:
1076:
1075:
1067:
1060:
1055:
1054:
1049:
1048:
1040:
1014:
1012:
1011:
1006:
1004:
983:
981:
980:
975:
973:
958:
956:
955:
950:
949:
941:
934:
933:
928:
927:
919:
914:
913:
898:
887:
886:
881:
880:
872:
865:
864:
859:
858:
850:
836:
819:
805:
804:
799:
798:
790:
783:
782:
777:
776:
768:
763:
762:
747:
739:
736:
731:
716:
705:
701:
667:
666:
658:
655:
653:
642:
634:
629:
589:
584:parametric curve
581:
563:
560:
549:
545:
541:
529:
527:
526:
521:
519:
518:
458:
454:
447:
432:depending on an
428:
426:
425:
420:
418:
417:
346:
345:
320:
319:
311:
295:
291:
287:
283:
279:
231:symmetric tensor
223:symmetric matrix
219:coordinate basis
182:
174:
170:
162:
158:
154:
142:
138:
119:
116:A metric tensor
112:
108:
104:
100:
84:
72:
68:
52:
22790:
22789:
22785:
22784:
22783:
22781:
22780:
22779:
22745:
22744:
22743:
22738:
22677:Banach manifold
22670:Generalizations
22665:
22620:
22557:
22454:
22416:Ricci curvature
22372:Cotangent space
22350:
22288:
22130:
22124:
22083:Exponential map
22047:
21992:
21986:
21906:
21896:
21866:
21861:
21812:Albert Einstein
21779:
21760:Einstein tensor
21723:
21704:Ricci curvature
21684:Kronecker delta
21670:Notable tensors
21665:
21586:Connection form
21563:
21557:
21488:
21474:Tensor operator
21431:
21425:
21365:
21341:Computer vision
21334:
21316:
21312:Tensor calculus
21256:
21245:
21240:
21210:
21205:
21174:
21153:Generalizations
21148:
21102:
21071:
21006:Exponential map
20967:
20957:
20901:
20882:
20870:
20802:
20754:
20722:
20714:, McGraw-Hill,
20663:
20633:
20615:Tensor geometry
20609:
20604:
20603:
20596:
20592:
20579:
20575:
20567:
20563:
20555:
20551:
20542:
20538:
20529:
20525:
20516:
20512:
20507:
20459:
20446:
20438:
20413:
20412:
20400:
20396:
20390:
20386:
20381:
20376:
20371:
20365:
20364:
20359:
20353:
20349:
20344:
20339:
20333:
20332:
20327:
20322:
20313:
20299:
20295:
20291:
20280:
20278:
20271:
20267:
20266:
20261:
20255:
20254:
20249:
20244:
20239:
20225:
20221:
20217:
20206:
20204:
20197:
20193:
20186:
20185:
20173:
20169:
20167:
20164:
20163:
20098:
20094:
20085:
20081:
20072:
20068:
20059:
20055:
20054:
20050:
20048:
20045:
20044:
20009:
20005:
19996:
19992:
19980:
19976:
19967:
19963:
19954:
19950:
19938:
19934:
19922:
19918:
19906:
19902:
19890:
19886:
19877:
19873:
19864:
19860:
19855:
19852:
19851:
19808:
19807:
19802:
19797:
19792:
19786:
19785:
19780:
19775:
19770:
19764:
19763:
19758:
19753:
19748:
19742:
19741:
19736:
19731:
19726:
19713:
19712:
19700:
19692:
19691:
19683:
19678:
19673:
19667:
19666:
19661:
19653:
19648:
19642:
19641:
19636:
19631:
19623:
19617:
19616:
19611:
19606:
19601:
19591:
19590:
19582:
19579:
19578:
19509:
19505:
19496:
19492:
19483:
19479:
19470:
19466:
19465:
19461:
19452:
19448:
19446:
19443:
19442:
19432:Minkowski space
19428:
19422:
19396:
19392:
19376:
19372:
19363:
19359:
19347:
19343:
19338:
19335:
19334:
19306:
19305:
19293:
19289:
19287:
19281:
19280:
19275:
19265:
19264:
19256:
19253:
19252:
19245:
19241:
19237:
19233:
19223:
19211:
19201:
19198:
19170:
19166:
19162:
19155:
19151:
19147:
19145:
19136:
19132:
19128:
19121:
19117:
19113:
19111:
19105:
19089:
19085:
19081:
19074:
19070:
19066:
19064:
19055:
19051:
19047:
19040:
19036:
19032:
19030:
19021:
19017:
19008:
18992:
18988:
18986:
18983:
18982:
18973:
18971:
18964:Kronecker delta
18952:
18949:Euclidean space
18942:
18911:
18910:
18904:
18900:
18898:
18892:
18891:
18886:
18876:
18875:
18865:
18864:
18852:
18848:
18842:
18838:
18823:
18819:
18813:
18809:
18807:
18756:
18755:
18705:
18693:
18689:
18674:
18670:
18663:
18662:
18649:
18648:
18644:
18636:
18633:
18632:
18612:
18611:
18600:
18599:
18585:
18573:
18572:
18555:
18539:
18538:
18531:
18525:
18524:
18505:
18499:
18498:
18479:
18472:
18470:
18467:
18466:
18451:
18420:
18416:
18398:
18394:
18383:
18377:
18372:
18360:
18357:
18356:
18328:
18327:
18322:
18316:
18315:
18310:
18300:
18299:
18291:
18288:
18287:
18272:
18258:
18253:
18236:
18226:
18209:
18205:
18187:
18183:
18165:
18161:
18159:
18151:
18148:
18147:
18135:
18120:
18105:
18099:
18091:
18080:
18076:
18072:
18040:
18036:
18034:
18016:
18012:
17991:
17987:
17978:
17974:
17961:
17957:
17946:
17943:
17942:
17930:
17924:
17916:
17909:
17905:
17901:
17886:
17880:
17876:
17850:
17846:
17842:
17839:
17806:
17747:
17743:
17733:
17728:
17727:
17723:
17697:
17693:
17683:
17678:
17677:
17673:
17646:
17642:
17636:
17619:
17609:
17604:
17590:
17582:
17579:
17578:
17568:
17505:
17501:
17491:
17486:
17485:
17481:
17455:
17451:
17441:
17436:
17435:
17431:
17404:
17400:
17394:
17377:
17372:
17368:
17366:
17360:
17355:
17343:
17340:
17339:
17328:
17318:
17310:
17286:
17282:
17273:
17269:
17248:
17244:
17238:
17221:
17208:
17204:
17199:
17196:
17195:
17133:
17129:
17119:
17114:
17113:
17109:
17083:
17079:
17069:
17064:
17063:
17059:
17032:
17028:
17022:
17005:
16999:
16993:
16988:
16976:
16973:
16972:
16951:
16947:
16932:
16926:
16918:
16912:
16898:
16892:
16888:
16884:
16881:
16854:
16843:
16841:
16838:
16837:
16811:
16799:
16794:
16793:
16781:
16776:
16775:
16773:
16770:
16769:
16758:
16754:
16717:
16712:
16707:
16703:
16677:
16672:
16667:
16663:
16661:
16658:
16657:
16632:
16620:
16615:
16614:
16602:
16597:
16591:
16588:
16587:
16579:
16574:
16573:
16572:
16568:
16554:
16551:
16543:
16539:
16535:
16532:
16524:
16520:
16516:
16500:
16497:
16489:
16485:
16481:
16470:
16465:
16464:
16463:
16461:
16455:
16449:
16443:
16437:
16433:
16406:
16402:
16393:
16389:
16371:
16367:
16358:
16354:
16345:
16341:
16339:
16336:
16335:
16326:
16320:
16312:
16307:
16306:
16305:
16300:
16291:
16285:
16277:
16274:
16266:
16243:
16239:
16230:
16226:
16220:
16216:
16201:
16197:
16188:
16184:
16178:
16174:
16169:
16166:
16165:
16155:
16147:
16146:. Furthermore,
16140:
16132:
16125:
16117:
16105:
16100:
16099:
16098:
16096:
16090:
16084:
16077:
16069:
16066:
16058:
16055:
16047:
16024:
16020:
16011:
16007:
15998:
15994:
15982:
15978:
15969:
15965:
15959:
15955:
15950:
15947:
15946:
15935:
15930:
15929:
15928:
15926:
15917:
15911:
15907:
15898:
15889:
15881:
15877:
15874:
15866:
15860:
15854:
15818:
15814:
15799:
15794:
15792:
15791:
15784:
15780:
15774:
15770:
15768:
15765:
15764:
15755:
15749:
15741:
15722:
15716:
15700:
15670:
15661:
15657:
15648:
15644:
15635:
15631:
15629:
15626:
15625:
15616:
15612:
15589:
15577:
15573:
15562:
15559:
15558:
15551:
15547:
15540:
15534:
15522:
15516:
15512:
15505:
15491:
15488:
15482:
15456:
15451:
15450:
15438:
15433:
15432:
15423:
15419:
15411:
15400:
15395:
15392:
15391:
15381:
15337:
15332:
15330:
15329:
15321:
15310:
15302:
15299:
15298:
15275:
15271:
15256:
15252:
15250:
15247:
15246:
15239:
15228:
15172:
15168:
15166:
15163:
15162:
15148:
15145:
15139:
15111:
15107:
15099:
15088:
15084:
15080:
15068:
15064:
15062:
15059:
15058:
15048:
15037:
15031:
15011:
15002:
14958:
14946:
14941:
14940:
14928:
14923:
14922:
14913:
14909:
14907:
14904:
14903:
14896:
14890:
14889:with itself to
14886:
14846:
14835:
14829:
14825:
14817:
14809:
14806:
14805:
14785:
14759:
14758:
14749:
14745:
14741:
14734:
14730:
14726:
14724:
14722:
14717:
14708:
14704:
14700:
14693:
14689:
14685:
14683:
14681:
14672:
14668:
14664:
14657:
14653:
14649:
14647:
14644:
14643:
14638:
14633:
14628:
14622:
14621:
14612:
14608:
14604:
14597:
14593:
14589:
14587:
14585:
14580:
14571:
14567:
14563:
14556:
14552:
14548:
14546:
14544:
14535:
14531:
14527:
14520:
14516:
14512:
14510:
14507:
14506:
14497:
14493:
14489:
14482:
14478:
14474:
14472:
14470:
14465:
14456:
14452:
14448:
14441:
14437:
14433:
14431:
14429:
14420:
14416:
14412:
14405:
14401:
14397:
14395:
14388:
14387:
14376:
14373:
14372:
14365:
14332:
14331:
14327:
14307:
14299:
14296:
14295:
14286:
14254:
14250:
14232:
14228:
14205:
14202:
14201:
14194:
14190:
14186:
14182:
14154:
14149:
14148:
14139:
14135:
14131:
14124:
14120:
14116:
14114:
14108:
14104:
14098:
14087:
14077:
14066:
14044:
14040:
14038:
14035:
14034:
14027:
14023:
14019:
14015:
14009:
14006:
13998:
13978:
13973:
13972:
13966:
13962:
13947:
13942:
13941:
13935:
13931:
13923:
13920:
13919:
13912:
13908:
13897:
13891:
13877:
13873:
13860:
13853:
13845:
13844:. The image of
13841:
13829:
13819:
13813:
13811:Euclidean space
13806:
13798:
13792:
13784:
13781:
13754:
13745:
13741:
13733:
13721:
13717:
13711:
13700:
13685:
13676:
13672:
13670:
13667:
13666:
13647:
13641:
13631:
13625:
13612:
13587:
13572:
13568:
13554:
13546:
13543:
13542:
13505:
13504:
13500:
13495:
13481:
13469:
13465:
13454:
13446:
13443:
13442:
13431:
13427:
13418:
13408:
13405:raise the index
13375:
13363:
13359:
13351:
13342:
13338:
13332:
13321:
13306:
13297:
13293:
13291:
13288:
13287:
13280:
13271:
13261:
13246:
13242:
13229:
13204:
13191:
13190:
13186:
13181:
13161:
13148:
13147:
13143:
13136:
13135:
13122:
13118:
13114:
13113:
13106:
13105:
13101:
13096:
13076:
13063:
13062:
13058:
13050:
13042:
13039:
13038:
13022:
12997:
12984:
12983:
12979:
12974:
12957:
12949:
12946:
12945:
12931:
12925:
12924:has components
12921:
12915:
12911:
12902:
12894:
12890:
12887:
12879:
12875:
12868:
12860:
12837:
12833:
12824:
12820:
12811:
12807:
12798:
12794:
12776:
12772:
12763:
12759:
12757:
12754:
12753:
12747:
12739:
12733:
12725:
12719:
12715:
12714:(by the matrix
12707:
12694:
12681:
12659:
12645:
12625:
12608:
12594:
12591:
12590:
12572:
12571:is replaced by
12566:
12563:
12555:
12551:
12542:
12533:
12503:
12502:
12494:
12480:
12465:
12464:
12456:
12444:
12443:
12440:
12439:
12431:
12422:
12418:
12416:
12411:
12403:
12394:
12390:
12388:
12380:
12371:
12367:
12363:
12357:
12356:
12347:
12346:
12338:
12329:
12325:
12317:
12308:
12304:
12287:
12278:
12274:
12266:
12257:
12253:
12242:
12233:
12229:
12221:
12212:
12208:
12201:
12194:
12192:
12189:
12188:
12180:
12176:
12151:
12136:
12132:
12118:
12110:
12107:
12106:
12102:transforms via
12097:
12091:
12078:
12054:
12053:
12045:
12036:
12032:
12029:
12028:
12022:
12021:
12013:
12004:
12000:
11997:
11996:
11988:
11979:
11975:
11968:
11967:
11956:
11948:
11945:
11944:
11938:Kronecker delta
11934:
11925:
11913:
11893:
11892:
11869:
11867:
11861:
11860:
11840:
11838:
11828:
11827:
11815:
11811:
11800:
11791:
11787:
11785:
11782:
11781:
11766:
11755:
11746:
11736:
11727:
11721:
11717:
11714:contravariantly
11700:
11674:
11663:
11649:
11635:
11627:
11624:
11623:
11608:
11604:
11598:
11588:
11553:
11542:
11532:
11531:
11523:
11514:
11510:
11507:
11506:
11500:
11499:
11491:
11482:
11478:
11475:
11474:
11466:
11457:
11453:
11446:
11445:
11440:
11431:
11427:
11419:
11410:
11406:
11391:
11387:
11379:
11370:
11366:
11357:
11353:
11345:
11336:
11332:
11324:
11321:
11320:
11311:
11307:
11298:
11288:
11285:
11279:
11264:
11260:
11254:
11253:are denoted by
11248:
11239:
11233:
11210:
11209:
11199:
11198:
11194:
11189:
11174:
11170:
11165:
11151:
11143:
11141:
11135:
11134:
11122:
11121:
11117:
11112:
11099:
11098:
11094:
11093:
11089:
11077:
11076:
11063:
11059:
11055:
11054:
11045:
11041:
11036:
11021:
11017:
11016:
11012:
10996:
10989:
10985:
10983:
10981:
10975:
10974:
10967:
10966:
10962:
10954:
10939:
10935:
10927:
10910:
10899:
10897:
10894:
10893:
10881:
10875:
10869:
10832:
10831:
10827:
10822:
10807:
10803:
10798:
10784:
10749:
10748:
10746:
10743:
10742:
10733:
10729:
10716:
10687:
10667:
10659:
10656:
10655:
10646:
10642:
10636:
10612:
10611:
10608:
10607:
10601:
10597:
10595:
10590:
10584:
10580:
10578:
10572:
10568:
10564:
10558:
10557:
10546:
10538:
10535:
10534:
10467:
10463:
10459:
10447:
10443:
10441:
10438:
10437:
10428:
10424:
10417:
10413:
10389:
10385:
10381:
10375:
10371:
10363:
10360:
10359:
10353:smooth function
10348:
10344:
10316:
10312:
10308:
10302:
10298:
10282:
10278:
10274:
10268:
10264:
10247:
10243:
10231:
10227:
10223:
10219:
10213:
10209:
10207:
10204:
10203:
10196:
10192:
10189:
10181:
10178:
10170:
10162:
10159:
10151:
10147:
10143:
10139:
10124:
10120:contravariantly
10081:
10080:
10067:
10063:
10059:
10058:
10049:
10045:
10040:
10025:
10021:
10009:
10005:
9997:
9989:
9986:
9985:
9976:
9970:
9952:
9920:
9916:
9907:
9903:
9902:
9898:
9884:
9872:
9868:
9866:
9863:
9862:
9853:
9849:
9840:
9830:
9827:
9818:ultrahyperbolic
9805:
9801:
9794:
9790:
9779:
9771:
9767:
9759:
9755:
9751:
9740:
9736:
9728:
9724:
9716:
9712:
9691:
9687:
9684:
9676:
9660:
9650:
9646:
9630:
9626:
9622:
9618:
9614:
9610:
9606:
9602:
9582:
9572:
9568:
9564:
9563:
9548:
9532:
9528:
9524:
9523:
9514:
9504:
9500:
9496:
9495:
9480:
9470:
9466:
9462:
9461:
9452:
9442:
9438:
9434:
9433:
9419:
9415:
9409:
9405:
9399:
9394:
9390:
9384:
9380:
9378:
9375:
9374:
9368:
9360:
9349:
9346:
9342:
9334:
9326:
9322:
9318:
9317:independent of
9311:
9303:
9295:
9285:
9281:
9274:
9266:
9263:
9255:
9229:
9225:
9216:
9212:
9198:
9194:
9185:
9181:
9172:
9168:
9156:
9152:
9143:
9139:
9137:
9134:
9133:
9123:
9117:
9109:Jacobian matrix
9104:
9081:
9077:
9059:
9055:
9045:
9044:
9030:
9026:
9016:
9012:
9011:
8995:
8994:
8990:
8985:
8982:
8981:
8974:
8962:
8958:
8946:
8920:
8916:
8912:
8905:
8901:
8897:
8895:
8886:
8882:
8873:
8869:
8860:
8856:
8852:
8845:
8841:
8837:
8835:
8829:
8812:
8792:
8791:
8787:
8778:
8774:
8772:
8769:
8768:
8742:
8738:
8734:
8729:
8720:
8716:
8712:
8705:
8701:
8697:
8695:
8689:
8678:
8662:
8658:
8654:
8649:
8647:
8644:
8643:
8629:
8621:
8590:
8586:
8582:
8577:
8565:
8561:
8557:
8552:
8551:
8547:
8528:
8527:
8523:
8514:
8510:
8508:
8505:
8504:
8447:
8443:
8428:
8424:
8415:
8411:
8402:
8398:
8389:
8385:
8383:
8380:
8379:
8344:
8340:
8336:
8331:
8319:
8315:
8311:
8306:
8305:
8301:
8286:
8282:
8273:
8269:
8267:
8264:
8263:
8256:
8224:
8220:
8216:
8211:
8202:
8198:
8180:
8176:
8172:
8167:
8158:
8154:
8153:
8149:
8141:
8139:
8136:
8135:
8129:
8125:
8121:
8102:
8098:
8095:
8084:
8081:
8073:
8046:
8042:
8034:
8022:
8018:
8009:
8005:
7999:
7982:
7964:
7952:
7948:
7946:
7943:
7942:
7914:
7901:
7900:
7896:
7882:
7874:
7871:
7870:
7863:
7859:
7847:
7837:
7812:
7795:
7791:
7785:
7781:
7775:
7753:
7749:
7743:
7739:
7733:
7728:
7724:
7712:
7711:
7703:
7701:
7698:
7697:
7691:change of basis
7684:
7678:
7677:of the vectors
7668:
7662:
7640:
7632:
7624:
7611:
7610:
7606:
7601:
7593:
7582:
7574:
7566:
7553:
7552:
7548:
7543:
7535:
7512:
7509:
7508:
7499:
7493:
7486:
7482:
7476:
7472:
7463:
7438:
7426:
7422:
7416:
7412:
7406:
7402:
7396:
7379:
7361:
7357:
7348:
7344:
7343:
7339:
7330:
7326:
7320:
7316:
7310:
7293:
7266:
7263:
7262:
7249:
7245:
7235:
7215:
7211:
7205:
7201:
7195:
7184:
7163:
7159:
7153:
7149:
7143:
7132:
7120:
7117:
7116:
7107:
7094:
7091:
7083:
7077:
7036:
7032:
7023:
7019:
7018:
7014:
7000:
6988:
6984:
6982:
6979:
6978:
6969:
6960:
6950:
6935:
6928:
6919:
6899:
6895:
6886:
6882:
6873:
6869:
6837:
6834:
6833:
6829:
6825:
6821:
6814:
6810:
6803:
6795:
6792:
6784:
6780:
6776:
6772:
6768:
6760:
6751:
6742:
6734:
6731:
6723:
6722:there exists a
6719:
6711:
6706:to vary is not
6704:
6696:
6693:
6685:
6665:
6661:
6652:
6648:
6639:
6635:
6626:
6622:
6620:
6617:
6616:
6615:, the function
6612:
6604:
6597:
6589:
6564:
6560:
6551:
6547:
6538:
6534:
6522:
6518:
6509:
6505:
6496:
6492:
6490:
6487:
6486:
6483:
6475:
6472:
6464:
6457:
6449:
6434:
6433:
6420:
6416:
6407:
6403:
6394:
6390:
6375:
6371:
6362:
6358:
6349:
6345:
6335:
6326:
6322:
6310:
6306:
6294:
6290:
6281:
6277:
6274:
6273:
6268:
6254:
6250:
6241:
6237:
6228:
6224:
6209:
6205:
6196:
6192:
6183:
6179:
6169:
6160:
6156:
6147:
6143:
6131:
6127:
6115:
6111:
6107:
6105:
6102:
6101:
6097:
6093:
6089:
6086:
6078:
6075:
6067:
6064:
6056:
6049:
6041:
6026:
6023:
6015:
6012:
6004:
6000:
5991:
5982:
5974:
5970:
5966:
5956:
5950:
5937:
5914:
5909:
5908:
5906:
5903:
5902:
5900:Cartesian space
5888:
5880:
5877:smooth manifold
5872:
5869:
5856:
5839:
5838:
5815:
5814:
5809:
5803:
5802:
5797:
5787:
5786:
5781:
5775:
5771:
5767:
5765:
5759:
5758:
5736:
5732:
5721:
5715:
5711:
5707:
5705:
5699:
5698:
5676:
5665:
5654:
5653:
5652:
5643:
5632:
5631:
5630:
5629:
5625:
5624:
5610:
5599:
5598:
5597:
5588:
5577:
5576:
5575:
5574:
5570:
5559:
5548:
5547:
5546:
5537:
5526:
5525:
5524:
5523:
5519:
5517:
5511:
5507:
5500:
5498:
5495:
5494:
5479:
5440:
5429:
5428:
5427:
5418:
5407:
5406:
5405:
5404:
5400:
5394:
5390:
5388:
5385:
5384:
5377:
5373:
5369:
5353:
5349:
5345:
5338:
5306:
5302:
5293:
5289:
5288:
5279:
5271:
5264:
5262:
5242:
5239:
5238:
5229:
5223:
5219:
5195:
5187:
5179:
5167:
5163:
5161:
5158:
5157:
5148:
5141:
5137:
5133:
5126:
5120:
5113:
5107:
5090:
5089:
5075:
5074:
5066:
5065:
5061:
5044:
5036:
5020:
5006:
5005:
4994:
4983:
4982:
4978:
4972:
4971:
4966:
4952:
4940:
4939:
4938:
4934:
4917:
4909:
4893:
4883:
4871:
4870:
4859:
4855:
4851:
4844:
4842:
4839:
4838:
4829:
4823:
4789:
4781:
4764:
4756:
4748:
4745:
4744:
4740:, meaning that
4735:
4729:
4698:
4694:
4688:
4684:
4672:
4668:
4662:
4658:
4646:
4642:
4636:
4632:
4620:
4616:
4610:
4606:
4595:
4587:
4579:
4576:
4575:
4568:
4561:
4554:
4551:
4545:
4542:
4536:
4533:
4527:
4524:
4518:
4501:
4500:
4489:
4488:
4482:
4478:
4475:
4474:
4468:
4464:
4457:
4456:
4449:
4448:
4443:
4437:
4436:
4431:
4421:
4420:
4413:
4412:
4406:
4402:
4400:
4394:
4390:
4383:
4382:
4373:
4372:
4360:
4356:
4350:
4346:
4334:
4330:
4324:
4320:
4308:
4304:
4298:
4294:
4282:
4278:
4272:
4268:
4259:
4258:
4252:
4241:
4240:
4239:
4230:
4219:
4218:
4217:
4211:
4207:
4201:
4197:
4188:
4177:
4176:
4175:
4166:
4155:
4154:
4153:
4147:
4143:
4137:
4133:
4124:
4113:
4112:
4111:
4102:
4091:
4090:
4089:
4083:
4079:
4073:
4069:
4060:
4049:
4048:
4047:
4038:
4027:
4026:
4025:
4019:
4015:
4009:
4005:
3998:
3993:
3985:
3981:
3979:
3976:
3975:
3967:then using the
3951:
3950:
3944:
3933:
3932:
3931:
3925:
3921:
3912:
3901:
3900:
3899:
3893:
3889:
3882:
3877:
3874:
3873:
3867:
3856:
3855:
3854:
3848:
3844:
3835:
3824:
3823:
3822:
3816:
3812:
3805:
3800:
3796:
3794:
3791:
3790:
3784:
3778:
3775:
3769:
3749:
3738:
3737:
3736:
3730:
3726:
3717:
3706:
3705:
3704:
3698:
3694:
3686:
3684:
3681:
3680:
3673:
3666:tangent vectors
3662:
3643:
3642:
3632:
3628:
3620:
3605:
3604:
3597:
3596:
3588:
3582:
3581:
3573:
3563:
3562:
3555:
3554:
3546:
3544:
3536:
3533:
3532:
3524:
3522:
3514:
3507:
3506:
3499:
3498:
3490:
3485:
3477:
3467:
3466:
3457:
3456:
3449:
3448:
3440:
3434:
3433:
3425:
3415:
3414:
3407:
3406:
3394:
3390:
3382:
3379:
3377:
3365:
3361:
3353:
3350:
3347:
3346:
3334:
3330:
3322:
3319:
3317:
3305:
3301:
3293:
3290:
3283:
3282:
3275:
3274:
3269:
3263:
3262:
3257:
3247:
3246:
3239:
3238:
3231:
3230:
3218:
3214:
3206:
3203:
3201:
3189:
3185:
3177:
3174:
3171:
3170:
3158:
3154:
3146:
3143:
3141:
3129:
3125:
3117:
3114:
3107:
3106:
3105:
3098:
3097:
3089:
3084:
3076:
3066:
3065:
3056:
3055:
3048:
3047:
3038:
3037:
3024:
3023:
3016:
3015:
3010:
3004:
3003:
2998:
2988:
2987:
2980:
2979:
2971:
2958:
2957:
2950:
2944:
2940:
2933:
2931:
2928:
2927:
2903:
2902:
2894:
2888:
2887:
2879:
2869:
2868:
2861:
2860:
2848:
2844:
2836:
2833:
2831:
2819:
2815:
2807:
2804:
2801:
2800:
2788:
2784:
2776:
2773:
2771:
2759:
2755:
2747:
2744:
2737:
2736:
2726:
2725:
2716:
2715:
2702:
2701:
2699:
2696:
2695:
2688:
2684:
2680:
2666:
2662:
2658:
2652:
2620:
2619:
2614:
2608:
2607:
2602:
2592:
2591:
2589:
2586:
2585:
2553:
2552:
2541:
2537:
2529:
2527:
2525:
2514:
2510:
2502:
2500:
2497:
2496:
2485:
2481:
2473:
2471:
2469:
2458:
2454:
2446:
2444:
2437:
2436:
2428:
2425:
2424:
2418:Jacobian matrix
2413:
2409:
2405:
2367:
2366:
2355:
2351:
2343:
2341:
2339:
2328:
2324:
2316:
2314:
2311:
2310:
2299:
2295:
2287:
2285:
2283:
2272:
2268:
2260:
2258:
2251:
2250:
2243:
2242:
2237:
2231:
2230:
2225:
2215:
2214:
2207:
2206:
2199:
2198:
2187:
2183:
2175:
2173:
2171:
2160:
2156:
2148:
2146:
2143:
2142:
2131:
2127:
2119:
2117:
2115:
2104:
2100:
2092:
2090:
2083:
2082:
2081:
2071:
2070:
2062:
2060:
2052:
2049:
2048:
2040:
2038:
2030:
2023:
2022:
2020:
2017:
2016:
2003:
1999:
1995:
1988:
1981:
1974:
1930:
1929:
1918:
1917:
1916:
1903:
1902:
1891:
1890:
1889:
1878:
1864:
1863:
1852:
1851:
1850:
1837:
1836:
1825:
1824:
1823:
1812:
1798:
1797:
1786:
1785:
1784:
1771:
1770:
1759:
1758:
1757:
1746:
1744:
1741:
1740:
1722:
1715:
1711:
1707:
1704:
1681:
1680:
1671:
1670:
1657:
1656:
1649:
1648:
1643:
1637:
1636:
1631:
1621:
1620:
1613:
1612:
1604:
1591:
1590:
1581:
1577:
1572:
1569:
1568:
1558:
1554:
1550:
1546:
1530:
1526:
1518:
1508:
1494:
1458:
1447:
1446:
1445:
1436:
1425:
1424:
1423:
1407:
1396:
1395:
1394:
1385:
1374:
1373:
1372:
1356:
1345:
1344:
1343:
1334:
1323:
1322:
1321:
1313:
1310:
1309:
1268:
1264:
1219:
1215:
1193:
1189:
1178:
1175:
1174:
1134:
1122:
1121:
1117:
1115:
1106:
1095:
1094:
1093:
1078:
1066:
1065:
1061:
1059:
1050:
1039:
1038:
1037:
1035:
1032:
1031:
1015:represents the
994:
992:
989:
988:
971:
970:
951:
940:
939:
938:
929:
918:
917:
916:
909:
905:
891:
882:
871:
870:
869:
860:
849:
848:
847:
829:
812:
800:
789:
788:
787:
778:
767:
766:
765:
758:
754:
740:
738:
732:
727:
714:
713:
657:
656:
646:
641:
640:
636:
630:
625:
614:
607:
605:
602:
601:
587:
559:
555:
547:
543:
539:
536:
513:
512:
507:
501:
500:
495:
485:
484:
482:
479:
478:
456:
452:
437:
413:
412:
341:
340:
310:
309:
307:
304:
303:
293:
289:
285:
281:
277:
254:
180:
172:
168:
160:
156:
152:
140:
125:
117:
110:
106:
102:
98:
91:tangent vectors
82:
77:defined on the
70:
66:
63:Euclidean space
50:
24:
17:
12:
11:
5:
22788:
22778:
22777:
22775:Metric tensors
22772:
22767:
22762:
22757:
22740:
22739:
22737:
22736:
22731:
22726:
22721:
22716:
22715:
22714:
22704:
22699:
22694:
22689:
22684:
22679:
22673:
22671:
22667:
22666:
22664:
22663:
22658:
22653:
22648:
22643:
22638:
22632:
22630:
22626:
22625:
22622:
22621:
22619:
22618:
22613:
22608:
22603:
22598:
22593:
22588:
22583:
22578:
22573:
22567:
22565:
22559:
22558:
22556:
22555:
22550:
22545:
22540:
22535:
22530:
22525:
22515:
22510:
22505:
22495:
22490:
22485:
22480:
22475:
22470:
22464:
22462:
22456:
22455:
22453:
22452:
22447:
22442:
22441:
22440:
22430:
22425:
22424:
22423:
22413:
22408:
22403:
22398:
22397:
22396:
22386:
22381:
22380:
22379:
22369:
22364:
22358:
22356:
22352:
22351:
22349:
22348:
22343:
22338:
22333:
22332:
22331:
22321:
22316:
22311:
22305:
22303:
22296:
22290:
22289:
22287:
22286:
22281:
22271:
22266:
22252:
22247:
22242:
22237:
22232:
22230:Parallelizable
22227:
22222:
22217:
22216:
22215:
22205:
22200:
22195:
22190:
22185:
22180:
22175:
22170:
22165:
22160:
22150:
22140:
22134:
22132:
22126:
22125:
22123:
22122:
22117:
22112:
22110:Lie derivative
22107:
22105:Integral curve
22102:
22097:
22092:
22091:
22090:
22080:
22075:
22074:
22073:
22066:Diffeomorphism
22063:
22057:
22055:
22049:
22048:
22046:
22045:
22040:
22035:
22030:
22025:
22020:
22015:
22010:
22005:
21999:
21997:
21988:
21987:
21985:
21984:
21979:
21974:
21969:
21964:
21959:
21954:
21949:
21944:
21943:
21942:
21937:
21927:
21926:
21925:
21914:
21912:
21911:Basic concepts
21908:
21907:
21895:
21894:
21887:
21880:
21872:
21863:
21862:
21860:
21859:
21854:
21852:Woldemar Voigt
21849:
21844:
21839:
21834:
21829:
21824:
21819:
21817:Leonhard Euler
21814:
21809:
21804:
21799:
21793:
21791:
21789:Mathematicians
21785:
21784:
21781:
21780:
21778:
21777:
21772:
21767:
21762:
21757:
21752:
21747:
21742:
21737:
21731:
21729:
21725:
21724:
21722:
21721:
21716:
21714:Torsion tensor
21711:
21706:
21701:
21696:
21691:
21686:
21680:
21678:
21671:
21667:
21666:
21664:
21663:
21658:
21653:
21648:
21643:
21638:
21633:
21628:
21623:
21618:
21613:
21608:
21603:
21598:
21593:
21588:
21583:
21578:
21573:
21567:
21565:
21559:
21558:
21556:
21555:
21549:
21547:Tensor product
21544:
21539:
21537:Symmetrization
21534:
21529:
21527:Lie derivative
21524:
21519:
21514:
21509:
21504:
21498:
21496:
21490:
21489:
21487:
21486:
21481:
21476:
21471:
21466:
21461:
21456:
21451:
21449:Tensor density
21446:
21441:
21435:
21433:
21427:
21426:
21424:
21423:
21421:Voigt notation
21418:
21413:
21408:
21406:Ricci calculus
21403:
21398:
21393:
21391:Index notation
21388:
21383:
21377:
21375:
21371:
21370:
21367:
21366:
21364:
21363:
21358:
21353:
21348:
21343:
21337:
21335:
21333:
21332:
21327:
21321:
21318:
21317:
21315:
21314:
21309:
21307:Tensor algebra
21304:
21299:
21294:
21289:
21287:Dyadic algebra
21284:
21279:
21273:
21271:
21262:
21258:
21257:
21250:
21247:
21246:
21239:
21238:
21231:
21224:
21216:
21207:
21206:
21204:
21203:
21198:
21193:
21188:
21182:
21180:
21176:
21175:
21173:
21172:
21170:Sub-Riemannian
21167:
21162:
21156:
21154:
21150:
21149:
21147:
21146:
21141:
21136:
21131:
21126:
21121:
21116:
21110:
21108:
21104:
21103:
21101:
21100:
21095:
21090:
21085:
21079:
21077:
21073:
21072:
21070:
21069:
21064:
21059:
21054:
21049:
21048:
21047:
21038:
21033:
21028:
21018:
21013:
21008:
21003:
21002:
21001:
20996:
20991:
20986:
20975:
20973:
20972:Basic concepts
20969:
20968:
20956:
20955:
20948:
20941:
20933:
20927:
20926:
20916:Wells, Raymond
20912:
20899:
20873:
20868:
20852:
20827:(1): 125–201,
20805:
20800:
20784:Thorne, Kip S.
20776:
20758:
20752:
20739:
20726:
20720:
20707:
20694:
20667:
20661:
20644:
20631:
20608:
20605:
20602:
20601:
20598:Sternberg 1983
20590:
20573:
20561:
20549:
20536:
20523:
20509:
20508:
20506:
20503:
20502:
20501:
20495:
20493:Ricci calculus
20490:
20485:
20480:
20475:
20470:
20465:
20458:
20455:
20435:
20434:
20423:
20417:
20411:
20408:
20403:
20399:
20393:
20389:
20385:
20382:
20380:
20377:
20375:
20372:
20370:
20367:
20366:
20363:
20360:
20356:
20352:
20348:
20345:
20343:
20340:
20338:
20335:
20334:
20331:
20328:
20326:
20323:
20319:
20316:
20311:
20302:
20298:
20294:
20289:
20286:
20283:
20277:
20274:
20270:
20265:
20262:
20260:
20257:
20256:
20253:
20250:
20248:
20245:
20243:
20240:
20237:
20228:
20224:
20220:
20215:
20212:
20209:
20203:
20200:
20196:
20192:
20191:
20189:
20184:
20179:
20176:
20172:
20157:
20156:
20145:
20141:
20138:
20135:
20132:
20129:
20126:
20123:
20120:
20117:
20114:
20111:
20107:
20101:
20097:
20093:
20088:
20084:
20080:
20075:
20071:
20067:
20062:
20058:
20053:
20030:
20029:
20018:
20012:
20008:
20004:
19999:
19995:
19991:
19986:
19983:
19979:
19975:
19970:
19966:
19962:
19957:
19953:
19949:
19946:
19941:
19937:
19933:
19930:
19925:
19921:
19917:
19914:
19909:
19905:
19901:
19898:
19893:
19889:
19885:
19880:
19876:
19872:
19867:
19863:
19859:
19847:is written as
19830:
19829:
19818:
19812:
19806:
19803:
19801:
19798:
19796:
19793:
19791:
19788:
19787:
19784:
19781:
19779:
19776:
19774:
19771:
19769:
19766:
19765:
19762:
19759:
19757:
19754:
19752:
19749:
19747:
19744:
19743:
19740:
19737:
19735:
19732:
19730:
19727:
19725:
19722:
19719:
19718:
19716:
19711:
19708:
19696:
19690:
19687:
19684:
19682:
19679:
19677:
19674:
19672:
19669:
19668:
19665:
19662:
19660:
19657:
19654:
19652:
19649:
19647:
19644:
19643:
19640:
19637:
19635:
19632:
19630:
19627:
19624:
19622:
19619:
19618:
19615:
19612:
19610:
19607:
19605:
19602:
19600:
19597:
19596:
19594:
19589:
19586:
19568:
19567:
19556:
19552:
19549:
19546:
19543:
19540:
19537:
19534:
19531:
19528:
19525:
19522:
19518:
19512:
19508:
19504:
19499:
19495:
19491:
19486:
19482:
19478:
19473:
19469:
19464:
19460:
19455:
19451:
19424:Main article:
19421:
19418:
19417:
19416:
19405:
19399:
19395:
19391:
19387:
19384:
19379:
19375:
19371:
19366:
19362:
19358:
19355:
19350:
19346:
19342:
19328:
19327:
19316:
19310:
19304:
19301:
19296:
19292:
19288:
19286:
19283:
19282:
19279:
19276:
19274:
19271:
19270:
19268:
19263:
19260:
19197:
19194:
19193:
19192:
19181:
19173:
19169:
19165:
19158:
19154:
19150:
19139:
19135:
19131:
19124:
19120:
19116:
19108:
19104:
19100:
19092:
19088:
19084:
19077:
19073:
19069:
19058:
19054:
19050:
19043:
19039:
19035:
19027:
19024:
19020:
19014:
19011:
19007:
19003:
18998:
18995:
18991:
18967:
18929:
18928:
18915:
18907:
18903:
18899:
18897:
18894:
18893:
18890:
18887:
18885:
18882:
18881:
18879:
18874:
18869:
18863:
18860:
18855:
18851:
18845:
18841:
18837:
18834:
18831:
18826:
18822:
18816:
18812:
18808:
18806:
18803:
18800:
18797:
18794:
18791:
18788:
18785:
18782:
18779:
18776:
18773:
18770:
18767:
18764:
18761:
18758:
18757:
18754:
18751:
18748:
18745:
18742:
18739:
18736:
18733:
18730:
18727:
18724:
18721:
18718:
18715:
18712:
18709:
18706:
18704:
18701:
18696:
18692:
18688:
18685:
18682:
18677:
18673:
18669:
18668:
18666:
18661:
18658:
18652:
18647:
18643:
18640:
18626:
18625:
18610:
18604:
18598:
18595:
18592:
18589:
18586:
18584:
18581:
18578:
18575:
18574:
18571:
18568:
18565:
18562:
18559:
18556:
18554:
18551:
18548:
18545:
18544:
18542:
18537:
18534:
18532:
18530:
18527:
18526:
18523:
18520:
18517:
18514:
18511:
18508:
18506:
18504:
18501:
18500:
18497:
18494:
18491:
18488:
18485:
18482:
18480:
18478:
18475:
18474:
18443:
18442:
18431:
18423:
18419:
18415:
18412:
18409:
18406:
18401:
18397:
18393:
18390:
18387:
18380:
18375:
18371:
18367:
18364:
18350:
18349:
18338:
18332:
18326:
18323:
18321:
18318:
18317:
18314:
18311:
18309:
18306:
18305:
18303:
18298:
18295:
18257:
18254:
18252:
18249:
18212:
18208:
18204:
18201:
18198:
18195:
18190:
18186:
18182:
18175:
18171:
18168:
18164:
18158:
18155:
18112:by means of a
18103:
18060:
18057:
18050:
18046:
18043:
18039:
18033:
18030:
18027:
18022:
18019:
18015:
18011:
18008:
18003:
18000:
17997:
17994:
17990:
17986:
17981:
17977:
17973:
17969:
17964:
17960:
17956:
17953:
17950:
17920:
17884:
17872:, by giving a
17838:
17835:
17811:kinetic energy
17795:
17794:
17783:
17779:
17776:
17771:
17767:
17764:
17761:
17758:
17755:
17750:
17746:
17739:
17736:
17732:
17726:
17721:
17717:
17714:
17711:
17708:
17705:
17700:
17696:
17689:
17686:
17682:
17676:
17672:
17669:
17666:
17663:
17660:
17657:
17652:
17649:
17645:
17639:
17634:
17631:
17628:
17625:
17622:
17618:
17612:
17607:
17603:
17597:
17594:
17589:
17586:
17574:of the curve:
17567:
17564:
17560:
17559:
17548:
17544:
17541:
17534:
17529:
17525:
17522:
17519:
17516:
17513:
17508:
17504:
17497:
17494:
17490:
17484:
17479:
17475:
17472:
17469:
17466:
17463:
17458:
17454:
17447:
17444:
17440:
17434:
17430:
17427:
17424:
17421:
17418:
17415:
17410:
17407:
17403:
17397:
17392:
17389:
17386:
17383:
17380:
17376:
17371:
17363:
17358:
17354:
17350:
17347:
17305:is called the
17303:
17302:
17289:
17285:
17281:
17276:
17272:
17268:
17265:
17262:
17259:
17254:
17251:
17247:
17241:
17236:
17233:
17230:
17227:
17224:
17220:
17216:
17211:
17207:
17203:
17183:
17182:
17171:
17167:
17164:
17157:
17153:
17150:
17147:
17144:
17141:
17136:
17132:
17125:
17122:
17118:
17112:
17107:
17103:
17100:
17097:
17094:
17091:
17086:
17082:
17075:
17072:
17068:
17062:
17058:
17055:
17052:
17049:
17046:
17043:
17038:
17035:
17031:
17025:
17020:
17017:
17014:
17011:
17008:
17004:
16996:
16991:
16987:
16983:
16980:
16922:
16880:
16877:
16876:
16875:
16864:
16861:
16857:
16853:
16850:
16846:
16827:
16826:
16814:
16810:
16807:
16802:
16797:
16792:
16789:
16784:
16779:
16751:
16750:
16738:
16734:
16731:
16728:
16723:
16720:
16715:
16711:
16706:
16702:
16698:
16694:
16691:
16688:
16683:
16680:
16675:
16671:
16666:
16651:
16650:
16639:
16635:
16631:
16628:
16623:
16618:
16613:
16608:
16605:
16600:
16596:
16575:
16547:
16528:
16523:. The mapping
16493:
16466:
16451:
16439:
16430:
16429:
16418:
16414:
16409:
16405:
16401:
16396:
16392:
16388:
16385:
16382:
16379:
16374:
16370:
16366:
16361:
16357:
16353:
16348:
16344:
16322:
16308:
16296:
16281:
16270:
16263:
16262:
16251:
16246:
16242:
16238:
16233:
16229:
16223:
16219:
16215:
16212:
16209:
16204:
16200:
16196:
16191:
16187:
16181:
16177:
16173:
16151:
16136:
16121:
16101:
16086:
16073:
16068:. The mapping
16062:
16051:
16044:
16043:
16032:
16027:
16023:
16019:
16014:
16010:
16006:
16001:
15997:
15993:
15990:
15985:
15981:
15977:
15972:
15968:
15962:
15958:
15954:
15931:
15913:
15903:
15894:
15885:
15870:
15856:
15847:
15846:
15835:
15832:
15829:
15826:
15821:
15817:
15813:
15810:
15797:
15787:
15783:
15777:
15773:
15751:
15745:
15730:tangent bundle
15715:
15712:
15697:tensor product
15689:
15688:
15677:
15673:
15669:
15664:
15660:
15656:
15651:
15647:
15643:
15638:
15634:
15605:
15604:
15592:
15588:
15585:
15580:
15576:
15572:
15569:
15566:
15533:
15530:
15520:
15486:
15479:
15478:
15467:
15464:
15459:
15454:
15449:
15446:
15441:
15436:
15431:
15426:
15422:
15418:
15414:
15410:
15407:
15403:
15399:
15371:
15370:
15359:
15356:
15353:
15350:
15347:
15340:
15335:
15328:
15324:
15320:
15317:
15313:
15309:
15306:
15292:
15291:
15278:
15274:
15270:
15267:
15264:
15259:
15255:
15225:
15224:
15213:
15210:
15207:
15204:
15201:
15198:
15195:
15192:
15189:
15186:
15183:
15180:
15175:
15171:
15143:
15136:
15135:
15124:
15120:
15114:
15110:
15106:
15102:
15098:
15095:
15091:
15087:
15083:
15079:
15076:
15071:
15067:
15035:
15010:
15007:
14977:
14976:
14965:
14961:
14957:
14954:
14949:
14944:
14939:
14936:
14931:
14926:
14921:
14916:
14912:
14883:tangent bundle
14873:
14872:
14863:
14861:
14849:
14845:
14842:
14838:
14832:
14828:
14824:
14820:
14816:
14813:
14799:is a function
14793:vector bundles
14784:
14781:
14780:
14779:
14768:
14763:
14752:
14748:
14744:
14737:
14733:
14729:
14723:
14721:
14718:
14711:
14707:
14703:
14696:
14692:
14688:
14682:
14675:
14671:
14667:
14660:
14656:
14652:
14646:
14645:
14642:
14639:
14637:
14634:
14632:
14629:
14627:
14624:
14623:
14615:
14611:
14607:
14600:
14596:
14592:
14586:
14584:
14581:
14574:
14570:
14566:
14559:
14555:
14551:
14545:
14538:
14534:
14530:
14523:
14519:
14515:
14509:
14508:
14500:
14496:
14492:
14485:
14481:
14477:
14471:
14469:
14466:
14459:
14455:
14451:
14444:
14440:
14436:
14430:
14423:
14419:
14415:
14408:
14404:
14400:
14394:
14393:
14391:
14386:
14383:
14380:
14362:
14361:
14350:
14347:
14344:
14341:
14335:
14330:
14326:
14323:
14320:
14317:
14314:
14310:
14306:
14303:
14283:
14282:
14271:
14268:
14265:
14262:
14257:
14253:
14249:
14246:
14243:
14240:
14235:
14231:
14227:
14224:
14221:
14218:
14215:
14212:
14209:
14175:
14174:
14163:
14157:
14152:
14142:
14138:
14134:
14127:
14123:
14119:
14111:
14107:
14101:
14096:
14093:
14090:
14086:
14080:
14075:
14072:
14069:
14065:
14061:
14058:
14055:
14052:
14047:
14043:
14018:is applied to
14002:
13995:
13994:
13981:
13976:
13969:
13965:
13961:
13958:
13955:
13950:
13945:
13938:
13934:
13930:
13927:
13902:induced metric
13828:. The mapping
13805:function from
13780:
13779:Induced metric
13777:
13776:
13775:
13764:
13761:
13757:
13753:
13748:
13744:
13740:
13736:
13732:
13727:
13724:
13720:
13714:
13709:
13706:
13703:
13699:
13695:
13692:
13688:
13684:
13679:
13675:
13609:
13608:
13597:
13594:
13590:
13586:
13583:
13578:
13575:
13571:
13567:
13564:
13561:
13557:
13553:
13550:
13534:
13533:
13524:
13522:
13508:
13503:
13498:
13494:
13491:
13488:
13484:
13480:
13475:
13472:
13468:
13464:
13461:
13457:
13453:
13450:
13423:
13416:
13394:
13393:
13382:
13378:
13374:
13369:
13366:
13362:
13358:
13354:
13350:
13345:
13341:
13335:
13330:
13327:
13324:
13320:
13316:
13313:
13309:
13305:
13300:
13296:
13276:
13269:
13226:
13225:
13214:
13211:
13207:
13203:
13200:
13194:
13189:
13184:
13180:
13177:
13174:
13171:
13168:
13164:
13160:
13157:
13151:
13146:
13139:
13133:
13128:
13125:
13121:
13117:
13109:
13104:
13099:
13095:
13092:
13089:
13086:
13083:
13079:
13075:
13072:
13066:
13061:
13057:
13053:
13049:
13046:
13019:
13018:
13007:
13004:
13000:
12996:
12993:
12987:
12982:
12977:
12973:
12970:
12967:
12964:
12960:
12956:
12953:
12907:
12898:
12883:
12864:
12857:
12856:
12845:
12840:
12836:
12832:
12827:
12823:
12819:
12814:
12810:
12806:
12801:
12797:
12793:
12790:
12787:
12784:
12779:
12775:
12771:
12766:
12762:
12743:
12729:
12678:
12677:
12666:
12662:
12658:
12655:
12652:
12648:
12644:
12641:
12638:
12635:
12632:
12628:
12624:
12621:
12618:
12615:
12611:
12607:
12604:
12601:
12598:
12559:
12547:
12528:
12527:
12518:
12516:
12501:
12497:
12493:
12490:
12487:
12483:
12479:
12476:
12473:
12470:
12468:
12466:
12463:
12459:
12455:
12452:
12447:
12438:
12434:
12430:
12425:
12421:
12417:
12415:
12412:
12410:
12406:
12402:
12397:
12393:
12389:
12387:
12383:
12379:
12374:
12370:
12366:
12365:
12360:
12355:
12352:
12350:
12348:
12345:
12341:
12337:
12332:
12328:
12324:
12320:
12316:
12311:
12307:
12303:
12300:
12297:
12294:
12290:
12286:
12281:
12277:
12273:
12269:
12265:
12260:
12256:
12252:
12249:
12245:
12241:
12236:
12232:
12228:
12224:
12220:
12215:
12211:
12207:
12204:
12202:
12200:
12197:
12196:
12173:
12172:
12161:
12158:
12154:
12150:
12147:
12142:
12139:
12135:
12131:
12128:
12125:
12121:
12117:
12114:
12075:
12074:
12063:
12058:
12052:
12048:
12044:
12039:
12035:
12031:
12030:
12027:
12024:
12023:
12020:
12016:
12012:
12007:
12003:
11999:
11998:
11995:
11991:
11987:
11982:
11978:
11974:
11973:
11971:
11966:
11963:
11959:
11955:
11952:
11930:
11921:
11910:
11909:
11896:
11891:
11888:
11885:
11882:
11875:
11872:
11868:
11866:
11863:
11862:
11859:
11856:
11853:
11846:
11843:
11839:
11837:
11834:
11833:
11831:
11826:
11823:
11818:
11814:
11810:
11807:
11803:
11799:
11794:
11790:
11751:
11744:
11699:Consequently,
11697:
11696:
11685:
11681:
11677:
11673:
11670:
11666:
11662:
11659:
11655:
11652:
11648:
11645:
11641:
11638:
11634:
11631:
11583:
11582:
11573:
11571:
11560:
11556:
11552:
11549:
11545:
11541:
11536:
11530:
11526:
11522:
11517:
11513:
11509:
11508:
11505:
11502:
11501:
11498:
11494:
11490:
11485:
11481:
11477:
11476:
11473:
11469:
11465:
11460:
11456:
11452:
11451:
11449:
11443:
11439:
11434:
11430:
11426:
11422:
11418:
11413:
11409:
11405:
11402:
11399:
11394:
11390:
11386:
11382:
11378:
11373:
11369:
11365:
11360:
11356:
11352:
11348:
11344:
11339:
11335:
11331:
11328:
11303:
11296:
11278:
11275:
11224:
11223:
11208:
11202:
11197:
11192:
11188:
11185:
11180:
11177:
11173:
11168:
11164:
11161:
11158:
11154:
11150:
11147:
11144:
11140:
11137:
11136:
11132:
11125:
11120:
11115:
11111:
11108:
11102:
11097:
11092:
11087:
11080:
11074:
11069:
11066:
11062:
11058:
11051:
11048:
11044:
11039:
11035:
11032:
11027:
11024:
11020:
11015:
11010:
11006:
11003:
10999:
10995:
10992:
10988:
10984:
10980:
10977:
10976:
10970:
10965:
10961:
10957:
10953:
10950:
10945:
10942:
10938:
10934:
10930:
10926:
10923:
10920:
10917:
10913:
10909:
10906:
10903:
10901:
10864:
10863:
10854:
10852:
10841:
10835:
10830:
10825:
10821:
10818:
10813:
10810:
10806:
10801:
10797:
10794:
10791:
10787:
10783:
10780:
10777:
10774:
10771:
10768:
10765:
10762:
10756:
10753:
10713:
10712:
10701:
10697:
10694:
10690:
10686:
10683:
10680:
10677:
10674:
10670:
10666:
10663:
10633:
10632:
10621:
10615:
10604:
10600:
10596:
10594:
10591:
10587:
10583:
10579:
10575:
10571:
10567:
10566:
10561:
10556:
10553:
10549:
10545:
10542:
10524:
10523:
10512:
10508:
10505:
10502:
10499:
10496:
10493:
10490:
10487:
10484:
10480:
10475:
10470:
10466:
10462:
10458:
10455:
10450:
10446:
10410:
10409:
10397:
10392:
10388:
10384:
10378:
10374:
10370:
10367:
10341:
10340:
10329:
10324:
10319:
10315:
10311:
10305:
10301:
10297:
10294:
10290:
10285:
10281:
10277:
10271:
10267:
10263:
10260:
10256:
10250:
10246:
10242:
10239:
10234:
10230:
10226:
10222:
10216:
10212:
10185:
10174:
10155:
10113:
10112:
10103:
10101:
10090:
10084:
10078:
10073:
10070:
10066:
10062:
10055:
10052:
10048:
10043:
10039:
10036:
10031:
10028:
10024:
10020:
10015:
10012:
10008:
10004:
10000:
9996:
9993:
9959:inverse metric
9950:inverse matrix
9946:
9945:
9934:
9929:
9923:
9919:
9915:
9910:
9906:
9901:
9897:
9894:
9891:
9887:
9883:
9878:
9875:
9871:
9845:
9838:
9826:
9825:Inverse metric
9823:
9822:
9821:
9804:has signature
9787:
9748:
9715:has signature
9680:
9675:if the matrix
9605:between 1 and
9599:
9598:
9585:
9580:
9575:
9571:
9567:
9562:
9559:
9556:
9551:
9546:
9541:
9538:
9535:
9531:
9527:
9522:
9517:
9512:
9507:
9503:
9499:
9494:
9491:
9488:
9483:
9478:
9473:
9469:
9465:
9460:
9455:
9450:
9445:
9441:
9437:
9432:
9428:
9422:
9418:
9412:
9408:
9402:
9398:
9393:
9387:
9383:
9364:
9344:
9313:have constant
9307:
9270:
9259:
9252:
9251:
9240:
9237:
9232:
9228:
9224:
9219:
9215:
9210:
9206:
9201:
9197:
9193:
9188:
9184:
9180:
9175:
9171:
9167:
9164:
9159:
9155:
9151:
9146:
9142:
9127:quadratic form
9119:Main article:
9116:
9113:
9101:
9100:
9087:
9084:
9080:
9076:
9073:
9070:
9066:
9062:
9058:
9054:
9048:
9042:
9036:
9033:
9029:
9025:
9022:
9019:
9015:
9010:
9006:
9002:
8998:
8993:
8989:
8970:
8954:
8943:
8942:
8931:
8923:
8919:
8915:
8908:
8904:
8900:
8893:
8889:
8885:
8879:
8876:
8872:
8863:
8859:
8855:
8848:
8844:
8840:
8832:
8827:
8824:
8821:
8818:
8815:
8811:
8807:
8803:
8799:
8795:
8790:
8784:
8781:
8777:
8762:
8761:
8745:
8741:
8737:
8733:
8723:
8719:
8715:
8708:
8704:
8700:
8692:
8687:
8684:
8681:
8677:
8673:
8665:
8661:
8657:
8653:
8625:
8618:
8617:
8606:
8602:
8593:
8589:
8585:
8581:
8576:
8568:
8564:
8560:
8556:
8550:
8546:
8543:
8539:
8535:
8531:
8526:
8520:
8517:
8513:
8498:
8497:
8486:
8483:
8480:
8477:
8474:
8471:
8468:
8465:
8462:
8458:
8455:
8450:
8446:
8442:
8439:
8436:
8431:
8427:
8423:
8418:
8414:
8410:
8405:
8401:
8397:
8392:
8388:
8373:
8372:
8361:
8356:
8347:
8343:
8339:
8335:
8330:
8322:
8318:
8314:
8310:
8304:
8300:
8297:
8293:
8289:
8285:
8279:
8276:
8272:
8253:
8252:
8241:
8236:
8227:
8223:
8219:
8215:
8210:
8205:
8201:
8197:
8194:
8191:
8183:
8179:
8175:
8171:
8166:
8161:
8157:
8152:
8148:
8144:
8094:
8091:
8077:
8070:
8069:
8058:
8052:
8049:
8045:
8041:
8037:
8033:
8028:
8025:
8021:
8015:
8012:
8008:
8002:
7997:
7994:
7991:
7988:
7985:
7981:
7977:
7974:
7971:
7967:
7963:
7958:
7955:
7951:
7936:
7935:
7924:
7921:
7917:
7913:
7910:
7904:
7899:
7895:
7892:
7889:
7885:
7881:
7878:
7855:
7831:
7830:
7819:
7815:
7811:
7807:
7801:
7798:
7794:
7788:
7784:
7778:
7774:
7770:
7767:
7764:
7759:
7756:
7752:
7746:
7742:
7736:
7732:
7727:
7723:
7719:
7715:
7710:
7706:
7659:
7658:
7647:
7643:
7639:
7635:
7631:
7627:
7623:
7620:
7614:
7609:
7604:
7600:
7596:
7592:
7589:
7585:
7581:
7577:
7573:
7569:
7565:
7562:
7556:
7551:
7546:
7542:
7538:
7534:
7531:
7528:
7525:
7522:
7519:
7516:
7491:column vectors
7468:
7457:
7456:
7445:
7441:
7437:
7432:
7429:
7425:
7419:
7415:
7409:
7405:
7399:
7394:
7391:
7388:
7385:
7382:
7378:
7374:
7370:
7364:
7360:
7356:
7351:
7347:
7342:
7338:
7333:
7329:
7323:
7319:
7313:
7308:
7305:
7302:
7299:
7296:
7292:
7288:
7285:
7282:
7279:
7276:
7273:
7270:
7232:
7231:
7218:
7214:
7208:
7204:
7198:
7193:
7190:
7187:
7183:
7179:
7176:
7172:
7166:
7162:
7156:
7152:
7146:
7141:
7138:
7135:
7131:
7127:
7124:
7087:
7072:
7071:
7062:
7060:
7049:
7045:
7039:
7035:
7031:
7026:
7022:
7017:
7013:
7010:
7007:
7003:
6999:
6994:
6991:
6987:
6965:
6958:
6927:
6924:
6907:
6902:
6898:
6894:
6889:
6885:
6881:
6876:
6872:
6868:
6865:
6862:
6859:
6856:
6853:
6850:
6847:
6844:
6841:
6788:
6765:
6764:
6756:
6747:
6738:
6727:
6715:
6700:
6689:
6673:
6668:
6664:
6660:
6655:
6651:
6647:
6642:
6638:
6634:
6629:
6625:
6608:
6593:
6587:
6576:
6572:
6567:
6563:
6559:
6554:
6550:
6546:
6541:
6537:
6533:
6530:
6525:
6521:
6517:
6512:
6508:
6504:
6499:
6495:
6479:
6468:
6453:
6447:
6432:
6428:
6423:
6419:
6415:
6410:
6406:
6402:
6397:
6393:
6389:
6386:
6383:
6378:
6374:
6370:
6365:
6361:
6357:
6352:
6348:
6344:
6341:
6338:
6336:
6334:
6329:
6325:
6321:
6318:
6313:
6309:
6305:
6302:
6297:
6293:
6289:
6284:
6280:
6276:
6275:
6266:
6262:
6257:
6253:
6249:
6244:
6240:
6236:
6231:
6227:
6223:
6220:
6217:
6212:
6208:
6204:
6199:
6195:
6191:
6186:
6182:
6178:
6175:
6172:
6170:
6168:
6163:
6159:
6155:
6150:
6146:
6142:
6139:
6134:
6130:
6126:
6123:
6118:
6114:
6110:
6109:
6082:
6071:
6060:
6045:
6019:
6008:
5996:
5987:
5978:
5973:is a function
5952:
5923:
5920:
5917:
5912:
5868:
5865:
5853:
5852:
5837:
5834:
5830:
5827:
5819:
5813:
5810:
5808:
5805:
5804:
5801:
5798:
5796:
5793:
5792:
5790:
5785:
5778:
5774:
5770:
5768:
5764:
5761:
5760:
5757:
5754:
5750:
5747:
5739:
5735:
5731:
5728:
5725:
5718:
5714:
5710:
5708:
5704:
5701:
5700:
5697:
5694:
5690:
5687:
5679:
5674:
5668:
5661:
5658:
5651:
5646:
5639:
5636:
5628:
5623:
5619:
5613:
5606:
5603:
5596:
5591:
5584:
5581:
5573:
5568:
5562:
5555:
5552:
5545:
5540:
5533:
5530:
5522:
5514:
5510:
5506:
5504:
5502:
5476:
5475:
5464:
5461:
5457:
5454:
5449:
5443:
5436:
5433:
5426:
5421:
5414:
5411:
5403:
5397:
5393:
5337:
5334:
5333:
5332:
5321:
5313:
5309:
5305:
5300:
5296:
5292:
5286:
5282:
5278:
5274:
5270:
5267:
5261:
5258:
5255:
5252:
5249:
5246:
5218:and the angle
5216:
5215:
5202:
5198:
5194:
5190:
5186:
5183:
5178:
5174:
5170:
5166:
5104:
5103:
5087:
5082:
5078:
5073:
5069:
5064:
5060:
5057:
5054:
5051:
5047:
5043:
5039:
5035:
5032:
5029:
5026:
5023:
5021:
5018:
5013:
5009:
5004:
5001:
4997:
4993:
4990:
4986:
4981:
4977:
4974:
4973:
4964:
4960:
4955:
4951:
4947:
4943:
4937:
4933:
4930:
4927:
4924:
4920:
4916:
4912:
4908:
4905:
4902:
4899:
4896:
4894:
4891:
4886:
4882:
4878:
4874:
4869:
4866:
4862:
4858:
4854:
4850:
4847:
4846:
4812:
4811:
4800:
4796:
4792:
4788:
4784:
4780:
4777:
4774:
4771:
4767:
4763:
4759:
4755:
4752:
4722:
4721:
4710:
4706:
4701:
4697:
4691:
4687:
4683:
4680:
4675:
4671:
4665:
4661:
4657:
4654:
4649:
4645:
4639:
4635:
4631:
4628:
4623:
4619:
4613:
4609:
4605:
4602:
4598:
4594:
4590:
4586:
4583:
4549:
4540:
4531:
4522:
4515:
4514:
4499:
4493:
4485:
4481:
4477:
4476:
4471:
4467:
4463:
4462:
4460:
4453:
4447:
4444:
4442:
4439:
4438:
4435:
4432:
4430:
4427:
4426:
4424:
4417:
4409:
4405:
4401:
4397:
4393:
4389:
4388:
4386:
4381:
4378:
4376:
4374:
4371:
4368:
4363:
4359:
4353:
4349:
4345:
4342:
4337:
4333:
4327:
4323:
4319:
4316:
4311:
4307:
4301:
4297:
4293:
4290:
4285:
4281:
4275:
4271:
4267:
4264:
4262:
4260:
4255:
4248:
4245:
4238:
4233:
4226:
4223:
4214:
4210:
4204:
4200:
4196:
4191:
4184:
4181:
4174:
4169:
4162:
4159:
4150:
4146:
4140:
4136:
4132:
4127:
4120:
4117:
4110:
4105:
4098:
4095:
4086:
4082:
4076:
4072:
4068:
4063:
4056:
4053:
4046:
4041:
4034:
4031:
4022:
4018:
4012:
4008:
4004:
4001:
3999:
3996:
3992:
3988:
3984:
3983:
3965:
3964:
3947:
3940:
3937:
3928:
3924:
3920:
3915:
3908:
3905:
3896:
3892:
3888:
3885:
3883:
3880:
3876:
3875:
3870:
3863:
3860:
3851:
3847:
3843:
3838:
3831:
3828:
3819:
3815:
3811:
3808:
3806:
3803:
3799:
3798:
3782:
3773:
3766:
3765:
3752:
3745:
3742:
3733:
3729:
3725:
3720:
3713:
3710:
3701:
3697:
3693:
3689:
3661:
3658:
3657:
3656:
3641:
3635:
3631:
3626:
3623:
3619:
3616:
3613:
3610:
3608:
3606:
3601:
3594:
3591:
3587:
3584:
3583:
3579:
3576:
3572:
3569:
3568:
3566:
3559:
3552:
3549:
3545:
3542:
3539:
3535:
3534:
3530:
3527:
3523:
3520:
3517:
3513:
3512:
3510:
3503:
3496:
3493:
3489:
3486:
3483:
3480:
3476:
3473:
3472:
3470:
3465:
3462:
3460:
3458:
3453:
3446:
3443:
3439:
3436:
3435:
3431:
3428:
3424:
3421:
3420:
3418:
3411:
3400:
3397:
3393:
3388:
3385:
3378:
3371:
3368:
3364:
3359:
3356:
3349:
3348:
3340:
3337:
3333:
3328:
3325:
3318:
3311:
3308:
3304:
3299:
3296:
3289:
3288:
3286:
3279:
3273:
3270:
3268:
3265:
3264:
3261:
3258:
3256:
3253:
3252:
3250:
3242:
3235:
3224:
3221:
3217:
3212:
3209:
3202:
3195:
3192:
3188:
3183:
3180:
3173:
3172:
3164:
3161:
3157:
3152:
3149:
3142:
3135:
3132:
3128:
3123:
3120:
3113:
3112:
3110:
3102:
3095:
3092:
3088:
3085:
3082:
3079:
3075:
3072:
3071:
3069:
3064:
3061:
3059:
3057:
3052:
3046:
3043:
3040:
3039:
3036:
3033:
3030:
3029:
3027:
3020:
3014:
3011:
3009:
3006:
3005:
3002:
2999:
2997:
2994:
2993:
2991:
2984:
2978:
2975:
2972:
2970:
2967:
2964:
2963:
2961:
2956:
2953:
2951:
2947:
2943:
2939:
2936:
2935:
2921:
2920:
2907:
2900:
2897:
2893:
2890:
2889:
2885:
2882:
2878:
2875:
2874:
2872:
2865:
2854:
2851:
2847:
2842:
2839:
2832:
2825:
2822:
2818:
2813:
2810:
2803:
2802:
2794:
2791:
2787:
2782:
2779:
2772:
2765:
2762:
2758:
2753:
2750:
2743:
2742:
2740:
2735:
2730:
2724:
2721:
2718:
2717:
2714:
2711:
2708:
2707:
2705:
2651:
2648:
2638:
2637:
2624:
2618:
2615:
2613:
2610:
2609:
2606:
2603:
2601:
2598:
2597:
2595:
2575:
2574:
2563:
2557:
2547:
2544:
2540:
2535:
2532:
2526:
2520:
2517:
2513:
2508:
2505:
2499:
2498:
2491:
2488:
2484:
2479:
2476:
2470:
2464:
2461:
2457:
2452:
2449:
2443:
2442:
2440:
2435:
2432:
2396:
2395:
2386:
2384:
2371:
2361:
2358:
2354:
2349:
2346:
2340:
2334:
2331:
2327:
2322:
2319:
2313:
2312:
2305:
2302:
2298:
2293:
2290:
2284:
2278:
2275:
2271:
2266:
2263:
2257:
2256:
2254:
2247:
2241:
2238:
2236:
2233:
2232:
2229:
2226:
2224:
2221:
2220:
2218:
2210:
2203:
2193:
2190:
2186:
2181:
2178:
2172:
2166:
2163:
2159:
2154:
2151:
2145:
2144:
2137:
2134:
2130:
2125:
2122:
2116:
2110:
2107:
2103:
2098:
2095:
2089:
2088:
2086:
2080:
2075:
2068:
2065:
2061:
2058:
2055:
2051:
2050:
2046:
2043:
2039:
2036:
2033:
2029:
2028:
2026:
1965:
1964:
1955:
1953:
1942:
1936:
1933:
1925:
1922:
1915:
1909:
1906:
1898:
1895:
1888:
1884:
1881:
1876:
1870:
1867:
1859:
1856:
1849:
1843:
1840:
1832:
1829:
1822:
1818:
1815:
1810:
1804:
1801:
1793:
1790:
1783:
1777:
1774:
1766:
1763:
1756:
1752:
1749:
1703:
1700:
1699:
1698:
1685:
1679:
1676:
1673:
1672:
1669:
1666:
1663:
1662:
1660:
1653:
1647:
1644:
1642:
1639:
1638:
1635:
1632:
1630:
1627:
1626:
1624:
1617:
1611:
1608:
1605:
1603:
1600:
1597:
1596:
1594:
1589:
1584:
1580:
1576:
1523:principal part
1513:is called the
1489:
1488:
1479:
1477:
1466:
1461:
1454:
1451:
1444:
1439:
1432:
1429:
1422:
1419:
1415:
1410:
1403:
1400:
1393:
1388:
1381:
1378:
1371:
1368:
1364:
1359:
1352:
1349:
1342:
1337:
1330:
1327:
1320:
1317:
1299:
1298:
1289:
1287:
1276:
1271:
1267:
1263:
1260:
1257:
1253:
1250:
1247:
1244:
1240:
1237:
1233:
1230:
1227:
1222:
1218:
1214:
1211:
1208:
1204:
1201:
1196:
1192:
1188:
1185:
1182:
1159:
1158:
1147:
1140:
1137:
1129:
1126:
1120:
1114:
1109:
1102:
1099:
1091:
1084:
1081:
1073:
1070:
1064:
1058:
1053:
1046:
1043:
1017:Euclidean norm
1003:
1000:
997:
985:
984:
969:
965:
962:
954:
947:
944:
937:
932:
925:
922:
912:
908:
904:
901:
897:
894:
890:
885:
878:
875:
868:
863:
856:
853:
845:
842:
839:
835:
832:
828:
825:
822:
818:
815:
811:
808:
803:
796:
793:
786:
781:
774:
771:
761:
757:
753:
750:
746:
743:
735:
730:
726:
722:
719:
717:
715:
712:
709:
704:
700:
697:
694:
691:
688:
685:
682:
679:
676:
673:
670:
664:
661:
652:
649:
645:
639:
633:
628:
624:
620:
617:
615:
613:
610:
609:
535:
532:
517:
511:
508:
506:
503:
502:
499:
496:
494:
491:
490:
488:
430:
429:
416:
411:
408:
404:
401:
398:
395:
391:
388:
385:
381:
378:
375:
372:
368:
365:
362:
358:
355:
352:
349:
344:
339:
336:
333:
329:
326:
323:
317:
314:
271:parametrically
253:
250:
15:
9:
6:
4:
3:
2:
22787:
22776:
22773:
22771:
22768:
22766:
22763:
22761:
22758:
22756:
22753:
22752:
22750:
22735:
22732:
22730:
22729:Supermanifold
22727:
22725:
22722:
22720:
22717:
22713:
22710:
22709:
22708:
22705:
22703:
22700:
22698:
22695:
22693:
22690:
22688:
22685:
22683:
22680:
22678:
22675:
22674:
22672:
22668:
22662:
22659:
22657:
22654:
22652:
22649:
22647:
22644:
22642:
22639:
22637:
22634:
22633:
22631:
22627:
22617:
22614:
22612:
22609:
22607:
22604:
22602:
22599:
22597:
22594:
22592:
22589:
22587:
22584:
22582:
22579:
22577:
22574:
22572:
22569:
22568:
22566:
22564:
22560:
22554:
22551:
22549:
22546:
22544:
22541:
22539:
22536:
22534:
22531:
22529:
22526:
22524:
22520:
22516:
22514:
22511:
22509:
22506:
22504:
22500:
22496:
22494:
22491:
22489:
22486:
22484:
22481:
22479:
22476:
22474:
22471:
22469:
22466:
22465:
22463:
22461:
22457:
22451:
22450:Wedge product
22448:
22446:
22443:
22439:
22436:
22435:
22434:
22431:
22429:
22426:
22422:
22419:
22418:
22417:
22414:
22412:
22409:
22407:
22404:
22402:
22399:
22395:
22394:Vector-valued
22392:
22391:
22390:
22387:
22385:
22382:
22378:
22375:
22374:
22373:
22370:
22368:
22365:
22363:
22360:
22359:
22357:
22353:
22347:
22344:
22342:
22339:
22337:
22334:
22330:
22327:
22326:
22325:
22324:Tangent space
22322:
22320:
22317:
22315:
22312:
22310:
22307:
22306:
22304:
22300:
22297:
22295:
22291:
22285:
22282:
22280:
22276:
22272:
22270:
22267:
22265:
22261:
22257:
22253:
22251:
22248:
22246:
22243:
22241:
22238:
22236:
22233:
22231:
22228:
22226:
22223:
22221:
22218:
22214:
22211:
22210:
22209:
22206:
22204:
22201:
22199:
22196:
22194:
22191:
22189:
22186:
22184:
22181:
22179:
22176:
22174:
22171:
22169:
22166:
22164:
22161:
22159:
22155:
22151:
22149:
22145:
22141:
22139:
22136:
22135:
22133:
22127:
22121:
22118:
22116:
22113:
22111:
22108:
22106:
22103:
22101:
22098:
22096:
22093:
22089:
22088:in Lie theory
22086:
22085:
22084:
22081:
22079:
22076:
22072:
22069:
22068:
22067:
22064:
22062:
22059:
22058:
22056:
22054:
22050:
22044:
22041:
22039:
22036:
22034:
22031:
22029:
22026:
22024:
22021:
22019:
22016:
22014:
22011:
22009:
22006:
22004:
22001:
22000:
21998:
21995:
21991:Main results
21989:
21983:
21980:
21978:
21975:
21973:
21972:Tangent space
21970:
21968:
21965:
21963:
21960:
21958:
21955:
21953:
21950:
21948:
21945:
21941:
21938:
21936:
21933:
21932:
21931:
21928:
21924:
21921:
21920:
21919:
21916:
21915:
21913:
21909:
21904:
21900:
21893:
21888:
21886:
21881:
21879:
21874:
21873:
21870:
21858:
21855:
21853:
21850:
21848:
21845:
21843:
21840:
21838:
21835:
21833:
21830:
21828:
21825:
21823:
21820:
21818:
21815:
21813:
21810:
21808:
21805:
21803:
21800:
21798:
21795:
21794:
21792:
21790:
21786:
21776:
21773:
21771:
21768:
21766:
21763:
21761:
21758:
21756:
21753:
21751:
21748:
21746:
21743:
21741:
21738:
21736:
21733:
21732:
21730:
21726:
21720:
21717:
21715:
21712:
21710:
21707:
21705:
21702:
21700:
21697:
21695:
21694:Metric tensor
21692:
21690:
21687:
21685:
21682:
21681:
21679:
21675:
21672:
21668:
21662:
21659:
21657:
21654:
21652:
21649:
21647:
21644:
21642:
21639:
21637:
21634:
21632:
21629:
21627:
21624:
21622:
21619:
21617:
21614:
21612:
21609:
21607:
21606:Exterior form
21604:
21602:
21599:
21597:
21594:
21592:
21589:
21587:
21584:
21582:
21579:
21577:
21574:
21572:
21569:
21568:
21566:
21560:
21553:
21550:
21548:
21545:
21543:
21540:
21538:
21535:
21533:
21530:
21528:
21525:
21523:
21520:
21518:
21515:
21513:
21510:
21508:
21505:
21503:
21500:
21499:
21497:
21495:
21491:
21485:
21482:
21480:
21479:Tensor bundle
21477:
21475:
21472:
21470:
21467:
21465:
21462:
21460:
21457:
21455:
21452:
21450:
21447:
21445:
21442:
21440:
21437:
21436:
21434:
21428:
21422:
21419:
21417:
21414:
21412:
21409:
21407:
21404:
21402:
21399:
21397:
21394:
21392:
21389:
21387:
21384:
21382:
21379:
21378:
21376:
21372:
21362:
21359:
21357:
21354:
21352:
21349:
21347:
21344:
21342:
21339:
21338:
21336:
21331:
21328:
21326:
21323:
21322:
21319:
21313:
21310:
21308:
21305:
21303:
21300:
21298:
21295:
21293:
21290:
21288:
21285:
21283:
21280:
21278:
21275:
21274:
21272:
21270:
21266:
21263:
21259:
21255:
21254:
21248:
21244:
21237:
21232:
21230:
21225:
21223:
21218:
21217:
21214:
21202:
21199:
21197:
21194:
21192:
21189:
21187:
21184:
21183:
21181:
21177:
21171:
21168:
21166:
21163:
21161:
21158:
21157:
21155:
21151:
21145:
21144:Schur's lemma
21142:
21140:
21137:
21135:
21132:
21130:
21127:
21125:
21122:
21120:
21119:Gauss's lemma
21117:
21115:
21112:
21111:
21109:
21105:
21099:
21096:
21094:
21091:
21089:
21086:
21084:
21081:
21080:
21078:
21074:
21068:
21065:
21063:
21060:
21058:
21055:
21053:
21050:
21046:
21042:
21039:
21037:
21034:
21032:
21029:
21027:
21024:
21023:
21022:
21021:Metric tensor
21019:
21017:
21016:Inner product
21014:
21012:
21009:
21007:
21004:
21000:
20997:
20995:
20992:
20990:
20987:
20985:
20982:
20981:
20980:
20977:
20976:
20974:
20970:
20965:
20961:
20954:
20949:
20947:
20942:
20940:
20935:
20934:
20931:
20923:
20922:
20917:
20913:
20910:
20906:
20902:
20896:
20892:
20888:
20881:
20880:
20874:
20871:
20869:0-8218-1385-4
20865:
20861:
20857:
20856:Sternberg, S.
20853:
20850:
20846:
20842:
20838:
20834:
20830:
20826:
20822:
20818:
20814:
20810:
20806:
20803:
20801:0-7167-0344-0
20797:
20793:
20789:
20785:
20781:
20777:
20774:
20768:
20764:
20759:
20755:
20749:
20745:
20740:
20735:
20731:
20730:Kline, Morris
20727:
20723:
20717:
20713:
20708:
20703:
20699:
20698:Hawking, S.W.
20695:
20692:
20688:
20684:
20678:
20677:
20672:
20668:
20664:
20658:
20654:
20650:
20645:
20642:
20638:
20634:
20628:
20624:
20620:
20616:
20611:
20610:
20599:
20594:
20587:
20583:
20577:
20570:
20565:
20558:
20553:
20546:
20540:
20533:
20527:
20520:
20514:
20510:
20499:
20496:
20494:
20491:
20489:
20486:
20484:
20481:
20479:
20476:
20474:
20471:
20469:
20466:
20464:
20461:
20460:
20454:
20452:
20444:
20421:
20415:
20409:
20406:
20401:
20397:
20391:
20387:
20383:
20378:
20373:
20368:
20361:
20354:
20350:
20346:
20341:
20336:
20329:
20324:
20317:
20314:
20309:
20300:
20296:
20292:
20287:
20284:
20281:
20275:
20272:
20268:
20263:
20258:
20251:
20246:
20241:
20235:
20226:
20222:
20218:
20213:
20210:
20207:
20201:
20198:
20194:
20187:
20182:
20177:
20174:
20170:
20162:
20161:
20160:
20143:
20136:
20133:
20130:
20127:
20124:
20121:
20118:
20115:
20109:
20105:
20099:
20095:
20091:
20086:
20082:
20078:
20073:
20069:
20065:
20060:
20056:
20051:
20043:
20042:
20041:
20039:
20035:
20016:
20010:
20006:
20002:
19997:
19993:
19989:
19984:
19981:
19977:
19973:
19968:
19964:
19960:
19955:
19951:
19947:
19944:
19939:
19935:
19931:
19928:
19923:
19919:
19915:
19912:
19907:
19903:
19899:
19896:
19891:
19887:
19883:
19878:
19874:
19870:
19865:
19861:
19857:
19850:
19849:
19848:
19846:
19841:
19839:
19835:
19816:
19810:
19804:
19799:
19794:
19789:
19782:
19777:
19772:
19767:
19760:
19755:
19750:
19745:
19738:
19733:
19728:
19723:
19720:
19714:
19709:
19706:
19694:
19688:
19685:
19680:
19675:
19670:
19663:
19658:
19655:
19650:
19645:
19638:
19633:
19628:
19625:
19620:
19613:
19608:
19603:
19598:
19592:
19587:
19584:
19577:
19576:
19575:
19573:
19554:
19547:
19544:
19541:
19538:
19535:
19532:
19529:
19526:
19520:
19516:
19510:
19506:
19502:
19497:
19493:
19489:
19484:
19480:
19476:
19471:
19467:
19462:
19453:
19449:
19441:
19440:
19439:
19437:
19433:
19427:
19403:
19397:
19393:
19389:
19385:
19382:
19377:
19373:
19369:
19364:
19360:
19356:
19353:
19348:
19344:
19340:
19333:
19332:
19331:
19314:
19308:
19302:
19299:
19294:
19290:
19284:
19277:
19272:
19266:
19261:
19258:
19251:
19250:
19249:
19244:-axis in the
19231:
19226:
19219:
19215:
19209:
19204:
19179:
19171:
19167:
19156:
19152:
19137:
19133:
19122:
19118:
19106:
19102:
19098:
19090:
19086:
19075:
19071:
19056:
19052:
19041:
19037:
19025:
19022:
19018:
19012:
19009:
19005:
19001:
18996:
18993:
18989:
18981:
18980:
18979:
18976:
18970:
18965:
18961:
18956:
18950:
18945:
18941:
18936:
18934:
18913:
18905:
18901:
18895:
18888:
18883:
18877:
18872:
18867:
18861:
18858:
18853:
18849:
18843:
18839:
18835:
18832:
18829:
18824:
18820:
18814:
18810:
18804:
18801:
18798:
18795:
18792:
18789:
18786:
18783:
18780:
18777:
18774:
18771:
18768:
18765:
18762:
18759:
18752:
18749:
18746:
18743:
18740:
18737:
18734:
18731:
18728:
18725:
18722:
18719:
18716:
18713:
18710:
18707:
18702:
18699:
18694:
18690:
18686:
18683:
18680:
18675:
18671:
18664:
18659:
18656:
18645:
18641:
18638:
18631:
18630:
18629:
18608:
18602:
18596:
18593:
18590:
18587:
18582:
18579:
18576:
18569:
18566:
18563:
18560:
18557:
18552:
18549:
18546:
18540:
18535:
18533:
18528:
18521:
18518:
18515:
18512:
18509:
18507:
18502:
18495:
18492:
18489:
18486:
18483:
18481:
18476:
18465:
18464:
18463:
18459:
18455:
18450:
18446:
18429:
18421:
18413:
18410:
18404:
18399:
18391:
18388:
18378:
18373:
18369:
18365:
18362:
18355:
18354:
18353:
18336:
18330:
18324:
18319:
18312:
18307:
18301:
18296:
18293:
18286:
18285:
18284:
18280:
18276:
18271:
18267:
18263:
18248:
18246:
18242:
18234:
18229:
18210:
18206:
18202:
18199:
18196:
18193:
18188:
18184:
18180:
18173:
18169:
18162:
18156:
18153:
18143:
18139:
18134:
18130:
18126:
18117:
18115:
18109:
18102:
18097:
18089:
18084:
18075:supported in
18058:
18055:
18048:
18044:
18037:
18028:
18020:
18017:
18013:
18009:
18006:
17998:
17992:
17988:
17984:
17979:
17975:
17971:
17967:
17962:
17958:
17954:
17951:
17938:
17934:
17929:
17923:
17919:
17915:
17914:Borel measure
17899:
17896:
17890:
17883:
17879:on the space
17875:
17871:
17866:
17864:
17860:
17859:Borel measure
17856:
17853:-dimensional
17834:
17832:
17828:
17824:
17818:
17816:
17812:
17804:
17800:
17781:
17777:
17774:
17769:
17762:
17756:
17753:
17748:
17744:
17737:
17734:
17730:
17724:
17719:
17712:
17706:
17703:
17698:
17694:
17687:
17684:
17680:
17674:
17664:
17658:
17650:
17647:
17643:
17637:
17632:
17629:
17626:
17623:
17620:
17616:
17610:
17605:
17601:
17595:
17592:
17587:
17584:
17577:
17576:
17575:
17573:
17563:
17546:
17542:
17539:
17532:
17527:
17520:
17514:
17511:
17506:
17502:
17495:
17492:
17488:
17482:
17477:
17470:
17464:
17461:
17456:
17452:
17445:
17442:
17438:
17432:
17422:
17416:
17408:
17405:
17401:
17395:
17390:
17387:
17384:
17381:
17378:
17374:
17369:
17361:
17356:
17352:
17348:
17345:
17338:
17337:
17336:
17332:
17326:
17321:
17316:
17308:
17287:
17283:
17279:
17274:
17270:
17266:
17260:
17252:
17249:
17245:
17239:
17234:
17231:
17228:
17225:
17222:
17218:
17214:
17209:
17205:
17201:
17194:
17193:
17192:
17191:
17188:
17169:
17165:
17162:
17155:
17148:
17142:
17139:
17134:
17130:
17123:
17120:
17116:
17110:
17105:
17098:
17092:
17089:
17084:
17080:
17073:
17070:
17066:
17060:
17050:
17044:
17036:
17033:
17029:
17023:
17018:
17015:
17012:
17009:
17006:
17002:
16994:
16989:
16985:
16981:
16978:
16971:
16970:
16969:
16967:
16962:
16958:
16954:
16945:
16939:
16935:
16929:
16925:
16921:
16915:
16910:
16905:
16901:
16895:
16883:Suppose that
16862:
16859:
16851:
16848:
16836:
16835:
16834:
16832:
16805:
16800:
16790:
16787:
16782:
16768:
16767:
16766:
16764:
16736:
16732:
16729:
16726:
16721:
16718:
16713:
16709:
16704:
16700:
16696:
16692:
16689:
16686:
16681:
16678:
16673:
16669:
16664:
16656:
16655:
16654:
16637:
16626:
16621:
16611:
16606:
16603:
16598:
16594:
16586:
16585:
16584:
16578:
16571:
16565:
16562:
16558:
16553:is a mapping
16550:
16546:
16531:
16527:
16514:
16508:
16504:
16496:
16492:
16478:
16475:
16469:
16458:
16454:
16446:
16442:
16416:
16407:
16403:
16399:
16394:
16390:
16386:
16380:
16372:
16368:
16364:
16359:
16355:
16346:
16342:
16334:
16333:
16332:
16329:
16325:
16317:
16311:
16303:
16299:
16294:
16288:
16284:
16280:
16273:
16269:
16244:
16240:
16236:
16231:
16227:
16221:
16217:
16210:
16202:
16198:
16194:
16189:
16185:
16179:
16175:
16164:
16163:
16162:
16160:
16154:
16150:
16145:
16139:
16135:
16130:
16124:
16120:
16115:
16110:
16104:
16093:
16089:
16082:
16076:
16072:
16065:
16061:
16054:
16050:
16025:
16021:
16017:
16012:
16008:
15999:
15995:
15991:
15983:
15979:
15975:
15970:
15966:
15960:
15956:
15945:
15944:
15943:
15940:
15934:
15925:
15920:
15916:
15906:
15902:
15897:
15893:
15888:
15884:
15873:
15869:
15863:
15859:
15852:
15833:
15827:
15824:
15819:
15815:
15808:
15795:
15785:
15781:
15775:
15771:
15763:
15762:
15761:
15758:
15754:
15748:
15744:
15739:
15735:
15731:
15727:
15721:
15711:
15707:
15703:
15698:
15694:
15675:
15662:
15658:
15654:
15649:
15645:
15641:
15636:
15632:
15624:
15623:
15622:
15619:
15610:
15609:fiber product
15583:
15578:
15574:
15570:
15567:
15564:
15557:
15556:
15555:
15545:
15544:vector bundle
15539:
15529:
15527:
15519:
15509:
15504:
15499:
15495:
15485:
15465:
15462:
15457:
15447:
15444:
15439:
15429:
15424:
15416:
15408:
15405:
15390:
15389:
15388:
15387:
15378:
15376:
15357:
15354:
15351:
15348:
15345:
15338:
15326:
15318:
15315:
15307:
15304:
15297:
15296:
15295:
15276:
15272:
15268:
15265:
15262:
15257:
15253:
15245:
15244:
15243:
15236:
15232:
15208:
15205:
15202:
15196:
15193:
15187:
15184:
15181:
15173:
15169:
15161:
15160:
15159:
15156:
15152:
15142:
15122:
15118:
15112:
15104:
15096:
15093:
15081:
15074:
15069:
15065:
15057:
15056:
15055:
15052:
15046:
15042:
15034:
15030:
15026:
15023:) gives rise
15022:
15021:
15016:
15006:
15000:
14999:real analytic
14996:
14992:
14988:
14984:
14983:
14979:The mapping (
14963:
14952:
14947:
14937:
14934:
14929:
14919:
14914:
14910:
14902:
14901:
14900:
14893:
14884:
14880:
14879:fiber product
14871:
14864:
14862:
14840:
14830:
14826:
14822:
14814:
14811:
14804:
14803:
14800:
14798:
14797:metric tensor
14794:
14790:
14789:fiber bundles
14766:
14761:
14750:
14746:
14735:
14731:
14719:
14709:
14705:
14694:
14690:
14673:
14669:
14658:
14654:
14640:
14635:
14630:
14625:
14613:
14609:
14598:
14594:
14582:
14572:
14568:
14557:
14553:
14536:
14532:
14521:
14517:
14498:
14494:
14483:
14479:
14467:
14457:
14453:
14442:
14438:
14421:
14417:
14406:
14402:
14389:
14384:
14381:
14378:
14371:
14370:
14369:
14345:
14342:
14324:
14321:
14315:
14301:
14294:
14293:
14292:
14289:
14269:
14263:
14255:
14251:
14247:
14241:
14233:
14229:
14225:
14219:
14216:
14213:
14207:
14200:
14199:
14198:
14180:
14161:
14155:
14140:
14136:
14125:
14121:
14109:
14105:
14099:
14094:
14091:
14088:
14084:
14078:
14073:
14070:
14067:
14063:
14059:
14053:
14045:
14041:
14033:
14032:
14031:
14022:, the vector
14012:
14005:
14001:
13979:
13967:
13963:
13959:
13956:
13953:
13948:
13936:
13932:
13928:
13925:
13918:
13917:
13916:
13907:Suppose that
13905:
13903:
13894:
13889:
13884:
13880:
13872:Suppose that
13870:
13868:
13863:
13856:
13851:
13848:is called an
13839:
13835:
13832:is called an
13826:
13822:
13816:
13812:
13804:
13795:
13790:
13762:
13746:
13742:
13725:
13722:
13718:
13712:
13707:
13704:
13701:
13697:
13693:
13677:
13673:
13665:
13664:
13663:
13661:
13660:
13655:
13650:
13644:
13639:
13638:
13628:
13622:
13619:
13615:
13595:
13581:
13576:
13573:
13569:
13565:
13559:
13548:
13541:
13540:
13539:
13532:
13525:
13523:
13489:
13473:
13470:
13466:
13462:
13448:
13441:
13440:
13437:
13434:
13426:
13422:
13415:
13411:
13406:
13401:
13399:
13367:
13364:
13360:
13343:
13339:
13333:
13328:
13325:
13322:
13318:
13314:
13298:
13294:
13286:
13285:
13284:
13279:
13275:
13268:
13264:
13257:
13253:
13249:
13239:
13236:
13232:
13212:
13198:
13175:
13172:
13169:
13155:
13144:
13131:
13126:
13123:
13119:
13115:
13090:
13087:
13081:
13070:
13055:
13044:
13037:
13036:
13035:
13032:
13029:
13025:
13005:
12991:
12968:
12965:
12951:
12944:
12943:
12942:
12938:
12934:
12928:
12918:
12910:
12906:
12901:
12897:
12886:
12882:
12873:
12867:
12863:
12838:
12834:
12830:
12825:
12821:
12812:
12808:
12799:
12795:
12791:
12785:
12782:
12777:
12773:
12764:
12760:
12752:
12751:
12750:
12746:
12742:
12736:
12732:
12728:
12722:
12713:
12704:
12701:
12697:
12691:
12688:
12684:
12653:
12639:
12636:
12630:
12619:
12613:
12602:
12599:
12596:
12589:
12588:
12587:
12585:
12584:
12578:
12575:
12569:
12562:
12558:
12550:
12546:
12541:
12536:
12526:
12519:
12517:
12488:
12474:
12471:
12469:
12450:
12423:
12419:
12413:
12395:
12391:
12372:
12368:
12353:
12351:
12330:
12326:
12309:
12305:
12301:
12298:
12295:
12279:
12275:
12258:
12254:
12250:
12234:
12230:
12213:
12209:
12205:
12203:
12198:
12187:
12186:
12183:
12159:
12145:
12140:
12137:
12133:
12129:
12123:
12112:
12105:
12104:
12103:
12100:
12094:
12088:
12085:
12081:
12061:
12056:
12037:
12033:
12025:
12005:
12001:
11980:
11976:
11969:
11964:
11950:
11943:
11942:
11941:
11939:
11933:
11929:
11924:
11920:
11916:
11889:
11886:
11883:
11880:
11864:
11857:
11854:
11851:
11835:
11829:
11824:
11816:
11812:
11792:
11788:
11780:
11779:
11778:
11774:
11770:
11765:
11761:
11754:
11750:
11743:
11739:
11733:
11730:
11724:
11715:
11710:
11707:
11703:
11683:
11668:
11660:
11643:
11632:
11629:
11622:
11621:
11620:
11618:
11617:
11611:
11601:
11595:
11591:
11581:
11574:
11572:
11547:
11539:
11534:
11515:
11511:
11503:
11483:
11479:
11458:
11454:
11447:
11437:
11432:
11428:
11411:
11407:
11403:
11400:
11397:
11392:
11388:
11371:
11367:
11363:
11358:
11354:
11337:
11333:
11329:
11326:
11319:
11318:
11315:
11306:
11302:
11295:
11291:
11284:
11274:
11272:
11271:
11257:
11251:
11245:
11242:
11236:
11231:
11230:
11206:
11183:
11178:
11175:
11159:
11145:
11138:
11130:
11106:
11095:
11090:
11085:
11072:
11067:
11064:
11060:
11056:
11049:
11046:
11030:
11025:
11022:
11018:
11013:
11008:
11004:
10990:
10986:
10978:
10959:
10948:
10943:
10940:
10932:
10921:
10915:
10904:
10892:
10891:
10890:
10887:
10884:
10878:
10872:
10862:
10855:
10853:
10839:
10816:
10811:
10808:
10792:
10778:
10775:
10769:
10766:
10763:
10751:
10741:
10740:
10737:
10726:
10724:
10719:
10699:
10695:
10681:
10678:
10672:
10661:
10654:
10653:
10652:
10649:
10639:
10619:
10602:
10598:
10592:
10585:
10581:
10573:
10569:
10554:
10540:
10533:
10532:
10531:
10529:
10510:
10506:
10503:
10500:
10497:
10494:
10491:
10488:
10485:
10482:
10478:
10473:
10468:
10464:
10460:
10456:
10453:
10448:
10444:
10436:
10435:
10434:
10431:
10421:
10395:
10390:
10386:
10382:
10376:
10372:
10365:
10358:
10357:
10356:
10354:
10327:
10322:
10317:
10313:
10309:
10303:
10299:
10295:
10292:
10288:
10283:
10279:
10275:
10269:
10265:
10261:
10258:
10254:
10248:
10244:
10240:
10237:
10232:
10228:
10224:
10220:
10214:
10210:
10202:
10201:
10200:
10188:
10184:
10177:
10173:
10168:
10158:
10154:
10136:
10134:
10130:
10122:
10121:
10111:
10104:
10102:
10088:
10076:
10071:
10068:
10064:
10060:
10053:
10050:
10034:
10029:
10026:
10022:
10018:
10013:
10010:
10002:
9991:
9984:
9983:
9980:
9973:
9968:
9964:
9960:
9955:
9951:
9932:
9927:
9921:
9917:
9913:
9908:
9904:
9899:
9895:
9892:
9876:
9873:
9869:
9861:
9860:
9859:
9856:
9848:
9844:
9837:
9833:
9819:
9813:
9809:
9798:
9788:
9783:
9775:
9765:
9749:
9746:
9735:. Otherwise,
9734:
9720:
9710:
9709:
9708:
9705:
9703:
9698:
9694:
9690:positive and
9683:
9679:
9672:
9668:
9664:
9657:
9653:
9642:
9638:
9634:
9583:
9578:
9573:
9569:
9565:
9560:
9557:
9554:
9549:
9544:
9539:
9536:
9533:
9529:
9525:
9520:
9515:
9510:
9505:
9501:
9497:
9492:
9489:
9486:
9481:
9476:
9471:
9467:
9463:
9458:
9453:
9448:
9443:
9439:
9435:
9430:
9426:
9420:
9416:
9410:
9406:
9400:
9396:
9391:
9385:
9381:
9373:
9372:
9371:
9367:
9363:
9358:
9353:
9340:
9332:
9316:
9310:
9306:
9301:
9292:
9288:
9279:
9273:
9269:
9262:
9258:
9238:
9235:
9230:
9226:
9222:
9217:
9213:
9208:
9199:
9195:
9191:
9186:
9182:
9173:
9169:
9165:
9157:
9153:
9144:
9140:
9132:
9131:
9130:
9128:
9122:
9112:
9110:
9085:
9082:
9074:
9071:
9064:
9056:
9052:
9040:
9034:
9031:
9023:
9020:
9013:
9008:
9004:
9000:
8991:
8987:
8980:
8979:
8978:
8973:
8969:
8965:
8957:
8953:
8949:
8929:
8921:
8917:
8906:
8902:
8891:
8883:
8877:
8874:
8870:
8861:
8857:
8846:
8842:
8830:
8825:
8822:
8819:
8816:
8813:
8809:
8805:
8801:
8797:
8788:
8782:
8779:
8775:
8767:
8766:
8765:
8743:
8739:
8721:
8717:
8706:
8702:
8690:
8685:
8682:
8679:
8675:
8671:
8663:
8659:
8642:
8641:
8640:
8639:
8633:
8628:
8624:
8604:
8600:
8591:
8587:
8574:
8566:
8562:
8548:
8544:
8541:
8537:
8533:
8524:
8518:
8515:
8511:
8503:
8502:
8501:
8484:
8481:
8478:
8475:
8472:
8469:
8466:
8463:
8460:
8456:
8448:
8444:
8440:
8437:
8434:
8429:
8425:
8421:
8416:
8412:
8403:
8399:
8395:
8390:
8386:
8378:
8377:
8376:
8359:
8354:
8345:
8341:
8328:
8320:
8316:
8302:
8298:
8295:
8291:
8283:
8277:
8274:
8270:
8262:
8261:
8260:
8239:
8234:
8225:
8221:
8208:
8203:
8199:
8195:
8192:
8189:
8181:
8177:
8164:
8159:
8155:
8150:
8146:
8134:
8133:
8132:
8120:
8116:
8110:
8106:
8090:
8087:
8080:
8076:
8056:
8050:
8047:
8043:
8026:
8023:
8019:
8013:
8010:
8006:
8000:
7995:
7992:
7989:
7986:
7983:
7979:
7975:
7969:
7956:
7953:
7949:
7941:
7940:
7939:
7922:
7908:
7897:
7893:
7887:
7876:
7869:
7868:
7867:
7858:
7854:
7850:
7844:
7840:
7836:
7817:
7809:
7805:
7799:
7796:
7792:
7786:
7782:
7776:
7772:
7768:
7765:
7762:
7757:
7754:
7750:
7744:
7740:
7734:
7730:
7725:
7721:
7717:
7696:
7695:
7694:
7692:
7687:
7681:
7676:
7671:
7665:
7618:
7590:
7560:
7532:
7526:
7523:
7520:
7514:
7507:
7506:
7505:
7502:
7496:
7492:
7479:
7471:
7467:
7462:
7459:Denoting the
7430:
7427:
7423:
7417:
7413:
7407:
7403:
7397:
7392:
7389:
7386:
7383:
7380:
7376:
7372:
7368:
7362:
7358:
7354:
7349:
7345:
7340:
7336:
7331:
7327:
7321:
7317:
7311:
7306:
7303:
7300:
7297:
7294:
7290:
7286:
7280:
7277:
7274:
7268:
7261:
7260:
7259:
7257:
7256:
7242:
7238:
7216:
7212:
7206:
7202:
7196:
7191:
7188:
7185:
7181:
7177:
7174:
7170:
7164:
7160:
7154:
7150:
7144:
7139:
7136:
7133:
7129:
7125:
7122:
7115:
7114:
7113:
7110:
7105:
7101:
7097:
7090:
7086:
7080:
7070:
7063:
7061:
7047:
7043:
7037:
7033:
7029:
7024:
7020:
7015:
7011:
7008:
6992:
6989:
6985:
6977:
6976:
6973:
6972:are given by
6968:
6964:
6957:
6953:
6948:
6944:
6943:vector fields
6940:
6933:
6923:
6900:
6896:
6892:
6887:
6883:
6874:
6870:
6866:
6860:
6851:
6848:
6845:
6839:
6820:
6819:vector fields
6809:
6801:
6791:
6787:
6759:
6755:
6750:
6746:
6741:
6737:
6730:
6726:
6718:
6714:
6709:
6703:
6699:
6692:
6688:
6666:
6662:
6658:
6653:
6649:
6640:
6636:
6627:
6623:
6611:
6607:
6602:
6601:nondegenerate
6596:
6592:
6588:
6574:
6565:
6561:
6557:
6552:
6548:
6539:
6535:
6531:
6523:
6519:
6515:
6510:
6506:
6497:
6493:
6482:
6478:
6471:
6467:
6462:
6456:
6452:
6448:
6430:
6421:
6417:
6413:
6408:
6404:
6395:
6391:
6387:
6384:
6376:
6372:
6368:
6363:
6359:
6350:
6346:
6342:
6339:
6337:
6327:
6323:
6319:
6316:
6311:
6307:
6303:
6300:
6295:
6291:
6282:
6278:
6264:
6255:
6251:
6247:
6242:
6238:
6229:
6225:
6221:
6218:
6210:
6206:
6202:
6197:
6193:
6184:
6180:
6176:
6173:
6171:
6161:
6157:
6153:
6148:
6144:
6140:
6137:
6132:
6128:
6124:
6116:
6112:
6085:
6081:
6074:
6070:
6063:
6059:
6054:
6048:
6044:
6040:
6039:
6038:
6036:
6032:
6022:
6018:
6011:
6007:
5999:
5995:
5990:
5986:
5981:
5977:
5964:
5963:tangent space
5961:, called the
5959:
5955:
5949:
5944:
5940:
5921:
5918:
5915:
5901:
5897:
5891:
5887:(in the case
5886:
5879:of dimension
5878:
5864:
5862:
5835:
5832:
5828:
5825:
5817:
5811:
5806:
5799:
5794:
5788:
5776:
5772:
5769:
5762:
5755:
5752:
5748:
5745:
5737:
5733:
5729:
5726:
5723:
5716:
5712:
5709:
5702:
5695:
5692:
5688:
5685:
5677:
5672:
5666:
5656:
5649:
5644:
5634:
5626:
5621:
5617:
5611:
5601:
5594:
5589:
5579:
5571:
5566:
5560:
5550:
5543:
5538:
5528:
5520:
5512:
5508:
5505:
5493:
5492:
5491:
5489:
5485:
5484:cross product
5462:
5459:
5455:
5452:
5447:
5441:
5431:
5424:
5419:
5409:
5401:
5395:
5391:
5383:
5382:
5381:
5365:
5361:
5352:
5343:
5319:
5276:
5265:
5259:
5253:
5247:
5244:
5237:
5236:
5235:
5232:
5226:
5192:
5181:
5176:
5156:
5155:
5154:
5151:
5145:
5129:
5123:
5116:
5110:
5085:
5080:
5071:
5062:
5058:
5055:
5052:
5041:
5030:
5027:
5024:
5022:
5016:
5011:
5002:
4999:
4991:
4988:
4979:
4975:
4962:
4958:
4949:
4945:
4935:
4931:
4928:
4925:
4914:
4903:
4900:
4897:
4895:
4889:
4880:
4876:
4867:
4864:
4856:
4852:
4848:
4837:
4836:
4835:
4832:
4826:
4821:
4817:
4798:
4786:
4775:
4772:
4761:
4750:
4743:
4742:
4741:
4738:
4732:
4727:
4708:
4704:
4699:
4695:
4689:
4685:
4681:
4678:
4673:
4669:
4663:
4659:
4655:
4652:
4647:
4643:
4637:
4633:
4629:
4626:
4621:
4617:
4611:
4607:
4603:
4592:
4581:
4574:
4573:
4572:
4564:
4557:
4548:
4539:
4530:
4521:
4497:
4491:
4483:
4479:
4469:
4465:
4458:
4451:
4445:
4440:
4433:
4428:
4422:
4415:
4407:
4403:
4395:
4391:
4384:
4379:
4377:
4369:
4366:
4361:
4357:
4351:
4347:
4343:
4340:
4335:
4331:
4325:
4321:
4317:
4314:
4309:
4305:
4299:
4295:
4291:
4288:
4283:
4279:
4273:
4269:
4265:
4263:
4253:
4243:
4236:
4231:
4221:
4212:
4208:
4202:
4198:
4194:
4189:
4179:
4172:
4167:
4157:
4148:
4144:
4138:
4134:
4130:
4125:
4115:
4108:
4103:
4093:
4084:
4080:
4074:
4070:
4066:
4061:
4051:
4044:
4039:
4029:
4020:
4016:
4010:
4006:
4002:
4000:
3990:
3974:
3973:
3972:
3970:
3945:
3935:
3926:
3922:
3918:
3913:
3903:
3894:
3890:
3886:
3884:
3868:
3858:
3849:
3845:
3841:
3836:
3826:
3817:
3813:
3809:
3807:
3789:
3788:
3787:
3781:
3772:
3750:
3740:
3731:
3727:
3723:
3718:
3708:
3699:
3695:
3691:
3679:
3678:
3677:
3671:
3667:
3639:
3633:
3624:
3621:
3617:
3611:
3609:
3599:
3592:
3589:
3585:
3577:
3574:
3570:
3564:
3557:
3550:
3547:
3540:
3537:
3528:
3525:
3518:
3515:
3508:
3501:
3494:
3491:
3487:
3481:
3478:
3474:
3468:
3463:
3461:
3451:
3444:
3441:
3437:
3429:
3426:
3422:
3416:
3409:
3398:
3395:
3386:
3369:
3366:
3357:
3338:
3335:
3326:
3309:
3306:
3297:
3284:
3277:
3271:
3266:
3259:
3254:
3248:
3233:
3222:
3219:
3210:
3193:
3190:
3181:
3162:
3159:
3150:
3133:
3130:
3121:
3108:
3100:
3093:
3090:
3086:
3080:
3077:
3073:
3067:
3062:
3060:
3050:
3044:
3041:
3034:
3031:
3025:
3018:
3012:
3007:
3000:
2995:
2989:
2982:
2976:
2973:
2968:
2965:
2959:
2954:
2952:
2945:
2941:
2937:
2926:
2925:
2924:
2905:
2898:
2895:
2891:
2883:
2880:
2876:
2870:
2863:
2852:
2849:
2840:
2823:
2820:
2811:
2792:
2789:
2780:
2763:
2760:
2751:
2738:
2733:
2728:
2722:
2719:
2712:
2709:
2703:
2694:
2693:
2692:
2678:
2674:
2673:
2656:
2647:
2645:
2644:
2622:
2616:
2611:
2604:
2599:
2593:
2584:
2583:
2582:
2581:. The matrix
2580:
2561:
2555:
2545:
2542:
2533:
2518:
2515:
2506:
2489:
2486:
2477:
2462:
2459:
2450:
2438:
2433:
2430:
2423:
2422:
2421:
2419:
2403:
2394:
2387:
2385:
2369:
2359:
2356:
2347:
2332:
2329:
2320:
2303:
2300:
2291:
2276:
2273:
2264:
2252:
2245:
2239:
2234:
2227:
2222:
2216:
2201:
2191:
2188:
2179:
2164:
2161:
2152:
2135:
2132:
2123:
2108:
2105:
2096:
2084:
2078:
2073:
2066:
2063:
2056:
2053:
2044:
2041:
2034:
2031:
2024:
2015:
2014:
2011:
2009:
1991:
1984:
1977:
1972:
1963:
1956:
1954:
1940:
1934:
1931:
1920:
1913:
1907:
1904:
1893:
1886:
1882:
1879:
1874:
1868:
1865:
1854:
1847:
1841:
1838:
1827:
1820:
1816:
1813:
1808:
1802:
1799:
1788:
1781:
1775:
1772:
1761:
1754:
1750:
1747:
1739:
1738:
1735:
1733:
1732:
1725:
1718:
1683:
1677:
1674:
1667:
1664:
1658:
1651:
1645:
1640:
1633:
1628:
1622:
1615:
1609:
1606:
1601:
1598:
1592:
1587:
1582:
1578:
1574:
1567:
1566:
1565:
1562:
1542:
1538:
1529:
1524:
1516:
1511:
1506:
1502:
1501:
1493:The quantity
1487:
1480:
1478:
1464:
1459:
1449:
1442:
1437:
1427:
1420:
1417:
1413:
1408:
1398:
1391:
1386:
1376:
1369:
1366:
1362:
1357:
1347:
1340:
1335:
1325:
1318:
1315:
1308:
1307:
1304:
1297:
1290:
1288:
1274:
1269:
1261:
1258:
1251:
1248:
1245:
1242:
1238:
1235:
1231:
1228:
1225:
1220:
1212:
1209:
1202:
1199:
1194:
1186:
1183:
1173:
1172:
1169:
1168:
1164:
1145:
1138:
1124:
1112:
1107:
1097:
1089:
1082:
1068:
1056:
1051:
1041:
1030:
1029:
1028:
1026:
1022:
1018:
998:
967:
963:
960:
952:
942:
935:
930:
920:
910:
902:
895:
892:
888:
883:
873:
866:
861:
851:
840:
833:
830:
823:
816:
813:
809:
806:
801:
791:
784:
779:
769:
759:
751:
744:
741:
733:
728:
724:
720:
718:
710:
707:
692:
686:
683:
677:
671:
659:
650:
647:
643:
631:
626:
622:
618:
616:
611:
600:
599:
598:
597:
593:
585:
579:
575:
571:
567:
558:
553:
531:
515:
509:
504:
497:
492:
486:
475:
473:
469:
465:
460:
451:
445:
441:
435:
406:
402:
399:
393:
389:
383:
379:
376:
370:
366:
360:
356:
353:
347:
337:
331:
327:
324:
312:
302:
301:
300:
299:
276:
272:
268:
264:
263:
258:
249:
247:
243:
240:
239:nondegenerate
236:
232:
228:
224:
220:
215:
213:
209:
205:
201:
197:
192:
190:
186:
178:
166:
150:
149:infinitesimal
146:
136:
132:
128:
123:
114:
96:
92:
88:
80:
79:tangent space
76:
75:bilinear form
64:
60:
59:inner product
56:
49:
45:
41:
37:
36:metric tensor
33:
29:
22:
22656:Moving frame
22651:Morse theory
22641:Gauge theory
22433:Tensor field
22362:Closed/Exact
22341:Vector field
22309:Distribution
22250:Hypercomplex
22245:Quaternionic
21982:Vector field
21940:Smooth atlas
21857:Hermann Weyl
21693:
21661:Vector space
21646:Pseudotensor
21611:Fiber bundle
21564:abstractions
21459:Mixed tensor
21444:Tensor field
21251:
21179:Applications
21107:Main results
21020:
20920:
20878:
20859:
20824:
20820:
20791:
20772:
20762:
20743:
20733:
20711:
20701:
20690:
20686:
20675:
20652:
20614:
20593:
20576:
20564:
20552:
20545:Wells (1980)
20539:
20526:
20513:
20436:
20158:
20031:
19842:
19831:
19569:
19429:
19329:
19224:
19217:
19213:
19202:
19199:
18978:is given by
18974:
18968:
18954:
18943:
18937:
18930:
18627:
18457:
18453:
18447:
18444:
18351:
18278:
18274:
18259:
18239:denotes the
18227:
18141:
18137:
18118:
18107:
18100:
18082:
17936:
17932:
17921:
17917:
17888:
17881:
17867:
17840:
17819:
17796:
17571:
17569:
17561:
17333:
17319:
17315:line element
17304:
17184:
16960:
16956:
16952:
16937:
16933:
16930:
16923:
16919:
16913:
16903:
16899:
16893:
16882:
16828:
16752:
16652:
16576:
16569:
16567:The inverse
16566:
16560:
16556:
16548:
16544:
16529:
16525:
16506:
16502:
16494:
16490:
16484:varies over
16479:
16473:
16467:
16460:to the dual
16456:
16452:
16444:
16440:
16431:
16331:by means of
16327:
16323:
16315:
16309:
16301:
16297:
16292:
16289:
16282:
16278:
16271:
16267:
16264:
16152:
16148:
16137:
16133:
16122:
16118:
16108:
16102:
16091:
16087:
16074:
16070:
16063:
16059:
16052:
16048:
16045:
15938:
15932:
15918:
15914:
15904:
15900:
15895:
15891:
15886:
15882:
15871:
15867:
15861:
15857:
15848:
15756:
15752:
15746:
15742:
15723:
15705:
15701:
15690:
15617:
15606:
15541:
15517:
15507:
15497:
15493:
15483:
15480:
15379:
15375:braiding map
15372:
15293:
15234:
15230:
15226:
15154:
15150:
15140:
15138:The section
15137:
15054:with itself
15050:
15032:
15018:
15012:
14989:, and often
14980:
14978:
14891:
14876:
14865:
14796:
14786:
14363:
14291:is given by
14287:
14284:
14176:
14010:
14003:
13999:
13996:
13906:
13901:
13892:
13882:
13878:
13871:
13861:
13854:
13824:
13820:
13814:
13793:
13782:
13657:
13653:
13648:
13642:
13635:
13626:
13620:
13617:
13613:
13610:
13537:
13526:
13432:
13424:
13420:
13413:
13409:
13404:
13402:
13397:
13395:
13277:
13273:
13266:
13262:
13255:
13251:
13247:
13237:
13234:
13230:
13227:
13030:
13027:
13023:
13020:
12936:
12932:
12926:
12916:
12908:
12904:
12899:
12895:
12884:
12880:
12865:
12861:
12858:
12744:
12740:
12737:
12730:
12726:
12720:
12711:
12702:
12699:
12695:
12689:
12686:
12682:
12679:
12581:
12576:
12573:
12567:
12560:
12556:
12548:
12544:
12538:denotes the
12534:
12531:
12520:
12174:
12098:
12092:
12086:
12083:
12079:
12076:
11931:
11927:
11922:
11918:
11914:
11911:
11772:
11768:
11752:
11748:
11741:
11737:
11734:
11728:
11722:
11713:
11708:
11705:
11701:
11698:
11614:
11609:
11599:
11593:
11589:
11586:
11575:
11304:
11300:
11293:
11289:
11286:
11268:
11255:
11249:
11243:
11240:
11234:
11227:
11225:
10885:
10882:
10876:
10870:
10867:
10856:
10727:
10722:
10717:
10714:
10647:
10641:by a matrix
10637:
10634:
10525:
10429:
10422:
10411:
10342:
10186:
10182:
10175:
10171:
10156:
10152:
10137:
10118:
10116:
10105:
9971:
9966:
9962:
9958:
9953:
9947:
9854:
9846:
9842:
9835:
9831:
9828:
9811:
9807:
9796:
9781:
9773:
9743:is called a
9731:is called a
9718:
9706:
9696:
9692:
9681:
9677:
9670:
9666:
9662:
9655:
9651:
9640:
9636:
9632:
9600:
9365:
9361:
9354:
9329:is called a
9308:
9304:
9298:is called a
9290:
9286:
9271:
9267:
9260:
9256:
9253:
9124:
9107:denotes the
9102:
8971:
8967:
8963:
8955:
8951:
8947:
8944:
8763:
8631:
8626:
8622:
8619:
8499:
8374:
8254:
8108:
8104:
8097:A system of
8096:
8085:
8078:
8074:
8071:
7937:
7856:
7852:
7848:
7842:
7838:
7832:
7693:of the form
7685:
7679:
7669:
7663:
7660:
7500:
7494:
7477:
7469:
7465:
7458:
7253:
7240:
7236:
7233:
7108:
7099:
7095:
7088:
7084:
7078:
7075:
7064:
6966:
6962:
6955:
6951:
6936:
6813:of manifold
6789:
6785:
6766:
6757:
6753:
6748:
6744:
6739:
6735:
6728:
6724:
6716:
6712:
6701:
6697:
6690:
6686:
6609:
6605:
6594:
6590:
6480:
6476:
6469:
6465:
6454:
6450:
6083:
6079:
6072:
6068:
6061:
6057:
6046:
6042:
6020:
6016:
6009:
6005:
5997:
5993:
5988:
5984:
5979:
5975:
5957:
5953:
5948:vector space
5942:
5938:
5896:hypersurface
5889:
5870:
5854:
5482:denotes the
5477:
5363:
5359:
5350:
5342:surface area
5339:
5230:
5224:
5217:
5153:is given by
5149:
5146:
5127:
5121:
5114:
5108:
5105:
4830:
4824:
4813:
4736:
4730:
4723:
4562:
4555:
4546:
4537:
4528:
4519:
4516:
3966:
3779:
3770:
3767:
3663:
2922:
2676:
2670:
2653:
2641:
2639:
2576:
2399:
2388:
1989:
1982:
1975:
1968:
1957:
1729:
1723:
1716:
1705:
1563:
1540:
1536:
1527:
1509:
1505:line element
1498:
1492:
1481:
1302:
1291:
1167:differential
1160:
986:
577:
573:
569:
565:
556:
537:
476:
461:
443:
439:
434:ordered pair
431:
266:
260:
259:in his 1827
255:
252:Introduction
233:. From the
216:
212:tensor field
193:
185:metric space
148:
134:
130:
126:
121:
115:
95:real numbers
85:(that is, a
39:
35:
28:mathematical
25:
22601:Levi-Civita
22591:Generalized
22563:Connections
22513:Lie algebra
22445:Volume form
22346:Vector flow
22319:Pushforward
22314:Lie bracket
22213:Lie algebra
22178:G-structure
21967:Pushforward
21947:Submanifold
21797:Élie Cartan
21745:Spin tensor
21719:Weyl tensor
21677:Mathematics
21641:Multivector
21432:definitions
21330:Engineering
21269:Mathematics
20792:Gravitation
20569:Vaughn 2007
20451:mass–energy
19838:proper time
19236:-axis, and
18960:orthonormal
18129:volume form
18088:determinant
17325:pulled back
16902:= 1, 2, …,
16765:) to a map
14179:pushforward
13888:dot product
12889:at a point
12712:covariantly
11758:define the
10728:For a pair
10526:Denote the
9967:dual metric
9702:eigenvalues
8255:The metric
7673:denote the
6808:open subset
6031:real number
5946:there is a
5861:determinant
4814:It is also
3969:bilinearity
3670:dot product
1553:units, and
1019:. Here the
273:, with the
227:covariantly
53:(such as a
38:(or simply
22749:Categories
22724:Stratifold
22682:Diffeology
22478:Associated
22279:Symplectic
22264:Riemannian
22193:Hyperbolic
22120:Submersion
22028:Hopf–Rinow
21962:Submersion
21957:Smooth map
21626:Linear map
21494:Operations
21139:Ricci flow
21088:Hyperbolic
20607:References
20038:black hole
19230:colatitude
18225:where the
16829:or by the
15924:dual space
15718:See also:
15536:See also:
14987:continuous
13797:, and let
13396:is called
12870:defines a
12710:transform
12540:row vector
11777:such that
11762:to be the
11760:dual basis
11281:See also:
10528:row vector
9764:Lorentzian
8638:chain rule
7835:invertible
7082:functions
6733:such that
5867:Definition
4724:This is a
1971:chain rule
1021:chain rule
592:arc length
534:Arc length
189:derivative
22606:Principal
22581:Ehresmann
22538:Subbundle
22528:Principal
22503:Fibration
22483:Cotangent
22355:Covectors
22208:Lie group
22188:Hermitian
22131:manifolds
22100:Immersion
22095:Foliation
22033:Noether's
22018:Frobenius
22013:De Rham's
22008:Darboux's
21899:Manifolds
21765:EM tensor
21601:Dimension
21552:Transpose
21083:Hermitian
21036:Signature
20999:Sectional
20979:Curvature
20849:120009332
20841:1432-1807
20773:to appear
20586:Lee (1997
20410:θ
20407:
20384:−
20347:−
20315:−
20276:−
20264:−
20202:−
20178:ν
20175:μ
20137:φ
20131:θ
20011:ν
19998:μ
19985:ν
19982:μ
19969:μ
19956:μ
19929:−
19913:−
19897:−
19721:−
19686:−
19656:−
19626:−
19459:→
19454:μ
19394:φ
19386:θ
19383:
19361:θ
19303:θ
19300:
19164:∂
19149:∂
19130:∂
19115:∂
19103:∑
19083:∂
19068:∂
19049:∂
19034:∂
19019:δ
19006:∑
18862:θ
18859:
18833:θ
18830:
18805:θ
18802:
18796:θ
18793:
18781:θ
18778:
18772:θ
18769:
18760:−
18753:θ
18750:
18744:θ
18741:
18729:θ
18726:
18720:θ
18717:
18708:−
18703:θ
18700:
18684:θ
18681:
18597:θ
18594:
18583:θ
18580:
18570:θ
18567:
18558:−
18553:θ
18550:
18522:θ
18519:
18496:θ
18493:
18370:∫
18270:Cartesian
18266:Euclidean
18200:∧
18197:⋯
18194:∧
18154:ω
18018:−
18014:φ
18010:∘
17993:φ
17989:∫
17976:μ
17959:∫
17949:Λ
17757:γ
17754:∘
17707:γ
17704:∘
17659:γ
17617:∑
17602:∫
17515:γ
17512:∘
17465:γ
17462:∘
17417:γ
17375:∑
17353:∫
17219:∑
17187:quadratic
17143:γ
17140:∘
17093:γ
17090:∘
17045:γ
17003:∑
16986:∫
16966:arclength
16852:⊗
16809:→
16801:∗
16791:⊗
16783:∗
16733:α
16727:β
16719:−
16693:β
16687:α
16679:−
16630:→
16622:∗
16604:−
16295: : T
15828:−
15728:from the
15668:→
15655:×
15607:from the
15587:→
15575:×
15458:∗
15448:⊗
15440:∗
15430:≅
15425:∗
15409:⊗
15352:⊗
15339:≅
15334:→
15319:⊗
15305:τ
15277:⊗
15266:τ
15263:∘
15258:⊗
15185:⊗
15174:⊗
15113:∗
15097:⊗
15078:Γ
15075:∈
15070:⊗
15025:naturally
14956:→
14938:×
14877:from the
14844:→
14827:×
14743:∂
14732:φ
14728:∂
14720:…
14702:∂
14691:φ
14687:∂
14666:∂
14655:φ
14651:∂
14641:⋮
14636:⋱
14631:⋮
14626:⋮
14606:∂
14595:φ
14591:∂
14583:…
14565:∂
14554:φ
14550:∂
14529:∂
14518:φ
14514:∂
14491:∂
14480:φ
14476:∂
14468:…
14450:∂
14439:φ
14435:∂
14414:∂
14403:φ
14399:∂
14382:φ
14346:φ
14325:φ
14256:∗
14252:φ
14248:⋅
14234:∗
14230:φ
14133:∂
14122:φ
14118:∂
14085:∑
14064:∑
14046:∗
14042:φ
14030:given by
13957:⋯
13838:injective
13834:immersion
13809:into the
13698:∑
13574:−
13471:−
13319:∑
13124:−
12805:↦
12786:−
12654:θ
12620:θ
12597:α
12489:θ
12451:θ
12414:…
12327:θ
12299:⋯
12276:θ
12231:θ
12199:α
12146:θ
12138:−
12113:θ
12034:θ
12026:⋮
12002:θ
11977:θ
11951:θ
11912:That is,
11789:θ
11504:⋮
11401:⋯
11184:β
11176:−
11146:α
11107:β
11065:−
11047:−
11023:−
10991:α
10949:β
10941:−
10905:α
10817:β
10809:−
10779:α
10770:β
10764:α
10755:~
10723:covariant
10682:α
10662:α
10599:α
10593:…
10582:α
10570:α
10541:α
10501:…
10457:α
10445:α
10373:α
10369:↦
10300:α
10266:α
10211:α
10167:linearity
10069:−
10051:−
10027:−
10011:−
9963:conjugate
9700:negative
9601:for some
9570:ξ
9561:−
9558:⋯
9555:−
9530:ξ
9521:−
9502:ξ
9490:⋯
9468:ξ
9440:ξ
9407:ξ
9397:∑
9339:connected
9315:signature
9223:∈
9083:−
9032:−
8914:∂
8899:∂
8854:∂
8839:∂
8810:∑
8736:∂
8732:∂
8714:∂
8699:∂
8676:∑
8656:∂
8652:∂
8584:∂
8580:∂
8559:∂
8555:∂
8479:…
8438:…
8338:∂
8334:∂
8313:∂
8309:∂
8218:∂
8214:∂
8193:…
8174:∂
8170:∂
7980:∑
7833:for some
7773:∑
7766:…
7731:∑
7709:↦
7675:transpose
7377:∑
7291:∑
7182:∑
7130:∑
6633:↦
6461:symmetric
5773:∬
5730:−
5713:∬
5660:→
5650:⋅
5638:→
5622:−
5605:→
5595:⋅
5583:→
5554:→
5544:⋅
5532:→
5509:∬
5435:→
5425:×
5413:→
5392:∬
5254:θ
5248:
5056:μ
5028:λ
5003:μ
4992:λ
4929:μ
4901:λ
4868:μ
4857:λ
4247:→
4237:⋅
4225:→
4183:→
4173:⋅
4161:→
4119:→
4109:⋅
4097:→
4055:→
4045:⋅
4033:→
3991:⋅
3939:→
3907:→
3862:→
3830:→
3744:→
3712:→
3392:∂
3384:∂
3363:∂
3355:∂
3332:∂
3324:∂
3303:∂
3295:∂
3216:∂
3208:∂
3187:∂
3179:∂
3156:∂
3148:∂
3127:∂
3119:∂
2846:∂
2838:∂
2817:∂
2809:∂
2786:∂
2778:∂
2757:∂
2749:∂
2677:invariant
2539:∂
2531:∂
2512:∂
2504:∂
2483:∂
2475:∂
2456:∂
2448:∂
2353:∂
2345:∂
2326:∂
2318:∂
2297:∂
2289:∂
2270:∂
2262:∂
2185:∂
2177:∂
2158:∂
2150:∂
2129:∂
2121:∂
2102:∂
2094:∂
2010:equation
1924:→
1914:⋅
1897:→
1858:→
1848:⋅
1831:→
1792:→
1782:⋅
1765:→
1453:→
1443:⋅
1431:→
1402:→
1392:⋅
1380:→
1351:→
1341:⋅
1329:→
1163:quadratic
1136:∂
1128:→
1119:∂
1101:→
1080:∂
1072:→
1063:∂
1045:→
999:⋅
946:→
936:⋅
924:→
877:→
867:⋅
855:→
795:→
785:⋅
773:→
725:∫
663:→
623:∫
316:→
44:structure
30:field of
22702:Orbifold
22697:K-theory
22687:Diffiety
22411:Pullback
22225:Oriented
22203:Kenmotsu
22183:Hadamard
22129:Types of
22078:Geodesic
21903:Glossary
21631:Manifold
21616:Geodesic
21374:Notation
21098:Kenmotsu
21011:Geodesic
20964:Glossary
20918:(1980),
20858:(1983),
20815:(1900),
20790:(1973),
20732:(1990),
20673:(1827),
20571:, §3.4.3
20519:pullback
20457:See also
19834:timelike
19430:In flat
18251:Examples
18231:are the
18125:oriented
18123:is also
18071:for all
15922:and its
15481:so that
13818:, where
13789:open set
13283:, where
13228:so that
11884:≠
10725:vector.
10347:varies,
10129:covector
9001:′
8798:′
8764:so that
8534:′
8119:open set
7718:′
6800:smoothly
6053:bilinear
5312:‖
5304:‖
5299:‖
5291:‖
5173:‖
5165:‖
5081:′
5012:′
4946:′
4877:′
4816:bilinear
3625:′
3593:′
3578:′
3551:′
3541:′
3529:′
3519:′
3495:′
3482:′
3445:′
3430:′
3399:′
3370:′
3339:′
3310:′
3223:′
3194:′
3163:′
3134:′
3094:′
3081:′
2923:so that
2899:′
2884:′
2853:′
2824:′
2793:′
2764:′
2546:′
2519:′
2490:′
2463:′
2360:′
2333:′
2304:′
2277:′
2192:′
2165:′
2136:′
2109:′
2067:′
2057:′
2045:′
2035:′
2006:via the
1973:relates
1935:′
1908:′
1883:′
1869:′
1842:′
1817:′
1803:′
1776:′
1751:′
1507:, while
1002:‖
996:‖
896:′
834:′
817:′
745:′
703:‖
638:‖
596:integral
552:interval
450:open set
246:smoothly
167:between
165:distance
137:) > 0
48:manifold
22760:Tensors
22646:History
22629:Related
22543:Tangent
22521:)
22501:)
22468:Adjoint
22460:Bundles
22438:density
22336:Torsion
22302:Vectors
22294:Tensors
22277:)
22262:)
22258:,
22256:Pseudo−
22235:Poisson
22168:Finsler
22163:Fibered
22158:Contact
22156:)
22148:Complex
22146:)
22115:Section
21728:Physics
21562:Related
21325:Physics
21243:Tensors
21165:Hilbert
21160:Finsler
20909:2324500
20641:1223091
19222:, with
18140:, ...,
18086:is the
18079:. Here
17799:physics
17317:. When
17313:is the
15732:to the
15699:bundle
15695:of the
15693:section
15501:of the
15373:is the
15043:of the
15039:of the
15029:section
15013:By the
14881:of the
14014:. When
11771:, ...,
11747:, ...,
11592:, ...,
11299:, ...,
9841:, ...,
9784:− 1, 1)
9723:, then
9294:, then
8107:, ...,
7846:matrix
6961:, ...,
5898:in the
5885:surface
5859:is the
5372:in the
5132:in the
1561:units.
554:, then
455:in the
177:infimum
55:surface
26:In the
22611:Vector
22596:Koszul
22576:Cartan
22571:Affine
22553:Vector
22548:Tensor
22533:Spinor
22523:Normal
22519:Stable
22473:Affine
22377:bundle
22329:bundle
22275:Almost
22198:Kähler
22154:Almost
22144:Almost
22138:Closed
22038:Sard's
21994:(list)
21656:Vector
21651:Spinor
21636:Matrix
21430:Tensor
21093:Kähler
20989:Scalar
20984:tensor
20907:
20897:
20866:
20847:
20839:
20798:
20750:
20718:
20659:
20639:
20629:
20437:where
17855:volume
17572:energy
16964:. The
16950:, for
16909:matrix
16114:kernel
15380:Since
15294:where
14995:smooth
14364:where
14185:along
13997:where
13915:, say
13787:be an
12532:where
11940:. Let
11936:, the
11878:
11849:
10889:gives
9760:(3, 1)
9756:(1, 3)
9103:where
8117:on an
7661:where
7461:matrix
6035:scalar
5855:where
5478:where
5125:, and
4820:linear
4544:, and
2687:, and
2665:, and
2579:tensor
2412:, and
2008:matrix
2002:, and
1987:, and
1303:where
987:where
590:. The
284:, and
200:tensor
40:metric
22719:Sheaf
22493:Fiber
22269:Rizza
22240:Prime
22071:Local
22061:Curve
21923:Atlas
21576:Basis
21261:Scope
20994:Ricci
20883:(PDF)
20845:S2CID
20689:Vol.
20505:Notes
18953:∂ / ∂
18947:on a
16501:Hom(T
16157:is a
16142:is a
16083:from
16079:is a
15546:. If
15524:is a
15027:to a
14997:, or
13823:>
13801:be a
13662:) is
9333:. If
7489:into
7112:. If
6947:frame
6945:, or
6939:basis
6802:with
6761:) ≠ 0
5894:) or
5875:be a
2675:) is
1545:when
468:angle
196:Gauss
73:is a
61:on a
46:on a
22586:Form
22488:Dual
22421:flow
22284:Tame
22260:Sub−
22173:Flat
22053:Maps
20895:ISBN
20864:ISBN
20837:ISSN
20796:ISBN
20748:ISBN
20716:ISBN
20657:ISBN
20627:ISBN
20445:and
20032:The
19228:the
18958:are
18235:and
18081:det
16931:Let
16505:, T*
16276:and
16057:and
15849:the
15704:* ⊗
15496:⊗ T*
15041:dual
14791:and
14193:and
13783:Let
13419:...
13254:...
12939:, −)
12912:, −)
12543:...
11926:) =
11263:and
10732:and
10195:and
10180:and
9979:via
9961:(or
9829:Let
9776:− 1)
9772:(1,
9721:, 0)
8961:and
7683:and
7667:and
7498:and
7485:and
7248:and
7076:The
6954:= (
6824:and
6474:and
6096:and
6092:and
6014:and
5871:Let
5340:The
5336:Area
5228:and
5140:and
4828:and
4734:and
4560:and
3777:and
1969:The
1721:and
1710:and
1497:in (
542:and
472:area
292:and
206:and
171:and
159:and
34:, a
22508:Jet
20887:doi
20829:doi
20619:doi
20398:sin
19374:sin
19291:sin
18931:by
18850:cos
18821:sin
18799:sin
18790:cos
18775:sin
18766:cos
18747:cos
18738:sin
18723:cos
18714:sin
18691:sin
18672:cos
18628:So
18591:cos
18577:sin
18564:sin
18547:cos
18516:sin
18490:cos
18167:det
18119:If
18042:det
17900:on
17893:of
17323:is
16946:in
16559:→ T
16511:of
16480:As
16304:→ T
16116:of
16095:to
15880:to
15876:at
15853:on
15801:def
15750:∈ T
15615:to
15611:of
15233:⊗ T
15158:by
15153:⊗ T
15047:of
14885:of
14181:of
13890:in
13857:= 3
13791:in
13412:=
13403:To
13265:=
13250:=
11740:= (
11292:= (
11273:).
10343:As
9965:or
9834:= (
9793:is
9789:If
9778:or
9758:or
9750:If
9711:If
9617:of
9355:By
9337:is
9280:at
9254:If
8966:= (
8950:= (
8124:in
7851:= (
7475:by
6941:of
6828:on
6779:of
6771:on
6720:≠ 0
6613:≠ 0
6599:is
6459:is
6270:and
6051:is
6025:at
5892:= 2
5857:det
5784:det
5245:cos
4968:and
4728:in
1994:to
1517:of
572:),
124:if
120:is
105:of
93:to
81:at
69:of
22751::
22499:Co
20905:MR
20903:,
20893:,
20843:,
20835:,
20825:54
20823:,
20819:,
20811:;
20786:;
20782:;
20775:).
20765:,
20691:VI
20685:,
20637:MR
20635:,
20625:,
19702:or
19574:,
19246:xy
19216:,
18969:ij
18935:.
18462::
18456:,
18277:,
18228:dx
18116:.
17941:,
17935:,
17865:.
17320:ds
17311:ds
16959:≤
16955:≤
16924:ij
16897:,
16757:,
16555:T*
16488:,
16477:.
16287:.
16131:,
15942:,
15760:,
15710:.
15528:.
15506:T*
15492:T*
15377:.
15020:10
14993:,
14982:10
14868:10
14366:Dφ
13904:.
13881:⊂
13869:.
13616:=
13400:.
13272:…
13241::
13233:=
13026:↦
12698:=
12685:=
12096:,
12082:↦
11704:=
10645:,
10420:.
10199::
10146:,
10135:.
9810:,
9704:.
9695:−
9682:ij
9669:−
9665:,
9654:−
9639:−
9635:,
9352:.
9289:∈
9105:Dy
8977:,
8972:ij
8956:ij
8627:ij
8089:.
8079:ij
7857:ij
7841:×
7504:,
7470:ij
7239:∈
7106:,
7098:×
7089:ij
6949:,
6922:.
6752:,
6485:,
6077:,
6066:,
5992:,
5941:∈
5863:.
5374:uv
5362:,
5144:.
5134:uv
5119:,
5112:,
4569:uv
4565:=
4558:=
4535:,
4526:,
2683:,
2661:,
2408:,
1998:,
1980:,
1960:2'
1559:dv
1551:du
1539:,
1510:ds
1495:ds
1165:)
1027::
580:))
457:uv
442:,
280:,
214:.
183:a
133:,
113:.
22517:(
22497:(
22273:(
22254:(
22152:(
22142:(
21905:)
21901:(
21891:e
21884:t
21877:v
21235:e
21228:t
21221:v
21043:/
20966:)
20962:(
20952:e
20945:t
20938:v
20889::
20831::
20771:(
20757:.
20738:.
20725:.
20706:.
20666:.
20621::
20447:M
20439:G
20422:,
20416:]
20402:2
20392:2
20388:r
20379:0
20374:0
20369:0
20362:0
20355:2
20351:r
20342:0
20337:0
20330:0
20325:0
20318:1
20310:)
20301:2
20297:c
20293:r
20288:M
20285:G
20282:2
20273:1
20269:(
20259:0
20252:0
20247:0
20242:0
20236:)
20227:2
20223:c
20219:r
20214:M
20211:G
20208:2
20199:1
20195:(
20188:[
20183:=
20171:g
20144:,
20140:)
20134:,
20128:,
20125:r
20122:,
20119:t
20116:c
20113:(
20110:=
20106:)
20100:3
20096:x
20092:,
20087:2
20083:x
20079:,
20074:1
20070:x
20066:,
20061:0
20057:x
20052:(
20017:.
20007:r
20003:d
19994:r
19990:d
19978:g
19974:=
19965:r
19961:d
19952:r
19948:d
19945:=
19940:2
19936:z
19932:d
19924:2
19920:y
19916:d
19908:2
19904:x
19900:d
19892:2
19888:t
19884:d
19879:2
19875:c
19871:=
19866:2
19862:s
19858:d
19817:.
19811:]
19805:1
19800:0
19795:0
19790:0
19783:0
19778:1
19773:0
19768:0
19761:0
19756:0
19751:1
19746:0
19739:0
19734:0
19729:0
19724:1
19715:[
19710:=
19707:g
19695:]
19689:1
19681:0
19676:0
19671:0
19664:0
19659:1
19651:0
19646:0
19639:0
19634:0
19629:1
19621:0
19614:0
19609:0
19604:0
19599:1
19593:[
19588:=
19585:g
19555:,
19551:)
19548:z
19545:,
19542:y
19539:,
19536:x
19533:,
19530:t
19527:c
19524:(
19521:=
19517:)
19511:3
19507:x
19503:,
19498:2
19494:x
19490:,
19485:1
19481:x
19477:,
19472:0
19468:x
19463:(
19450:r
19434:(
19404:.
19398:2
19390:d
19378:2
19370:+
19365:2
19357:d
19354:=
19349:2
19345:s
19341:d
19315:.
19309:]
19295:2
19285:0
19278:0
19273:1
19267:[
19262:=
19259:g
19242:x
19238:φ
19234:z
19225:θ
19220:)
19218:φ
19214:θ
19212:(
19203:ℝ
19180:.
19172:j
19168:q
19157:k
19153:x
19138:i
19134:q
19123:k
19119:x
19107:k
19099:=
19091:j
19087:q
19076:l
19072:x
19057:i
19053:q
19042:k
19038:x
19026:l
19023:k
19013:l
19010:k
19002:=
18997:j
18994:i
18990:g
18975:q
18966:δ
18955:x
18944:x
18914:]
18906:2
18902:r
18896:0
18889:0
18884:1
18878:[
18873:=
18868:]
18854:2
18844:2
18840:r
18836:+
18825:2
18815:2
18811:r
18787:r
18784:+
18763:r
18735:r
18732:+
18711:r
18695:2
18687:+
18676:2
18665:[
18660:=
18657:J
18651:T
18646:J
18642:=
18639:g
18609:.
18603:]
18588:r
18561:r
18541:[
18536:=
18529:J
18513:r
18510:=
18503:y
18487:r
18484:=
18477:x
18460:)
18458:θ
18454:r
18452:(
18430:.
18422:2
18418:)
18414:y
18411:d
18408:(
18405:+
18400:2
18396:)
18392:x
18389:d
18386:(
18379:b
18374:a
18366:=
18363:L
18337:.
18331:]
18325:1
18320:0
18313:0
18308:1
18302:[
18297:=
18294:g
18281:)
18279:y
18275:x
18273:(
18237:∧
18211:n
18207:x
18203:d
18189:1
18185:x
18181:d
18174:|
18170:g
18163:|
18157:=
18144:)
18142:x
18138:x
18136:(
18121:M
18110:)
18108:M
18106:(
18104:0
18101:C
18092:Λ
18083:g
18077:U
18073:f
18059:x
18056:d
18049:|
18045:g
18038:|
18032:)
18029:x
18026:(
18021:1
18007:f
18002:)
17999:U
17996:(
17985:=
17980:g
17972:d
17968:f
17963:U
17955:=
17952:f
17939:)
17937:φ
17933:U
17931:(
17922:g
17918:μ
17910:g
17906:M
17902:M
17891:)
17889:M
17887:(
17885:0
17882:C
17877:Λ
17851:n
17847:M
17843:n
17807:E
17782:.
17778:t
17775:d
17770:)
17766:)
17763:t
17760:(
17749:j
17745:x
17738:t
17735:d
17731:d
17725:(
17720:)
17716:)
17713:t
17710:(
17699:i
17695:x
17688:t
17685:d
17681:d
17675:(
17671:)
17668:)
17665:t
17662:(
17656:(
17651:j
17648:i
17644:g
17638:n
17633:1
17630:=
17627:j
17624:,
17621:i
17611:b
17606:a
17596:2
17593:1
17588:=
17585:E
17547:.
17543:t
17540:d
17533:|
17528:)
17524:)
17521:t
17518:(
17507:j
17503:x
17496:t
17493:d
17489:d
17483:(
17478:)
17474:)
17471:t
17468:(
17457:i
17453:x
17446:t
17443:d
17439:d
17433:(
17429:)
17426:)
17423:t
17420:(
17414:(
17409:j
17406:i
17402:g
17396:n
17391:1
17388:=
17385:j
17382:,
17379:i
17370:|
17362:b
17357:a
17349:=
17346:L
17329:M
17288:j
17284:x
17280:d
17275:i
17271:x
17267:d
17264:)
17261:p
17258:(
17253:j
17250:i
17246:g
17240:n
17235:1
17232:=
17229:j
17226:,
17223:i
17215:=
17210:2
17206:s
17202:d
17170:.
17166:t
17163:d
17156:)
17152:)
17149:t
17146:(
17135:j
17131:x
17124:t
17121:d
17117:d
17111:(
17106:)
17102:)
17099:t
17096:(
17085:i
17081:x
17074:t
17071:d
17067:d
17061:(
17057:)
17054:)
17051:t
17048:(
17042:(
17037:j
17034:i
17030:g
17024:n
17019:1
17016:=
17013:j
17010:,
17007:i
16995:b
16990:a
16982:=
16979:L
16961:b
16957:t
16953:a
16948:M
16940:)
16938:t
16936:(
16934:γ
16920:g
16914:G
16904:n
16900:i
16894:x
16889:M
16885:g
16863:.
16860:M
16856:T
16849:M
16845:T
16813:R
16806:M
16796:T
16788:M
16778:T
16759:β
16755:α
16737:]
16730:,
16722:1
16714:g
16710:S
16705:[
16701:=
16697:]
16690:,
16682:1
16674:g
16670:S
16665:[
16638:M
16634:T
16627:M
16617:T
16612::
16607:1
16599:g
16595:S
16577:g
16570:S
16561:M
16557:M
16549:g
16545:S
16540:M
16536:M
16530:g
16526:S
16521:g
16517:g
16509:)
16507:M
16503:M
16495:g
16491:S
16486:M
16482:p
16474:M
16468:p
16462:T
16457:M
16453:p
16450:T
16445:M
16441:p
16438:T
16434:S
16417:.
16413:]
16408:p
16404:Y
16400:,
16395:p
16391:X
16387:S
16384:[
16381:=
16378:)
16373:p
16369:Y
16365:,
16360:p
16356:X
16352:(
16347:S
16343:g
16328:M
16324:p
16321:T
16316:M
16310:p
16302:M
16298:p
16293:S
16283:p
16279:Y
16272:p
16268:X
16250:]
16245:p
16241:X
16237:,
16232:p
16228:Y
16222:g
16218:S
16214:[
16211:=
16208:]
16203:p
16199:Y
16195:,
16190:p
16186:X
16180:g
16176:S
16172:[
16153:g
16149:S
16138:g
16134:S
16123:g
16119:S
16109:M
16103:p
16097:T
16092:M
16088:p
16085:T
16075:g
16071:S
16064:p
16060:Y
16053:p
16049:X
16031:)
16026:p
16022:Y
16018:,
16013:p
16009:X
16005:(
16000:p
15996:g
15992:=
15989:]
15984:p
15980:Y
15976:,
15971:p
15967:X
15961:g
15957:S
15953:[
15939:M
15933:p
15927:T
15919:M
15915:p
15912:T
15908:)
15905:p
15901:Y
15899:,
15896:p
15892:X
15890:(
15887:p
15883:g
15878:p
15872:p
15868:Y
15862:M
15858:p
15855:T
15834:,
15831:)
15825:,
15820:p
15816:X
15812:(
15809:g
15796:=
15786:p
15782:X
15776:g
15772:S
15757:M
15753:p
15747:p
15743:X
15708:*
15706:E
15702:E
15676:.
15672:R
15663:p
15659:E
15650:p
15646:E
15642::
15637:p
15633:g
15618:R
15613:E
15591:R
15584:E
15579:M
15571:E
15568::
15565:g
15552:M
15548:E
15521:⊗
15518:g
15513:g
15508:M
15498:M
15494:M
15487:⊗
15484:g
15466:,
15463:M
15453:T
15445:M
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6888:p
6884:X
6880:(
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6867:=
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6861:p
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6855:)
6852:Y
6849:,
6846:X
6843:(
6840:g
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6826:Y
6822:X
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6769:g
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6758:p
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6745:X
6743:(
6740:p
6736:g
6729:p
6725:Y
6717:p
6713:X
6702:p
6698:Y
6691:p
6687:X
6672:)
6667:p
6663:Y
6659:,
6654:p
6650:X
6646:(
6641:p
6637:g
6628:p
6624:Y
6610:p
6606:X
6595:p
6591:g
6575:.
6571:)
6566:p
6562:X
6558:,
6553:p
6549:Y
6545:(
6540:p
6536:g
6532:=
6529:)
6524:p
6520:Y
6516:,
6511:p
6507:X
6503:(
6498:p
6494:g
6481:p
6477:Y
6470:p
6466:X
6455:p
6451:g
6431:.
6427:)
6422:p
6418:V
6414:,
6409:p
6405:Y
6401:(
6396:p
6392:g
6388:b
6385:+
6382:)
6377:p
6373:U
6369:,
6364:p
6360:Y
6356:(
6351:p
6347:g
6343:a
6340:=
6333:)
6328:p
6324:V
6320:b
6317:+
6312:p
6308:U
6304:a
6301:,
6296:p
6292:Y
6288:(
6283:p
6279:g
6265:,
6261:)
6256:p
6252:Y
6248:,
6243:p
6239:V
6235:(
6230:p
6226:g
6222:b
6219:+
6216:)
6211:p
6207:Y
6203:,
6198:p
6194:U
6190:(
6185:p
6181:g
6177:a
6174:=
6167:)
6162:p
6158:Y
6154:,
6149:p
6145:V
6141:b
6138:+
6133:p
6129:U
6125:a
6122:(
6117:p
6113:g
6098:b
6094:a
6090:p
6084:p
6080:Y
6073:p
6069:V
6062:p
6058:U
6047:p
6043:g
6033:(
6027:p
6021:p
6017:Y
6010:p
6006:X
6001:)
5998:p
5994:Y
5989:p
5985:X
5983:(
5980:p
5976:g
5971:p
5967:p
5958:M
5954:p
5951:T
5943:M
5939:p
5922:1
5919:+
5916:n
5911:R
5890:n
5881:n
5873:M
5836:v
5833:d
5829:u
5826:d
5818:]
5812:G
5807:F
5800:F
5795:E
5789:[
5777:D
5763:=
5756:v
5753:d
5749:u
5746:d
5738:2
5734:F
5727:G
5724:E
5717:D
5703:=
5696:v
5693:d
5689:u
5686:d
5678:2
5673:)
5667:v
5657:r
5645:u
5635:r
5627:(
5618:)
5612:v
5602:r
5590:v
5580:r
5572:(
5567:)
5561:u
5551:r
5539:u
5529:r
5521:(
5513:D
5480:×
5463:v
5460:d
5456:u
5453:d
5448:|
5442:v
5432:r
5420:u
5410:r
5402:|
5396:D
5378:M
5370:D
5366:)
5364:v
5360:u
5358:(
5355:→
5351:r
5346:M
5320:.
5308:b
5295:a
5285:)
5281:b
5277:,
5273:a
5269:(
5266:g
5260:=
5257:)
5251:(
5231:b
5225:a
5220:θ
5201:)
5197:a
5193:,
5189:a
5185:(
5182:g
5177:=
5169:a
5150:a
5142:λ
5138:μ
5130:′
5128:b
5122:b
5117:′
5115:a
5109:a
5086:)
5077:b
5072:,
5068:a
5063:(
5059:g
5053:+
5050:)
5046:b
5042:,
5038:a
5034:(
5031:g
5025:=
5017:)
5008:b
5000:+
4996:b
4989:,
4985:a
4980:(
4976:g
4963:,
4959:)
4954:b
4950:,
4942:a
4936:(
4932:g
4926:+
4923:)
4919:b
4915:,
4911:a
4907:(
4904:g
4898:=
4890:)
4885:b
4881:,
4873:a
4865:+
4861:a
4853:(
4849:g
4831:b
4825:a
4799:.
4795:)
4791:a
4787:,
4783:b
4779:(
4776:g
4773:=
4770:)
4766:b
4762:,
4758:a
4754:(
4751:g
4737:b
4731:a
4709:.
4705:G
4700:2
4696:b
4690:2
4686:a
4682:+
4679:F
4674:1
4670:b
4664:2
4660:a
4656:+
4653:F
4648:2
4644:b
4638:1
4634:a
4630:+
4627:E
4622:1
4618:b
4612:1
4608:a
4604:=
4601:)
4597:b
4593:,
4589:a
4585:(
4582:g
4563:b
4556:a
4550:2
4547:b
4541:2
4538:a
4532:1
4529:b
4523:1
4520:a
4498:.
4492:]
4484:2
4480:b
4470:1
4466:b
4459:[
4452:]
4446:G
4441:F
4434:F
4429:E
4423:[
4416:]
4408:2
4404:a
4396:1
4392:a
4385:[
4380:=
4370:.
4367:G
4362:2
4358:b
4352:2
4348:a
4344:+
4341:F
4336:1
4332:b
4326:2
4322:a
4318:+
4315:F
4310:2
4306:b
4300:1
4296:a
4292:+
4289:E
4284:1
4280:b
4274:1
4270:a
4266:=
4254:v
4244:r
4232:v
4222:r
4213:2
4209:b
4203:2
4199:a
4195:+
4190:u
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4158:r
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4145:b
4139:2
4135:a
4131:+
4126:v
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4094:r
4085:2
4081:b
4075:1
4071:a
4067:+
4062:u
4052:r
4040:u
4030:r
4021:1
4017:b
4011:1
4007:a
4003:=
3995:b
3987:a
3946:v
3936:r
3927:2
3923:b
3919:+
3914:u
3904:r
3895:1
3891:b
3887:=
3879:b
3869:v
3859:r
3850:2
3846:a
3842:+
3837:u
3827:r
3818:1
3814:a
3810:=
3802:a
3783:2
3780:p
3774:1
3771:p
3751:v
3741:r
3732:2
3728:p
3724:+
3719:u
3709:r
3700:1
3696:p
3692:=
3688:p
3674:M
3640:.
3634:2
3630:)
3622:s
3618:d
3615:(
3612:=
3600:]
3590:v
3586:d
3575:u
3571:d
3565:[
3558:]
3548:G
3538:F
3526:F
3516:E
3509:[
3502:]
3492:v
3488:d
3479:u
3475:d
3469:[
3464:=
3452:]
3442:v
3438:d
3427:u
3423:d
3417:[
3410:]
3396:v
3387:v
3367:u
3358:v
3336:v
3327:u
3307:u
3298:u
3285:[
3278:]
3272:G
3267:F
3260:F
3255:E
3249:[
3241:T
3234:]
3220:v
3211:v
3191:u
3182:v
3160:v
3151:u
3131:u
3122:u
3109:[
3101:]
3091:v
3087:d
3078:u
3074:d
3068:[
3063:=
3051:]
3045:v
3042:d
3035:u
3032:d
3026:[
3019:]
3013:G
3008:F
3001:F
2996:E
2990:[
2983:]
2977:v
2974:d
2969:u
2966:d
2960:[
2955:=
2946:2
2942:s
2938:d
2906:]
2896:v
2892:d
2881:u
2877:d
2871:[
2864:]
2850:v
2841:v
2821:u
2812:v
2790:v
2781:u
2761:u
2752:u
2739:[
2734:=
2729:]
2723:v
2720:d
2713:u
2710:d
2704:[
2689:G
2685:F
2681:E
2672:1
2667:G
2663:F
2659:E
2643:3
2623:]
2617:G
2612:F
2605:F
2600:E
2594:[
2562:.
2556:]
2543:v
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2434:=
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2389:(
2370:]
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2253:[
2246:]
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2217:[
2209:T
2202:]
2189:v
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2162:u
2153:v
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2124:u
2106:u
2097:u
2085:[
2079:=
2074:]
2064:G
2054:F
2042:F
2032:E
2025:[
2004:G
2000:F
1996:E
1992:′
1990:G
1985:′
1983:F
1978:′
1976:E
1962:)
1958:(
1941:.
1932:v
1921:r
1905:v
1894:r
1887:=
1880:G
1875:,
1866:v
1855:r
1839:u
1828:r
1821:=
1814:F
1809:,
1800:u
1789:r
1773:u
1762:r
1755:=
1748:E
1731:2
1726:′
1724:v
1719:′
1717:u
1712:v
1708:u
1684:]
1678:v
1675:d
1668:u
1665:d
1659:[
1652:]
1646:G
1641:F
1634:F
1629:E
1623:[
1616:]
1610:v
1607:d
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1599:d
1593:[
1588:=
1583:2
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1543:)
1541:v
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1535:(
1532:→
1528:r
1519:M
1500:1
1486:)
1484:2
1482:(
1465:.
1460:v
1450:r
1438:v
1428:r
1421:=
1418:G
1414:,
1409:v
1399:r
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1377:r
1370:=
1367:F
1363:,
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1326:r
1319:=
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1294:1
1292:(
1275:,
1270:2
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1262:v
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1252:G
1249:+
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1226:+
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1207:(
1203:E
1200:=
1195:2
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1146:.
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1113:=
1108:v
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1057:=
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561:→
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127:g
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23:.
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