1289:
7069:
6750:
8622:
6248:
132:), together with all their "algebraic consequences". This means that, applying algebraic operations to this set (e.g. adding those polynomials to each other or multiplying them with any other polynomials) will give rise to the same zero locus. In other words, one can actually consider the zero locus of the algebraic
8386:
7241:
7064:{\displaystyle {\mathcal {C}}E({\mathcal {O}})=\{E_{r}^{p,q},{\text{d}}_{r}^{p,q}\}\qquad {\text{where}}\qquad E_{0}^{p,q}:={\frac {{\mathcal {C}}^{p}\Omega ^{p+q}({\mathcal {O}})}{{\mathcal {C}}^{p+1}\Omega ^{p+q}({\mathcal {O}})}},\qquad {\text{and}}\qquad E_{r+1}^{p,q}:=H(E_{r}^{p,q},d_{r}^{p,q}).}
7792:
In other words, secondary calculus provides substitutes for functions, vector fields, differential forms, differential operators, etc., on a (generically) very singular space where these objects cannot be defined in the usual (smooth) way on the space of solution. Furthermore, the space of these new
143:
the starting equations, obtaining new differential constraints. Therefore, the differential analogue of a variety should be the space of solutions of a system of differential equations, together with all their "differential consequences". Instead of considering the zero locus of an algebraic ideal,
5775:
6701:
5996:
2604:
8225:
3256:
8464:
4770:
6425:
6059:
7889:
5245:
1085:
8262:
5400:
3564:
5881:
7105:
6324:
5562:
5154:
4673:
4487:
281:
The formal definition of a diffiety, which relies on the geometric approach to differential equations and their solutions, requires the notions of jets of submanifolds, prolongations, and Cartan distribution, which are recalled below.
2458:
8109:
5667:
4837:
4137:
6597:
5682:
3151:
6608:
2950:. While jet bundles are enough to deal with many equations arising in geometry, jet spaces of submanifolds provide a greater generality, used to tackle several PDEs imposed on submanifolds of a given manifold, such as
1971:
5893:
5290:
2961:
As in the jet bundle case, the Cartan distribution is important in the algebro-geometric approach to differential equations because it allows to encode solutions in purely geometric terms. Indeed, a submanifold
5887:, they all vanish locally. In order to extract much more and even local information, one thus needs to take the Cartan distribution into account and introduce a more sophisticated sequence. To this end, let
8121:
8010:
3309:
8617:{\displaystyle {\overline {H}}^{\bullet }({\mathcal {O}},\wedge ^{p}{\mathcal {V}}^{*})=H^{\bullet }({\overline {\Omega }}^{\bullet }({\mathcal {O}}\otimes \wedge ^{p}{\mathcal {V}}^{*}),{\overline {d}})}
3065:
8433:
4003:
3943:
6475:
4881:
4404:
4332:
4288:
9012:
2453:
7451:
4678:
6243:{\displaystyle {\mathcal {C}}\Omega ^{p}({\mathcal {O}}):=\{w\in \Omega ^{p}({\mathcal {O}})\mid w(X_{1},\cdots ,X_{p})=0\quad \forall ~X_{1},\ldots ,X_{p}\in {\mathcal {C}}({\mathcal {O}})\}.}
4189:
3393:
3146:
2809:
239:
6333:
5042:
4997:
4957:
4593:
4554:
7394:
2900:
952:
2073:
7789:, to formalise in cohomological terms the idea of a differential calculus on the space of solutions of a given system of PDEs (i.e. the space of integral manifolds of a given diffiety).
504:
7802:
6738:
5324:
3438:
4912:
4244:
4054:
2635:
187:
3610:
2990:
2863:
1237:
6522:
2704:
2298:
1590:
3722:
3889:
3753:
2210:
7963:
7926:
7277:
8381:{\displaystyle {\overline {H}}^{\bullet }({\mathcal {O}},{\mathcal {V}})=H^{\bullet }({\overline {\Omega }}^{\bullet }({\mathcal {O}}\otimes {\mathcal {V}}),{\overline {d}})}
8041:
8250:
7767:
7733:
7586:
7525:
7346:
6051:
6027:
5590:
5483:
5449:
5066:
4515:
3803:
3333:
3018:
7497:
5461:
associated to a diffiety, which can be used to investigate certain properties of the formal solution space of a differential equation by exploiting its Cartan distribution
1405:
3851:
3658:
3091:
1868:
7666:
7630:
7318:
1764:
1721:
7708:
7236:{\displaystyle \Omega ^{k}({\mathcal {O}})\supset {\mathcal {C}}^{1}\Omega ^{k}({\mathcal {O}})\supset \cdots \supset {\mathcal {C}}^{k+1}\Omega ^{k}({\mathcal {O}})=0,}
5329:
2388:
2139:
1814:
1280:
865:
752:
5786:
5159:
2009:
1124:
717:
7098:
2341:
1334:
636:
416:
1360:
7554:
6256:
5491:
5083:
4602:
4416:
344:
3779:
1485:
1439:
1532:
8457:
6495:
4352:
4209:
4023:
3433:
3413:
2940:
2920:
2833:
2757:
2737:
2655:
2319:
2250:
2230:
2159:
2097:
1842:
1681:
1661:
1641:
1617:
1509:
1459:
1380:
1309:
1185:
1164:
1144:
946:
926:
906:
886:
818:
798:
777:
679:
657:
611:
591:
568:
547:
525:
460:
436:
391:
368:
312:
1875:
8046:
5597:
4775:
4075:
5770:{\displaystyle C^{\infty }({\mathcal {O}})\longrightarrow \Omega ^{1}({\mathcal {O}})\longrightarrow \Omega ^{2}({\mathcal {O}})\longrightarrow \cdots }
7279:
of the diffiety. One can therefore analyse the terms of the spectral sequence order by order to recover information on the original PDE. For instance:
8899:"A spectral sequence associated with a nonlinear differential equation and algebro-geometric foundations of Lagrangian field theory with constraints"
6527:
6696:{\displaystyle \Omega ({\mathcal {O}})\supset {\mathcal {C}}\Omega ({\mathcal {O}})\supset {\mathcal {C}}^{2}\Omega ({\mathcal {O}})\supset \cdots }
10163:
8660:
9354:
5991:{\displaystyle {\mathcal {C}}\Omega ({\mathcal {O}})=\sum _{i\geq 0}{\mathcal {C}}\Omega ^{i}({\mathcal {O}})\subseteq \Omega ({\mathcal {O}})}
2599:{\displaystyle {\mathcal {C}}:J^{k}(E,m)\rightarrow TJ^{k}(E,m),\qquad \theta \mapsto {\mathcal {C}}_{\theta }\subset T_{\theta }(J^{k}(E,m))}
10158:
8898:
8645:
7786:
8220:{\displaystyle {\overline {H}}^{\bullet }({\mathcal {O}})=H^{\bullet }({\overline {\Omega }}^{\bullet }({\mathcal {O}}),{\overline {d}})}
3251:{\displaystyle {\mathcal {C}}({\mathcal {E}}):=\{{\mathcal {C}}_{\theta }\cap T_{\theta }({\mathcal {E}})~|~\theta \in {\mathcal {E}}\}}
9445:
7744:
determines two spectral sequences: one of the two spectral sequences determined by the variational bicomplex is exactly the
Vinogradov
7968:
9469:
5250:
4359:
3949:
guarantees that, under minor regularity assumptions, checking the smoothness of a finite number of prolongations is enough. Then the
3946:
3261:
9664:
7773:
Similarly to the terms of the spectral sequence, many terms of the variational bicomplex can be given a physical interpretation in
3023:
10211:
8739:
8391:
3956:
3894:
6433:
4842:
4365:
4293:
4249:
9534:
9157:
8995:
8930:
8824:
8757:
4765:{\displaystyle ({\mathcal {E}}^{\infty },{\mathcal {F}}({\mathcal {E}}^{\infty }),{\mathcal {C}}({\mathcal {E}}^{\infty }))}
9760:
7777:: for example, one obtains cohomology classes corresponding to action functionals, conserved currents, gauge charges, etc.
2397:
2392:
242:
7403:
9813:
9341:
17:
6420:{\displaystyle {\text{d}}({\mathcal {C}}\Omega ^{i}({\mathcal {O}}))\subset {\mathcal {C}}\Omega ^{i+1}({\mathcal {O}})}
139:
When dealing with differential equations, apart from applying algebraic operations as above, one has also the option to
10097:
3724:, so that their intersection is well-defined. However, such an intersection is not necessarily a manifold again, hence
4142:
3346:
3099:
2762:
192:
8789:
97:
5006:
4965:
4925:
4561:
4522:
9862:
7351:
2868:
7884:{\displaystyle {\overline {\Omega }}^{\bullet }({\mathcal {O}}):=\Gamma (\wedge ^{\bullet }{\mathcal {C(O)}}^{*})}
2018:
9845:
9454:
8667:
7770:-spectral sequence. However, the variational bicomplex was developed independently from the Vinogradov sequence.
8925:. A. V. Bocharov, I. S. Krasilʹshchik, A. M. Vinogradov. Providence, R.I.: American Mathematical Society. 1999.
10216:
6709:
5299:
4886:
4218:
4028:
2609:
161:
10057:
9464:
3569:
2965:
2838:
1191:
85:
466:
10206:
10042:
9765:
9539:
6500:
2661:
2255:
1537:
10087:
7397:
4290:
its Cartan distribution. Note that, unlike in the finite case, one can show that the Cartan distribution
3663:
9297:
3863:
3727:
2172:
10092:
10062:
9770:
9726:
9707:
9474:
9418:
8954:
7935:
7898:
7793:
objects are naturally endowed with the same algebraic structures of the space of the original objects.
7249:
5293:
9231:
8650:
7929:
2943:
1080:{\displaystyle J^{k}(E,m):=\{_{p}^{k}~|~p\in M,~{\text{dim}}(M)=m,M\subset E\ {\text{ submanifold}}\}}
262:
258:
9629:
9494:
8819:. Adv. Stud. Contemp. Math., N. Y. Vol. 1. New York etc.: Gordon and Breach Science Publishers.
8019:
96:, that is, to encode the space of solutions in a more conceptual way. The term was coined in 1984 by
9174:
9144:. Contemporary Mathematics. Vol. 219. Providence, Rhode Island: American Mathematical Society.
8842:"The Profinite Dimensional Manifold Structure of Formal Solution Spaces of Formally Integrable PDEs"
8231:
7748:
7714:
7567:
7506:
7327:
6032:
6008:
5571:
5464:
5430:
5047:
4496:
3784:
3314:
2999:
10014:
9879:
9571:
9413:
8388:, which is naturally a Lie algebra; moreover, it forms a graded Lie-Rinehart algebra together with
7458:
6600:
5395:{\displaystyle d_{\theta }\Phi ({\mathcal {C}}_{\theta })\subseteq {\mathcal {C}}_{\Phi (\theta )}}
506:
if one can locally describe both submanifolds as zeroes of functions defined in a neighbourhood of
246:
128:) model the space of solutions of a system of algebraic equations (i.e. the zero locus of a set of
3808:
3615:
3559:{\displaystyle {\mathcal {E}}^{k}:=J^{k}({\mathcal {E}},m)\cap J^{k+l}(E,m)\subseteq J^{k+l}(E,m)}
3070:
1847:
9711:
9681:
9605:
9595:
9551:
9381:
9334:
8633:
7635:
7603:
7285:
5876:{\displaystyle H^{i}({\mathcal {O}}):={\text{ker}}({\text{d}}_{i})/{\text{im}}({\text{d}}_{i-1})}
5240:{\displaystyle ({\mathcal {O}}',{\mathcal {F}}({\mathcal {O}}'),{\mathcal {C}}({\mathcal {O'}}))}
4355:
1733:
1694:
347:
9314:
9100:"The b-spectral sequence, Lagrangian formalism, and conservation laws. II. The nonlinear theory"
9067:
7671:
2351:
2102:
1777:
1243:
828:
722:
10052:
9671:
9566:
9479:
9386:
7774:
5407:
2951:
1688:
6319:{\displaystyle {\mathcal {C}}\Omega ^{i}({\mathcal {O}})\subseteq \Omega ^{i}({\mathcal {O}})}
5557:{\displaystyle ({\mathcal {O}},{\mathcal {F}}({\mathcal {O}}),{\mathcal {C}}({\mathcal {O}}))}
5149:{\displaystyle ({\mathcal {O}},{\mathcal {F}}({\mathcal {O}}),{\mathcal {C}}({\mathcal {O}}))}
4668:{\displaystyle ({\mathcal {O}},{\mathcal {F}}({\mathcal {O}}),{\mathcal {C}}({\mathcal {O}}))}
4482:{\displaystyle ({\mathcal {O}},{\mathcal {F}}({\mathcal {O}}),{\mathcal {C}}({\mathcal {O}}))}
2946:
and their solutions, which provide a coordinate-free way to describe the analogous notions of
1978:
1091:
684:
9701:
9696:
7737:
7077:
2947:
2326:
9068:"The b-spectral sequence, Lagrangian formalism, and conservation laws. I. The linear theory"
1339:
10032:
9970:
9818:
9522:
9512:
9484:
9459:
9369:
9253:
8863:
7557:
7532:
317:
254:
8744:. London Mathematical Society Lecture Note Series. Cambridge: Cambridge University Press.
8:
10170:
9852:
9730:
9715:
9644:
9403:
8970:
3805:
to be "nice enough" such that at least its first prolongation is indeed a submanifold of
3758:
1684:
1464:
1418:
1385:
133:
89:
10143:
9257:
8867:
1514:
1314:
616:
396:
10221:
10112:
10067:
9964:
9835:
9639:
9327:
9277:
9243:
9212:
9186:
8948:
8879:
8853:
8717:
8655:
8442:
6480:
6327:
4337:
4194:
4008:
3418:
3398:
2925:
2905:
2818:
2742:
2722:
2640:
2304:
2235:
2215:
2144:
2082:
1827:
1666:
1646:
1626:
1602:
1494:
1444:
1365:
1294:
1170:
1149:
1129:
931:
911:
891:
871:
803:
783:
762:
664:
642:
596:
576:
553:
532:
510:
445:
421:
376:
353:
297:
145:
121:
93:
9649:
9099:
9042:
245:. Elementary diffieties play the same role in the theory of differential equations as
10047:
10027:
10022:
9929:
9840:
9654:
9634:
9489:
9428:
9303:
9269:
9216:
9204:
9153:
9120:
9116:
9083:
9024:
8991:
8936:
8926:
8820:
8795:
8785:
8753:
8721:
8709:
6741:
5565:
5458:
2993:
1724:
250:
125:
9281:
8883:
8227:, which is naturally a commutative DG algebra (it is actually the first page of the
5884:
10185:
9979:
9934:
9857:
9828:
9686:
9619:
9614:
9609:
9599:
9391:
9374:
9265:
9261:
9196:
9145:
9112:
9079:
8983:
8922:
Symmetries and conservation laws for differential equations of mathematical physics
8871:
8745:
8701:
8104:{\displaystyle {\mathcal {V}}:=T{\mathcal {O}}/{\mathcal {C(O)}}\to {\mathcal {O}}}
7500:
7321:
5781:
5662:{\displaystyle \Omega ({\mathcal {O}}):=\sum _{i\geq 0}\Omega ^{i}({\mathcal {O}})}
4919:
4832:{\displaystyle ({\mathcal {E}}^{\infty },{\mathcal {C}}({\mathcal {E}}^{\infty }))}
4132:{\displaystyle ({\mathcal {E}}^{\infty },{\mathcal {C}}({\mathcal {E}}^{\infty }))}
1620:
45:
10128:
10037:
9867:
9823:
9589:
8968:
Tulczyjew, W. M. (1980). García, P. L.; Pérez-Rendón, A.; Souriau, J. M. (eds.).
7892:
5673:
2955:
593:
is a coordinate-invariant notion and an equivalence relation. One says also that
9994:
9919:
9889:
9787:
9780:
9720:
9691:
9561:
9556:
9517:
8781:
Cohomological analysis of partial differential equations and secondary calculus
8086:
8080:
7950:
7944:
7913:
7907:
7867:
7861:
6592:{\displaystyle w_{1}\wedge \cdots \wedge w_{k},~w_{i}\in {\mathcal {C}}\Omega }
9200:
8940:
8841:
8688:
1723:. Then the notion of jet of submanifolds boils down to the standard notion of
10200:
10180:
10004:
9999:
9984:
9974:
9924:
9901:
9775:
9735:
9676:
9624:
9423:
9273:
9208:
9124:
9028:
8749:
8713:
3950:
8799:
10107:
10102:
9944:
9911:
9884:
9792:
9433:
9138:
Henneaux, Marc; Krasil′shchik, Joseph; Vinogradov, Alexandre, eds. (1998).
8920:
8875:
4358:. However, due to the infinite-dimensionality of the ambient manifold, the
1593:
1488:
8632:, i.e. the solution space of the Euler-Lagrange equations associated to a
9950:
9939:
9896:
9797:
9398:
8779:
8629:
3311:
encodes the information about the solutions of the differential equation
371:
101:
35:
9309:
1966:{\displaystyle j^{k}(M):M\rightarrow J^{k}(E,m),\quad p\mapsto _{p}^{k}}
10175:
10133:
9959:
9872:
9504:
9408:
9319:
9149:
8987:
8705:
8013:
5883:
contain some structural information about the PDE; however, due to the
1728:
140:
129:
28:
8111:, its cohomology is used to define the following "secondary objects":
9989:
9954:
9659:
9546:
8846:
SIGMA. Symmetry, Integrability and
Geometry: Methods and Applications
7741:
6002:
2012:
1771:
1288:
10153:
10148:
9529:
9350:
8816:
Geometry of jet spaces and nonlinear partial differential equations
8812:
5296:
preserves the Cartan distribution, i.e. such that, for every point
3945:
are smooth surjective submersions. Note that a suitable version of
1767:
81:
9248:
9191:
8858:
7246:
so that the spectral sequence converges to the de Rham cohomology
8813:
Krasil'shchik, I. S.; Lychagin, V. V.; Vinogradov, A. M. (1986).
7561:
8814:
8005:{\displaystyle {\overline {\Omega }}^{\bullet }({\mathcal {O}})}
6740:
are stable, this filtration completely determines the following
249:
do in the theory of algebraic equations. Accordingly, just like
9745:
9139:
9137:
9013:"On variation bicomplexes associated to differential equations"
5285:{\displaystyle \Phi :{\mathcal {O}}\rightarrow {\mathcal {O}}'}
3304:{\displaystyle ({\mathcal {E}},{\mathcal {C}}({\mathcal {E}}))}
8083:
7947:
7910:
7864:
7074:
The filtration above is finite in each degree, i.e. for every
72:
8666:
Another way of generalizing ideas from algebraic geometry is
9175:"On the strong homotopy Lie–Rinehart algebra of a foliation"
6330:
since it is stable w.r.t. to the de Rham differential, i.e.
5044:, with a capital D (to distinguish it from the dimension of
3060:{\displaystyle T_{\theta }S\subset {\mathcal {C}}_{\theta }}
7592:
Many higher-order terms do not have an interpretation yet.
63:
60:
51:
8428:{\displaystyle {\overline {H}}^{\bullet }({\mathcal {O}})}
3998:{\displaystyle \{{\mathcal {E}}^{k}\}_{k\in \mathbb {N} }}
3938:{\displaystyle {\mathcal {E}}^{k}\to {\mathcal {E}}^{k-1}}
7736:-spectral sequence one obtains the slightly less general
6470:{\displaystyle {\mathcal {C}}^{k}\Omega ({\mathcal {O}})}
241:, together with an extra structure provided by a special
84:
object which plays the same role in the modern theory of
8976:
Differential
Geometrical Methods in Mathematical Physics
8628:
Secondary calculus can also be related to the covariant
4876:{\displaystyle {\mathcal {F}}({\mathcal {E}}^{\infty })}
4399:{\displaystyle {\mathcal {C}}({\mathcal {E}}^{\infty })}
4327:{\displaystyle {\mathcal {C}}({\mathcal {E}}^{\infty })}
4283:{\displaystyle {\mathcal {C}}({\mathcal {E}}^{\infty })}
4005:
extends the definition of prolongation to the case when
1188:, one can use such functions to build local coordinates
7600:
As a particular case, starting with a fibred manifold
3096:
One can also look at the Cartan distribution of a PDE
8467:
8445:
8394:
8265:
8234:
8124:
8049:
8022:
7971:
7938:
7901:
7805:
7751:
7717:
7674:
7638:
7606:
7570:
7535:
7509:
7461:
7406:
7354:
7330:
7288:
7252:
7108:
7080:
6753:
6712:
6611:
6530:
6503:
6483:
6436:
6336:
6259:
6062:
6035:
6011:
5896:
5789:
5685:
5600:
5574:
5494:
5467:
5433:
5332:
5302:
5253:
5162:
5086:
5050:
5009:
4968:
4928:
4889:
4845:
4778:
4681:
4605:
4564:
4525:
4499:
4419:
4368:
4340:
4296:
4252:
4221:
4197:
4145:
4078:
4031:
4011:
3959:
3897:
3866:
3811:
3787:
3761:
3730:
3666:
3618:
3572:
3441:
3421:
3401:
3349:
3317:
3264:
3154:
3102:
3073:
3026:
3002:
2968:
2928:
2908:
2871:
2841:
2821:
2765:
2745:
2725:
2664:
2643:
2612:
2461:
2448:{\displaystyle {\mathcal {C}}\subseteq T(J^{k}(E,m))}
2400:
2354:
2329:
2307:
2258:
2238:
2218:
2175:
2147:
2105:
2085:
2021:
1981:
1878:
1850:
1830:
1780:
1736:
1697:
1669:
1649:
1629:
1605:
1540:
1517:
1497:
1467:
1447:
1421:
1388:
1368:
1342:
1317:
1297:
1246:
1194:
1173:
1152:
1132:
1126:
is locally determined by the derivatives up to order
1094:
955:
934:
914:
894:
874:
831:
806:
786:
765:
725:
687:
667:
645:
619:
599:
579:
556:
535:
513:
469:
448:
424:
399:
379:
356:
320:
300:
195:
164:
66:
57:
7446:{\displaystyle {\text{d}}_{1}^{0,n}{\mathcal {L}}=0}
5405:
Diffieties together with their morphisms define the
2164:
69:
48:
8784:. Providence, R.I.: American Mathematical Society.
54:
9232:"Secondary calculus and the covariant phase space"
9098:
8969:
8687:
8616:
8451:
8427:
8380:
8244:
8219:
8103:
8043:. Then, possibly tensoring with the normal bundle
8035:
8004:
7957:
7920:
7883:
7761:
7727:
7702:
7660:
7624:
7580:
7548:
7519:
7491:
7445:
7388:
7340:
7312:
7271:
7235:
7092:
7063:
6732:
6695:
6591:
6516:
6489:
6469:
6419:
6318:
6242:
6045:
6021:
5990:
5875:
5769:
5661:
5584:
5556:
5477:
5443:
5394:
5318:
5284:
5239:
5148:
5060:
5036:
4991:
4951:
4918:means a suitable localisation with respect to the
4906:
4875:
4831:
4764:
4667:
4587:
4548:
4509:
4481:
4398:
4346:
4326:
4282:
4238:
4203:
4183:
4131:
4048:
4017:
3997:
3937:
3883:
3845:
3797:
3773:
3747:
3716:
3652:
3604:
3558:
3427:
3407:
3387:
3327:
3303:
3250:
3140:
3085:
3059:
3012:
2984:
2934:
2914:
2894:
2857:
2827:
2803:
2751:
2731:
2698:
2649:
2629:
2598:
2447:
2382:
2335:
2313:
2292:
2244:
2224:
2204:
2153:
2133:
2091:
2067:
2003:
1965:
1862:
1836:
1808:
1758:
1715:
1675:
1655:
1635:
1611:
1584:
1526:
1503:
1479:
1453:
1433:
1399:
1374:
1354:
1328:
1303:
1274:
1231:
1179:
1158:
1138:
1118:
1079:
940:
920:
900:
880:
859:
812:
792:
771:
746:
711:
673:
651:
630:
605:
585:
562:
541:
519:
498:
454:
430:
410:
385:
362:
338:
306:
233:
181:
9105:Journal of Mathematical Analysis and Applications
9072:Journal of Mathematical Analysis and Applications
7891:of a diffiety, which can be seen as the leafwise
10198:
4184:{\displaystyle {\mathcal {E}}\subset J^{k}(E,m)}
3388:{\displaystyle {\mathcal {E}}\subset J^{l}(E,m)}
3141:{\displaystyle {\mathcal {E}}\subset J^{k}(E,m)}
2804:{\displaystyle {\mathcal {E}}\subset J^{k}(E,m)}
1819:
234:{\displaystyle {\mathcal {E}}\subset J^{k}(E,m)}
8840:Güneysu, Batu; Pflaum, Markus J. (2017-01-10).
8661:Differential calculus over commutative algebras
8624:, which is naturally a commutative DG algebra.
5037:{\displaystyle \mathrm {Dim} ({\mathcal {O}})}
4992:{\displaystyle {\mathcal {C}}({\mathcal {O}})}
4952:{\displaystyle {\mathcal {F}}({\mathcal {O}})}
4588:{\displaystyle {\mathcal {C}}({\mathcal {O}})}
4549:{\displaystyle {\mathcal {F}}({\mathcal {O}})}
9335:
8839:
8651:Partial differential equations on Jet bundles
7389:{\displaystyle {\mathcal {L}}\in E_{1}^{0,n}}
2944:partial differential equations on jet bundles
2895:{\displaystyle S^{k}\subseteq {\mathcal {E}}}
1283:with a natural structure of smooth manifold.
285:
136:generated by the initial set of polynomials.
9141:Secondary Calculus and Cohomological Physics
8646:Secondary calculus and cohomological physics
6830:
6780:
6234:
6096:
4493:a (generally infinite-dimensional) manifold
3978:
3960:
3245:
3178:
2068:{\displaystyle M^{k}:={\text{im}}(j^{k}(M))}
1074:
984:
573:One can show that being tangent up to order
4883:denotes the algebra of smooth functions on
4212:
4063:
9342:
9328:
9229:
9179:Communications in Contemporary Mathematics
9172:
9096:
9065:
9010:
8896:
8777:
8685:
5071:
9247:
9190:
8967:
8857:
6733:{\displaystyle {\mathcal {C}}^{k}\Omega }
5319:{\displaystyle \theta \in {\mathcal {O}}}
3989:
3891:are smooth manifolds and all projections
2710:
681:, and denotes their equivalence class by
9349:
8737:
8689:"Local symmetries and conservation laws"
7785:Vinogradov developed a theory, known as
7595:
4907:{\displaystyle {\mathcal {E}}^{\infty }}
4239:{\displaystyle {\mathcal {E}}^{\infty }}
4049:{\displaystyle {\mathcal {E}}^{\infty }}
3338:
2630:{\displaystyle {\mathcal {C}}_{\theta }}
1287:
182:{\displaystyle {\mathcal {E}}^{\infty }}
8982:. Berlin, Heidelberg: Springer: 22–48.
6497:-th power, i.e. the linear subspace of
3660:are viewed as embedded submanifolds of
3605:{\displaystyle J^{k}({\mathcal {E}},m)}
2985:{\displaystyle S\subset {\mathcal {E}}}
2858:{\displaystyle S\subset {\mathcal {E}}}
1232:{\displaystyle (x^{i},u_{\sigma }^{j})}
115:
14:
10199:
8016:together with a suitable differential
6029:whose restriction to the distribution
5420:
2992:is a solution if and only if it is an
2252:whose prolongation contains the point
1534:. More generally, all the projections
499:{\displaystyle p\in M\cap M'\subset E}
9323:
7780:
6517:{\displaystyle {\mathcal {C}}\Omega }
3947:Cartan–Kuranishi prolongation theorem
3856:Below we will assume that the PDE is
2699:{\displaystyle \theta \in J^{k}(E,m)}
2293:{\displaystyle \theta \in J^{k}(E,m)}
8773:
8771:
8769:
8733:
8731:
8253:-spectral sequence discussed above);
4519:the algebra of its smooth functions
1585:{\displaystyle J^{k}(E)\to J^{k-1}E}
276:
3717:{\displaystyle J^{k}(J^{l}(E,m),m)}
2323:(or jet plane, or Cartan plane) at
1844:-jet prolongation of a submanifold
1663:, one can consider submanifolds of
144:one needs therefore to work with a
24:
9310:Geometry of Differential Equations
8909:: 144–148 – via Math-Net.Ru.
8584:
8563:
8544:
8511:
8490:
8417:
8354:
8344:
8325:
8298:
8288:
8237:
8193:
8174:
8147:
8096:
8077:
8065:
8052:
7994:
7975:
7941:
7904:
7858:
7839:
7828:
7809:
7754:
7720:
7573:
7512:
7432:
7357:
7333:
7261:
7216:
7202:
7184:
7164:
7150:
7138:
7124:
7110:
6953:
6933:
6915:
6902:
6882:
6870:
6769:
6756:
6727:
6716:
6679:
6671:
6660:
6646:
6638:
6633:
6620:
6612:
6586:
6581:
6511:
6506:
6459:
6451:
6440:
6409:
6389:
6383:
6367:
6353:
6347:
6308:
6294:
6282:
6268:
6262:
6226:
6216:
6176:
6120:
6106:
6085:
6071:
6065:
6038:
6014:
5980:
5972:
5961:
5947:
5941:
5912:
5904:
5899:
5805:
5753:
5739:
5727:
5713:
5701:
5691:
5651:
5637:
5609:
5601:
5577:
5543:
5533:
5520:
5510:
5500:
5470:
5436:
5378:
5372:
5352:
5343:
5311:
5273:
5262:
5254:
5222:
5211:
5194:
5183:
5169:
5135:
5125:
5112:
5102:
5092:
5053:
5026:
5017:
5014:
5011:
4981:
4971:
4941:
4931:
4899:
4893:
4865:
4859:
4848:
4818:
4812:
4801:
4791:
4785:
4751:
4745:
4734:
4721:
4715:
4704:
4694:
4688:
4654:
4644:
4631:
4621:
4611:
4577:
4567:
4558:a finite-dimensional distribution
4538:
4528:
4502:
4468:
4458:
4445:
4435:
4425:
4388:
4382:
4371:
4316:
4310:
4299:
4272:
4266:
4255:
4231:
4225:
4148:
4118:
4112:
4101:
4091:
4085:
4041:
4035:
3966:
3918:
3901:
3884:{\displaystyle {\mathcal {E}}^{k}}
3870:
3790:
3748:{\displaystyle {\mathcal {E}}^{k}}
3734:
3588:
3474:
3445:
3352:
3320:
3290:
3280:
3270:
3240:
3213:
3184:
3167:
3157:
3105:
3046:
3005:
2977:
2887:
2850:
2768:
2616:
2541:
2464:
2403:
2205:{\displaystyle T_{\theta }(M^{k})}
198:
174:
168:
25:
10233:
9291:
8766:
8728:
7958:{\displaystyle {\mathcal {C(O)}}}
7921:{\displaystyle {\mathcal {C(O)}}}
7272:{\displaystyle H({\mathcal {O}})}
3781:. One therefore usually requires
2165:Cartan distribution on jet spaces
265:, one defines a (non-elementary)
98:Alexandre Mikhailovich Vinogradov
9097:Vinogradov, A. M. (1984-04-30).
8978:. Lecture Notes in Mathematics.
8686:Vinogradov, A. M. (March 1984).
8012:becomes naturally a commutative
4025:goes to infinity, and the space
3755:may not be an equation of order
44:
9236:Journal of Geometry and Physics
9230:Vitagliano, Luca (2009-04-01).
9223:
9166:
9131:
9090:
9066:Vinogradov, A.M. (1984-04-30).
9059:
9043:"variational bicomplex in nLab"
9035:
8971:"The Euler-Lagrange resolution"
8668:differential algebraic geometry
8461:are elements of the cohomology
8259:are elements of the cohomology
8118:are elements of the cohomology
8036:{\displaystyle {\overline {d}}}
7895:of the involutive distribution
6973:
6967:
6839:
6833:
6175:
5415:
2531:
1935:
868:, is defined as the set of all
10212:Partial differential equations
9382:Differentiable/Smooth manifold
9315:Differential Geometry and PDEs
9266:10.1016/j.geomphys.2008.12.001
9004:
8961:
8913:
8890:
8833:
8806:
8679:
8611:
8595:
8558:
8538:
8522:
8485:
8422:
8412:
8375:
8359:
8339:
8319:
8303:
8283:
8245:{\displaystyle {\mathcal {C}}}
8214:
8198:
8188:
8168:
8152:
8142:
8091:
7999:
7989:
7878:
7842:
7833:
7823:
7762:{\displaystyle {\mathcal {C}}}
7728:{\displaystyle {\mathcal {C}}}
7697:
7685:
7655:
7649:
7616:
7581:{\displaystyle {\mathcal {E}}}
7520:{\displaystyle {\mathcal {E}}}
7341:{\displaystyle {\mathcal {E}}}
7266:
7256:
7221:
7211:
7169:
7159:
7129:
7119:
7055:
7007:
6958:
6948:
6907:
6897:
6774:
6764:
6684:
6674:
6651:
6641:
6625:
6615:
6464:
6454:
6414:
6404:
6375:
6372:
6362:
6342:
6313:
6303:
6287:
6277:
6231:
6221:
6166:
6134:
6125:
6115:
6090:
6080:
6046:{\displaystyle {\mathcal {C}}}
6022:{\displaystyle {\mathcal {O}}}
5985:
5975:
5966:
5956:
5917:
5907:
5870:
5849:
5836:
5821:
5810:
5800:
5761:
5758:
5748:
5735:
5732:
5722:
5709:
5706:
5696:
5656:
5646:
5614:
5604:
5585:{\displaystyle {\mathcal {O}}}
5551:
5548:
5538:
5525:
5515:
5495:
5478:{\displaystyle {\mathcal {C}}}
5444:{\displaystyle {\mathcal {C}}}
5387:
5381:
5363:
5346:
5267:
5234:
5231:
5216:
5203:
5188:
5163:
5143:
5140:
5130:
5117:
5107:
5087:
5061:{\displaystyle {\mathcal {O}}}
5031:
5021:
4986:
4976:
4946:
4936:
4870:
4853:
4839:is an elementary diffiety and
4826:
4823:
4806:
4779:
4759:
4756:
4739:
4726:
4709:
4682:
4662:
4659:
4649:
4636:
4626:
4606:
4582:
4572:
4543:
4533:
4510:{\displaystyle {\mathcal {O}}}
4476:
4473:
4463:
4450:
4440:
4420:
4393:
4376:
4321:
4304:
4277:
4260:
4246:its infinite prolongation and
4178:
4166:
4126:
4123:
4106:
4079:
3912:
3840:
3828:
3798:{\displaystyle {\mathcal {E}}}
3711:
3702:
3690:
3677:
3647:
3635:
3599:
3583:
3553:
3541:
3519:
3507:
3485:
3469:
3435:-th prolongation is defined as
3382:
3370:
3343:Given a differential equation
3328:{\displaystyle {\mathcal {E}}}
3298:
3295:
3285:
3265:
3225:
3218:
3208:
3172:
3162:
3135:
3123:
3013:{\displaystyle {\mathcal {C}}}
2798:
2786:
2693:
2681:
2593:
2590:
2578:
2565:
2535:
2525:
2513:
2497:
2494:
2482:
2442:
2439:
2427:
2414:
2377:
2365:
2287:
2275:
2199:
2186:
2128:
2116:
2062:
2059:
2053:
2040:
1998:
1992:
1949:
1942:
1939:
1929:
1917:
1904:
1895:
1889:
1803:
1791:
1753:
1747:
1707:
1560:
1557:
1551:
1269:
1257:
1226:
1195:
1102:
1095:
1045:
1039:
1012:
994:
987:
978:
966:
854:
842:
695:
688:
333:
321:
228:
216:
155:will consist therefore of the
86:partial differential equations
13:
1:
8694:Acta Applicandae Mathematicae
8673:
7796:More precisely, consider the
7492:{\displaystyle E_{1}^{0,n-1}}
4922:corresponding to the algebra
2942:, one recovers the notion of
1820:Prolongations of submanifolds
908:-dimensional submanifolds of
9117:10.1016/0022-247X(84)90072-6
9084:10.1016/0022-247X(84)90071-4
9017:Osaka Journal of Mathematics
8606:
8547:
8474:
8401:
8370:
8328:
8272:
8209:
8177:
8131:
8028:
7978:
7812:
3846:{\displaystyle J^{k+1}(E,m)}
3653:{\displaystyle J^{k+l}(E,m)}
3148:more intrinsically, defining
3086:{\displaystyle \theta \in S}
1863:{\displaystyle M\subseteq E}
1441:one recovers just points in
1146:of the functions describing
257:are composed of irreducible
7:
10088:Classification of manifolds
8741:The Geometry of Jet Bundles
8639:
7661:{\displaystyle J^{k}(\pi )}
7625:{\displaystyle \pi :E\to X}
7313:{\displaystyle E_{1}^{0,n}}
6005:of differential forms over
1759:{\displaystyle J^{k}(\pi )}
1716:{\displaystyle \pi :E\to X}
1599:As a particular case, when
189:of a differential equation
124:the main objects of study (
10:
10238:
8897:Vinogradov, A. M. (1978).
8778:Vinogradov, A. M. (2001).
7798:horizontal De Rham complex
7703:{\displaystyle J^{k}(E,m)}
5564:, consider the algebra of
2922:is a fibred manifold over
2383:{\displaystyle J^{k}(E,m)}
2134:{\displaystyle J^{k}(E,m)}
1809:{\displaystyle J^{k}(E,m)}
1511:-dimensional subspaces of
1275:{\displaystyle J^{k}(E,m)}
860:{\displaystyle J^{k}(E,m)}
747:{\displaystyle j_{p}^{k}M}
286:Jet spaces of submanifolds
247:affine algebraic varieties
26:
10164:over commutative algebras
10121:
10080:
10013:
9910:
9806:
9753:
9744:
9580:
9503:
9442:
9362:
9306:(successor of above site)
9304:The Levi-Civita Institute
9201:10.1142/S0219199714500072
9173:Vitagliano, Luca (2014).
9011:Tsujishita, Toru (1982).
7668:instead of the jet space
5410:of differential equations
5247:consists of a smooth map
5001:dimension of the diffiety
4362:does not hold, therefore
3860:, i.e. all prolongations
2835:-dimensional submanifold
9880:Riemann curvature tensor
8750:10.1017/cbo9780511526411
8738:Saunders, D. J. (1989).
4064:Definition of a diffiety
3258:In this sense, the pair
2004:{\displaystyle j^{k}(M)}
1119:{\displaystyle _{p}^{k}}
712:{\displaystyle _{p}^{k}}
273:an elementary diffiety.
27:Not to be confused with
8634:Lagrangian field theory
8439:secondary differential
8257:secondary vector fields
7398:Euler-Lagrange equation
7324:constrained by the PDE
7093:{\displaystyle k\geq 0}
5080:between two diffieties
5072:Morphisms of diffieties
4675:is locally of the form
4056:has the structure of a
2952:Lagrangian submanifolds
2336:{\displaystyle \theta }
1336:have the same 1-jet at
528:, whose derivatives at
9672:Manifold with boundary
9387:Differential structure
9298:The Diffiety Institute
8953:: CS1 maint: others (
8876:10.3842/SIGMA.2017.003
8618:
8453:
8429:
8382:
8246:
8221:
8105:
8037:
8006:
7959:
7928:or, equivalently, the
7922:
7885:
7775:classical field theory
7763:
7740:. More precisely, any
7729:
7704:
7662:
7626:
7582:
7558:characteristic classes
7550:
7521:
7493:
7447:
7390:
7342:
7314:
7273:
7237:
7094:
7065:
6734:
6697:
6593:
6518:
6491:
6471:
6421:
6320:
6244:
6047:
6023:
5992:
5877:
5771:
5672:and the corresponding
5663:
5586:
5558:
5479:
5445:
5396:
5320:
5286:
5241:
5150:
5062:
5038:
4993:
4953:
4908:
4877:
4833:
4766:
4669:
4589:
4550:
4511:
4483:
4400:
4348:
4328:
4284:
4240:
4205:
4185:
4133:
4050:
4019:
3999:
3939:
3885:
3847:
3799:
3775:
3749:
3718:
3654:
3606:
3560:
3429:
3409:
3389:
3329:
3305:
3252:
3142:
3087:
3061:
3014:
2986:
2936:
2916:
2896:
2859:
2829:
2805:
2753:
2733:
2711:Differential equations
2700:
2651:
2631:
2600:
2449:
2384:
2337:
2315:
2294:
2246:
2232:is any submanifold of
2226:
2206:
2155:
2135:
2099:, is a submanifold of
2093:
2069:
2005:
1967:
1864:
1838:
1810:
1760:
1717:
1677:
1657:
1643:-dimensional manifold
1637:
1613:
1586:
1528:
1505:
1481:
1455:
1435:
1408:
1401:
1376:
1356:
1355:{\displaystyle p\in E}
1330:
1305:
1276:
1233:
1181:
1160:
1140:
1120:
1081:
942:
922:
902:
882:
861:
814:
794:
773:
748:
713:
675:
653:
632:
607:
587:
564:
543:
521:
500:
456:
432:
412:
387:
364:
340:
308:
235:
183:
10217:Differential geometry
8619:
8454:
8430:
8383:
8247:
8222:
8106:
8038:
8007:
7960:
7932:of the Lie algebroid
7930:Lie algebroid complex
7923:
7886:
7764:
7738:variational bicomplex
7730:
7705:
7663:
7627:
7596:Variational bicomplex
7583:
7551:
7549:{\displaystyle E_{2}}
7522:
7494:
7448:
7391:
7348:. In particular, for
7343:
7315:
7274:
7238:
7095:
7066:
6735:
6706:and since all ideals
6698:
6599:. Then one obtains a
6594:
6519:
6492:
6472:
6422:
6321:
6245:
6048:
6024:
5993:
5878:
5772:
5664:
5587:
5559:
5480:
5446:
5397:
5321:
5287:
5242:
5151:
5063:
5039:
4994:
4954:
4909:
4878:
4834:
4767:
4670:
4590:
4551:
4512:
4484:
4401:
4349:
4329:
4285:
4241:
4213:differential equation
4206:
4186:
4134:
4058:profinite-dimensional
4051:
4020:
4000:
3940:
3886:
3848:
3800:
3776:
3750:
3719:
3655:
3607:
3561:
3430:
3410:
3390:
3339:Prolongations of PDEs
3330:
3306:
3253:
3143:
3088:
3062:
3015:
2987:
2948:mathematical analysis
2937:
2917:
2897:
2860:
2830:
2806:
2754:
2734:
2717:differential equation
2701:
2652:
2632:
2601:
2450:
2385:
2338:
2316:
2295:
2247:
2227:
2207:
2156:
2136:
2094:
2070:
2006:
1968:
1865:
1839:
1811:
1761:
1727:of sections, and the
1718:
1678:
1658:
1638:
1614:
1587:
1529:
1506:
1482:
1456:
1436:
1402:
1377:
1357:
1331:
1306:
1291:
1277:
1234:
1182:
1161:
1141:
1121:
1082:
943:
923:
903:
883:
862:
815:
795:
774:
749:
714:
676:
654:
633:
608:
588:
565:
544:
522:
501:
457:
433:
413:
388:
365:
341:
339:{\displaystyle (m+e)}
309:
236:
184:
157:infinite prolongation
9819:Covariant derivative
9370:Topological manifold
8465:
8443:
8392:
8263:
8232:
8122:
8047:
8020:
7969:
7936:
7899:
7803:
7749:
7715:
7672:
7636:
7604:
7568:
7533:
7507:
7459:
7404:
7396:, the corresponding
7352:
7328:
7286:
7250:
7106:
7078:
6751:
6710:
6609:
6528:
6501:
6481:
6434:
6334:
6257:
6060:
6033:
6009:
5894:
5787:
5683:
5598:
5572:
5492:
5465:
5431:
5330:
5300:
5251:
5160:
5084:
5048:
5007:
4966:
4926:
4887:
4843:
4776:
4679:
4603:
4562:
4523:
4497:
4417:
4366:
4338:
4294:
4250:
4219:
4195:
4143:
4076:
4029:
4009:
3957:
3895:
3864:
3809:
3785:
3759:
3728:
3664:
3616:
3570:
3439:
3419:
3399:
3347:
3315:
3262:
3152:
3100:
3071:
3024:
3000:
2966:
2926:
2906:
2869:
2839:
2819:
2815:is defined to be an
2763:
2743:
2723:
2662:
2641:
2610:
2459:
2398:
2352:
2327:
2305:
2256:
2236:
2216:
2173:
2169:A space of the form
2145:
2103:
2083:
2019:
1979:
1876:
1848:
1828:
1778:
1734:
1695:
1667:
1647:
1627:
1603:
1538:
1515:
1495:
1465:
1445:
1419:
1407:have the same 3-jet.
1386:
1366:
1340:
1315:
1295:
1244:
1192:
1171:
1150:
1130:
1092:
953:
932:
912:
892:
872:
829:
804:
784:
763:
723:
685:
665:
643:
617:
597:
577:
554:
533:
511:
467:
446:
422:
397:
377:
354:
318:
298:
193:
162:
116:Intuitive definition
10207:Homological algebra
9853:Exterior derivative
9455:Atiyah–Singer index
9404:Riemannian manifold
9300:(frozen since 2010)
9258:2009JGP....59..426V
8868:2017SIGMA..13..003G
8116:secondary functions
7965:. Then the complex
7632:and its jet bundle
7488:
7429:
7385:
7309:
7054:
7030:
7000:
6860:
6829:
6803:
5455:Vinogradov sequence
5421:Vinogradov sequence
5003:and its denoted by
4070:elementary diffiety
3858:formally integrable
3774:{\displaystyle k+l}
2637:is the span of all
2345:Cartan distribution
2079:of the submanifold
1962:
1766:turns out to be an
1619:has a structure of
1480:{\displaystyle k=1}
1434:{\displaystyle k=1}
1400:{\displaystyle M''}
1225:
1115:
1007:
740:
708:
440:tangent up to order
153:elementary diffiety
94:algebraic equations
90:algebraic varieties
18:Vinogradov sequence
10159:Secondary calculus
10113:Singularity theory
10068:Parallel transport
9836:De Rham cohomology
9475:Generalized Stokes
8988:10.1007/BFb0089725
8903:Soviet Math. Dokl.
8706:10.1007/BF01405491
8656:Differential ideal
8614:
8449:
8425:
8378:
8242:
8217:
8101:
8033:
8002:
7955:
7918:
7881:
7787:secondary calculus
7781:Secondary calculus
7759:
7725:
7700:
7658:
7622:
7578:
7556:is interpreted as
7546:
7517:
7489:
7462:
7443:
7407:
7386:
7365:
7338:
7322:action functionals
7310:
7289:
7269:
7233:
7090:
7061:
7034:
7010:
6974:
6840:
6807:
6783:
6730:
6693:
6589:
6514:
6487:
6467:
6417:
6328:differential ideal
6316:
6240:
6043:
6019:
5988:
5938:
5873:
5767:
5659:
5635:
5582:
5566:differential forms
5554:
5475:
5451:-spectral sequence
5441:
5392:
5316:
5282:
5237:
5146:
5058:
5034:
4989:
4949:
4904:
4873:
4829:
4762:
4665:
4585:
4546:
4507:
4479:
4406:is not integrable
4396:
4344:
4324:
4280:
4236:
4215:on some manifold,
4201:
4181:
4129:
4046:
4015:
3995:
3935:
3881:
3843:
3795:
3771:
3745:
3714:
3650:
3602:
3556:
3425:
3405:
3385:
3325:
3301:
3248:
3138:
3083:
3057:
3010:
2982:
2932:
2912:
2892:
2855:
2825:
2801:
2749:
2729:
2696:
2647:
2627:
2596:
2445:
2380:
2333:
2311:
2290:
2242:
2222:
2202:
2151:
2131:
2089:
2065:
2001:
1963:
1948:
1860:
1834:
1806:
1756:
1713:
1673:
1653:
1633:
1609:
1582:
1527:{\displaystyle TE}
1524:
1501:
1477:
1451:
1431:
1415:For instance, for
1409:
1397:
1372:
1352:
1329:{\displaystyle M'}
1326:
1301:
1272:
1229:
1211:
1177:
1156:
1136:
1116:
1101:
1077:
993:
938:
918:
898:
878:
857:
810:
790:
769:
744:
726:
709:
694:
671:
649:
631:{\displaystyle M'}
628:
603:
583:
560:
550:agree up to order
539:
517:
496:
452:
428:
411:{\displaystyle M'}
408:
383:
360:
336:
304:
271:locally looks like
269:as an object that
231:
179:
146:differential ideal
122:algebraic geometry
10194:
10193:
10076:
10075:
9841:Differential form
9495:Whitney embedding
9429:Differential form
9159:978-0-8218-0828-3
8997:978-3-540-38405-2
8932:978-1-4704-4596-6
8826:978-2-88124-051-5
8759:978-0-521-36948-0
8609:
8550:
8477:
8452:{\displaystyle p}
8404:
8373:
8331:
8275:
8212:
8180:
8134:
8031:
7981:
7815:
7710:, instead of the
7503:for solutions of
7501:conservation laws
7411:
6971:
6962:
6837:
6811:
6742:spectral sequence
6565:
6490:{\displaystyle k}
6340:
6181:
5923:
5856:
5847:
5828:
5819:
5782:cohomology groups
5620:
5488:Given a diffiety
5459:spectral sequence
4962:The dimension of
4360:Frobenius theorem
4354:-dimensional and
4347:{\displaystyle m}
4204:{\displaystyle k}
4018:{\displaystyle k}
3428:{\displaystyle k}
3408:{\displaystyle l}
3231:
3223:
2994:integral manifold
2935:{\displaystyle X}
2915:{\displaystyle E}
2828:{\displaystyle m}
2759:is a submanifold
2752:{\displaystyle E}
2732:{\displaystyle k}
2650:{\displaystyle R}
2347:on the jet space
2314:{\displaystyle R}
2245:{\displaystyle E}
2225:{\displaystyle M}
2154:{\displaystyle M}
2141:diffeomorphic to
2092:{\displaystyle M}
2038:
1837:{\displaystyle k}
1676:{\displaystyle E}
1656:{\displaystyle X}
1636:{\displaystyle n}
1612:{\displaystyle E}
1504:{\displaystyle n}
1487:one recovers the
1454:{\displaystyle E}
1413:
1412:
1375:{\displaystyle M}
1304:{\displaystyle M}
1180:{\displaystyle p}
1159:{\displaystyle M}
1139:{\displaystyle k}
1088:As any given jet
1072:
1071: submanifold
1068:
1037:
1033:
1018:
1010:
941:{\displaystyle E}
928:at all points of
921:{\displaystyle E}
901:{\displaystyle m}
881:{\displaystyle k}
813:{\displaystyle E}
800:-submanifolds of
793:{\displaystyle k}
772:{\displaystyle k}
674:{\displaystyle p}
652:{\displaystyle k}
606:{\displaystyle M}
586:{\displaystyle k}
563:{\displaystyle k}
542:{\displaystyle p}
520:{\displaystyle p}
455:{\displaystyle k}
431:{\displaystyle E}
386:{\displaystyle M}
363:{\displaystyle m}
307:{\displaystyle E}
277:Formal definition
16:(Redirected from
10229:
10186:Stratified space
10144:Fréchet manifold
9858:Interior product
9751:
9750:
9448:
9344:
9337:
9330:
9321:
9320:
9286:
9285:
9251:
9227:
9221:
9220:
9194:
9170:
9164:
9163:
9150:10.1090/conm/219
9135:
9129:
9128:
9102:
9094:
9088:
9087:
9063:
9057:
9056:
9054:
9053:
9039:
9033:
9032:
9008:
9002:
9001:
8973:
8965:
8959:
8958:
8952:
8944:
8917:
8911:
8910:
8894:
8888:
8887:
8861:
8837:
8831:
8830:
8810:
8804:
8803:
8775:
8764:
8763:
8735:
8726:
8725:
8691:
8683:
8623:
8621:
8620:
8615:
8610:
8602:
8594:
8593:
8588:
8587:
8580:
8579:
8567:
8566:
8557:
8556:
8551:
8543:
8537:
8536:
8521:
8520:
8515:
8514:
8507:
8506:
8494:
8493:
8484:
8483:
8478:
8470:
8458:
8456:
8455:
8450:
8434:
8432:
8431:
8426:
8421:
8420:
8411:
8410:
8405:
8397:
8387:
8385:
8384:
8379:
8374:
8366:
8358:
8357:
8348:
8347:
8338:
8337:
8332:
8324:
8318:
8317:
8302:
8301:
8292:
8291:
8282:
8281:
8276:
8268:
8251:
8249:
8248:
8243:
8241:
8240:
8226:
8224:
8223:
8218:
8213:
8205:
8197:
8196:
8187:
8186:
8181:
8173:
8167:
8166:
8151:
8150:
8141:
8140:
8135:
8127:
8110:
8108:
8107:
8102:
8100:
8099:
8090:
8089:
8074:
8069:
8068:
8056:
8055:
8042:
8040:
8039:
8034:
8032:
8024:
8011:
8009:
8008:
8003:
7998:
7997:
7988:
7987:
7982:
7974:
7964:
7962:
7961:
7956:
7954:
7953:
7927:
7925:
7924:
7919:
7917:
7916:
7890:
7888:
7887:
7882:
7877:
7876:
7871:
7870:
7854:
7853:
7832:
7831:
7822:
7821:
7816:
7808:
7768:
7766:
7765:
7760:
7758:
7757:
7734:
7732:
7731:
7726:
7724:
7723:
7709:
7707:
7706:
7701:
7684:
7683:
7667:
7665:
7664:
7659:
7648:
7647:
7631:
7629:
7628:
7623:
7587:
7585:
7584:
7579:
7577:
7576:
7564:of solutions of
7555:
7553:
7552:
7547:
7545:
7544:
7526:
7524:
7523:
7518:
7516:
7515:
7498:
7496:
7495:
7490:
7487:
7470:
7452:
7450:
7449:
7444:
7436:
7435:
7428:
7417:
7412:
7409:
7395:
7393:
7392:
7387:
7384:
7373:
7361:
7360:
7347:
7345:
7344:
7339:
7337:
7336:
7319:
7317:
7316:
7311:
7308:
7297:
7278:
7276:
7275:
7270:
7265:
7264:
7242:
7240:
7239:
7234:
7220:
7219:
7210:
7209:
7200:
7199:
7188:
7187:
7168:
7167:
7158:
7157:
7148:
7147:
7142:
7141:
7128:
7127:
7118:
7117:
7099:
7097:
7096:
7091:
7070:
7068:
7067:
7062:
7053:
7042:
7029:
7018:
6999:
6988:
6972:
6969:
6963:
6961:
6957:
6956:
6947:
6946:
6931:
6930:
6919:
6918:
6910:
6906:
6905:
6896:
6895:
6880:
6879:
6874:
6873:
6865:
6859:
6848:
6838:
6835:
6828:
6817:
6812:
6809:
6802:
6791:
6773:
6772:
6760:
6759:
6739:
6737:
6736:
6731:
6726:
6725:
6720:
6719:
6702:
6700:
6699:
6694:
6683:
6682:
6670:
6669:
6664:
6663:
6650:
6649:
6637:
6636:
6624:
6623:
6598:
6596:
6595:
6590:
6585:
6584:
6575:
6574:
6563:
6559:
6558:
6540:
6539:
6523:
6521:
6520:
6515:
6510:
6509:
6496:
6494:
6493:
6488:
6476:
6474:
6473:
6468:
6463:
6462:
6450:
6449:
6444:
6443:
6426:
6424:
6423:
6418:
6413:
6412:
6403:
6402:
6387:
6386:
6371:
6370:
6361:
6360:
6351:
6350:
6341:
6338:
6325:
6323:
6322:
6317:
6312:
6311:
6302:
6301:
6286:
6285:
6276:
6275:
6266:
6265:
6249:
6247:
6246:
6241:
6230:
6229:
6220:
6219:
6210:
6209:
6191:
6190:
6179:
6165:
6164:
6146:
6145:
6124:
6123:
6114:
6113:
6089:
6088:
6079:
6078:
6069:
6068:
6052:
6050:
6049:
6044:
6042:
6041:
6028:
6026:
6025:
6020:
6018:
6017:
5997:
5995:
5994:
5989:
5984:
5983:
5965:
5964:
5955:
5954:
5945:
5944:
5937:
5916:
5915:
5903:
5902:
5882:
5880:
5879:
5874:
5869:
5868:
5857:
5854:
5848:
5845:
5843:
5835:
5834:
5829:
5826:
5820:
5817:
5809:
5808:
5799:
5798:
5776:
5774:
5773:
5768:
5757:
5756:
5747:
5746:
5731:
5730:
5721:
5720:
5705:
5704:
5695:
5694:
5668:
5666:
5665:
5660:
5655:
5654:
5645:
5644:
5634:
5613:
5612:
5591:
5589:
5588:
5583:
5581:
5580:
5563:
5561:
5560:
5555:
5547:
5546:
5537:
5536:
5524:
5523:
5514:
5513:
5504:
5503:
5484:
5482:
5481:
5476:
5474:
5473:
5453:(or, for short,
5450:
5448:
5447:
5442:
5440:
5439:
5401:
5399:
5398:
5393:
5391:
5390:
5376:
5375:
5362:
5361:
5356:
5355:
5342:
5341:
5325:
5323:
5322:
5317:
5315:
5314:
5291:
5289:
5288:
5283:
5281:
5277:
5276:
5266:
5265:
5246:
5244:
5243:
5238:
5230:
5229:
5228:
5215:
5214:
5202:
5198:
5197:
5187:
5186:
5177:
5173:
5172:
5155:
5153:
5152:
5147:
5139:
5138:
5129:
5128:
5116:
5115:
5106:
5105:
5096:
5095:
5068:as a manifold).
5067:
5065:
5064:
5059:
5057:
5056:
5043:
5041:
5040:
5035:
5030:
5029:
5020:
4998:
4996:
4995:
4990:
4985:
4984:
4975:
4974:
4958:
4956:
4955:
4950:
4945:
4944:
4935:
4934:
4920:Zariski topology
4913:
4911:
4910:
4905:
4903:
4902:
4897:
4896:
4882:
4880:
4879:
4874:
4869:
4868:
4863:
4862:
4852:
4851:
4838:
4836:
4835:
4830:
4822:
4821:
4816:
4815:
4805:
4804:
4795:
4794:
4789:
4788:
4771:
4769:
4768:
4763:
4755:
4754:
4749:
4748:
4738:
4737:
4725:
4724:
4719:
4718:
4708:
4707:
4698:
4697:
4692:
4691:
4674:
4672:
4671:
4666:
4658:
4657:
4648:
4647:
4635:
4634:
4625:
4624:
4615:
4614:
4594:
4592:
4591:
4586:
4581:
4580:
4571:
4570:
4555:
4553:
4552:
4547:
4542:
4541:
4532:
4531:
4516:
4514:
4513:
4508:
4506:
4505:
4489:, consisting of
4488:
4486:
4485:
4480:
4472:
4471:
4462:
4461:
4449:
4448:
4439:
4438:
4429:
4428:
4405:
4403:
4402:
4397:
4392:
4391:
4386:
4385:
4375:
4374:
4353:
4351:
4350:
4345:
4333:
4331:
4330:
4325:
4320:
4319:
4314:
4313:
4303:
4302:
4289:
4287:
4286:
4281:
4276:
4275:
4270:
4269:
4259:
4258:
4245:
4243:
4242:
4237:
4235:
4234:
4229:
4228:
4210:
4208:
4207:
4202:
4190:
4188:
4187:
4182:
4165:
4164:
4152:
4151:
4138:
4136:
4135:
4130:
4122:
4121:
4116:
4115:
4105:
4104:
4095:
4094:
4089:
4088:
4055:
4053:
4052:
4047:
4045:
4044:
4039:
4038:
4024:
4022:
4021:
4016:
4004:
4002:
4001:
3996:
3994:
3993:
3992:
3976:
3975:
3970:
3969:
3953:of the sequence
3944:
3942:
3941:
3936:
3934:
3933:
3922:
3921:
3911:
3910:
3905:
3904:
3890:
3888:
3887:
3882:
3880:
3879:
3874:
3873:
3852:
3850:
3849:
3844:
3827:
3826:
3804:
3802:
3801:
3796:
3794:
3793:
3780:
3778:
3777:
3772:
3754:
3752:
3751:
3746:
3744:
3743:
3738:
3737:
3723:
3721:
3720:
3715:
3689:
3688:
3676:
3675:
3659:
3657:
3656:
3651:
3634:
3633:
3611:
3609:
3608:
3603:
3592:
3591:
3582:
3581:
3565:
3563:
3562:
3557:
3540:
3539:
3506:
3505:
3478:
3477:
3468:
3467:
3455:
3454:
3449:
3448:
3434:
3432:
3431:
3426:
3414:
3412:
3411:
3406:
3394:
3392:
3391:
3386:
3369:
3368:
3356:
3355:
3334:
3332:
3331:
3326:
3324:
3323:
3310:
3308:
3307:
3302:
3294:
3293:
3284:
3283:
3274:
3273:
3257:
3255:
3254:
3249:
3244:
3243:
3229:
3228:
3221:
3217:
3216:
3207:
3206:
3194:
3193:
3188:
3187:
3171:
3170:
3161:
3160:
3147:
3145:
3144:
3139:
3122:
3121:
3109:
3108:
3092:
3090:
3089:
3084:
3066:
3064:
3063:
3058:
3056:
3055:
3050:
3049:
3036:
3035:
3019:
3017:
3016:
3011:
3009:
3008:
2991:
2989:
2988:
2983:
2981:
2980:
2956:minimal surfaces
2941:
2939:
2938:
2933:
2921:
2919:
2918:
2913:
2901:
2899:
2898:
2893:
2891:
2890:
2881:
2880:
2864:
2862:
2861:
2856:
2854:
2853:
2834:
2832:
2831:
2826:
2810:
2808:
2807:
2802:
2785:
2784:
2772:
2771:
2758:
2756:
2755:
2750:
2739:on the manifold
2738:
2736:
2735:
2730:
2705:
2703:
2702:
2697:
2680:
2679:
2656:
2654:
2653:
2648:
2636:
2634:
2633:
2628:
2626:
2625:
2620:
2619:
2605:
2603:
2602:
2597:
2577:
2576:
2564:
2563:
2551:
2550:
2545:
2544:
2512:
2511:
2481:
2480:
2468:
2467:
2454:
2452:
2451:
2446:
2426:
2425:
2407:
2406:
2389:
2387:
2386:
2381:
2364:
2363:
2342:
2340:
2339:
2334:
2320:
2318:
2317:
2312:
2299:
2297:
2296:
2291:
2274:
2273:
2251:
2249:
2248:
2243:
2231:
2229:
2228:
2223:
2211:
2209:
2208:
2203:
2198:
2197:
2185:
2184:
2160:
2158:
2157:
2152:
2140:
2138:
2137:
2132:
2115:
2114:
2098:
2096:
2095:
2090:
2074:
2072:
2071:
2066:
2052:
2051:
2039:
2036:
2031:
2030:
2010:
2008:
2007:
2002:
1991:
1990:
1972:
1970:
1969:
1964:
1961:
1956:
1916:
1915:
1888:
1887:
1869:
1867:
1866:
1861:
1843:
1841:
1840:
1835:
1815:
1813:
1812:
1807:
1790:
1789:
1765:
1763:
1762:
1757:
1746:
1745:
1722:
1720:
1719:
1714:
1682:
1680:
1679:
1674:
1662:
1660:
1659:
1654:
1642:
1640:
1639:
1634:
1618:
1616:
1615:
1610:
1591:
1589:
1588:
1583:
1578:
1577:
1550:
1549:
1533:
1531:
1530:
1525:
1510:
1508:
1507:
1502:
1486:
1484:
1483:
1478:
1460:
1458:
1457:
1452:
1440:
1438:
1437:
1432:
1406:
1404:
1403:
1398:
1396:
1381:
1379:
1378:
1373:
1361:
1359:
1358:
1353:
1335:
1333:
1332:
1327:
1325:
1310:
1308:
1307:
1302:
1281:
1279:
1278:
1273:
1256:
1255:
1238:
1236:
1235:
1230:
1224:
1219:
1207:
1206:
1186:
1184:
1183:
1178:
1165:
1163:
1162:
1157:
1145:
1143:
1142:
1137:
1125:
1123:
1122:
1117:
1114:
1109:
1086:
1084:
1083:
1078:
1073:
1070:
1066:
1038:
1035:
1031:
1016:
1015:
1008:
1006:
1001:
965:
964:
947:
945:
944:
939:
927:
925:
924:
919:
907:
905:
904:
899:
887:
885:
884:
879:
866:
864:
863:
858:
841:
840:
819:
817:
816:
811:
799:
797:
796:
791:
778:
776:
775:
770:
753:
751:
750:
745:
739:
734:
718:
716:
715:
710:
707:
702:
680:
678:
677:
672:
658:
656:
655:
650:
637:
635:
634:
629:
627:
612:
610:
609:
604:
592:
590:
589:
584:
569:
567:
566:
561:
548:
546:
545:
540:
526:
524:
523:
518:
505:
503:
502:
497:
489:
461:
459:
458:
453:
437:
435:
434:
429:
417:
415:
414:
409:
407:
392:
390:
389:
384:
369:
367:
366:
361:
345:
343:
342:
337:
313:
311:
310:
305:
290:
289:
259:affine varieties
240:
238:
237:
232:
215:
214:
202:
201:
188:
186:
185:
180:
178:
177:
172:
171:
79:
78:
75:
74:
71:
68:
65:
62:
59:
56:
53:
50:
21:
10237:
10236:
10232:
10231:
10230:
10228:
10227:
10226:
10197:
10196:
10195:
10190:
10129:Banach manifold
10122:Generalizations
10117:
10072:
10009:
9906:
9868:Ricci curvature
9824:Cotangent space
9802:
9740:
9582:
9576:
9535:Exponential map
9499:
9444:
9438:
9358:
9348:
9294:
9289:
9228:
9224:
9171:
9167:
9160:
9136:
9132:
9095:
9091:
9064:
9060:
9051:
9049:
9041:
9040:
9036:
9009:
9005:
8998:
8966:
8962:
8946:
8945:
8933:
8919:
8918:
8914:
8895:
8891:
8838:
8834:
8827:
8811:
8807:
8792:
8776:
8767:
8760:
8736:
8729:
8684:
8680:
8676:
8642:
8601:
8589:
8583:
8582:
8581:
8575:
8571:
8562:
8561:
8552:
8542:
8541:
8532:
8528:
8516:
8510:
8509:
8508:
8502:
8498:
8489:
8488:
8479:
8469:
8468:
8466:
8463:
8462:
8444:
8441:
8440:
8416:
8415:
8406:
8396:
8395:
8393:
8390:
8389:
8365:
8353:
8352:
8343:
8342:
8333:
8323:
8322:
8313:
8309:
8297:
8296:
8287:
8286:
8277:
8267:
8266:
8264:
8261:
8260:
8236:
8235:
8233:
8230:
8229:
8204:
8192:
8191:
8182:
8172:
8171:
8162:
8158:
8146:
8145:
8136:
8126:
8125:
8123:
8120:
8119:
8095:
8094:
8076:
8075:
8070:
8064:
8063:
8051:
8050:
8048:
8045:
8044:
8023:
8021:
8018:
8017:
7993:
7992:
7983:
7973:
7972:
7970:
7967:
7966:
7940:
7939:
7937:
7934:
7933:
7903:
7902:
7900:
7897:
7896:
7893:de Rham complex
7872:
7857:
7856:
7855:
7849:
7845:
7827:
7826:
7817:
7807:
7806:
7804:
7801:
7800:
7783:
7753:
7752:
7750:
7747:
7746:
7719:
7718:
7716:
7713:
7712:
7679:
7675:
7673:
7670:
7669:
7643:
7639:
7637:
7634:
7633:
7605:
7602:
7601:
7598:
7572:
7571:
7569:
7566:
7565:
7540:
7536:
7534:
7531:
7530:
7511:
7510:
7508:
7505:
7504:
7499:corresponds to
7471:
7466:
7460:
7457:
7456:
7431:
7430:
7418:
7413:
7408:
7405:
7402:
7401:
7374:
7369:
7356:
7355:
7353:
7350:
7349:
7332:
7331:
7329:
7326:
7325:
7320:corresponds to
7298:
7293:
7287:
7284:
7283:
7260:
7259:
7251:
7248:
7247:
7215:
7214:
7205:
7201:
7189:
7183:
7182:
7181:
7163:
7162:
7153:
7149:
7143:
7137:
7136:
7135:
7123:
7122:
7113:
7109:
7107:
7104:
7103:
7079:
7076:
7075:
7043:
7038:
7019:
7014:
6989:
6978:
6968:
6952:
6951:
6936:
6932:
6920:
6914:
6913:
6912:
6911:
6901:
6900:
6885:
6881:
6875:
6869:
6868:
6867:
6866:
6864:
6849:
6844:
6834:
6818:
6813:
6808:
6792:
6787:
6768:
6767:
6755:
6754:
6752:
6749:
6748:
6721:
6715:
6714:
6713:
6711:
6708:
6707:
6678:
6677:
6665:
6659:
6658:
6657:
6645:
6644:
6632:
6631:
6619:
6618:
6610:
6607:
6606:
6580:
6579:
6570:
6566:
6554:
6550:
6535:
6531:
6529:
6526:
6525:
6505:
6504:
6502:
6499:
6498:
6482:
6479:
6478:
6458:
6457:
6445:
6439:
6438:
6437:
6435:
6432:
6431:
6408:
6407:
6392:
6388:
6382:
6381:
6366:
6365:
6356:
6352:
6346:
6345:
6337:
6335:
6332:
6331:
6307:
6306:
6297:
6293:
6281:
6280:
6271:
6267:
6261:
6260:
6258:
6255:
6254:
6225:
6224:
6215:
6214:
6205:
6201:
6186:
6182:
6160:
6156:
6141:
6137:
6119:
6118:
6109:
6105:
6084:
6083:
6074:
6070:
6064:
6063:
6061:
6058:
6057:
6053:vanishes, i.e.
6037:
6036:
6034:
6031:
6030:
6013:
6012:
6010:
6007:
6006:
5979:
5978:
5960:
5959:
5950:
5946:
5940:
5939:
5927:
5911:
5910:
5898:
5897:
5895:
5892:
5891:
5858:
5853:
5852:
5844:
5839:
5830:
5825:
5824:
5816:
5804:
5803:
5794:
5790:
5788:
5785:
5784:
5752:
5751:
5742:
5738:
5726:
5725:
5716:
5712:
5700:
5699:
5690:
5686:
5684:
5681:
5680:
5674:de Rham complex
5650:
5649:
5640:
5636:
5624:
5608:
5607:
5599:
5596:
5595:
5576:
5575:
5573:
5570:
5569:
5542:
5541:
5532:
5531:
5519:
5518:
5509:
5508:
5499:
5498:
5493:
5490:
5489:
5469:
5468:
5466:
5463:
5462:
5435:
5434:
5432:
5429:
5428:
5423:
5418:
5377:
5371:
5370:
5369:
5357:
5351:
5350:
5349:
5337:
5333:
5331:
5328:
5327:
5310:
5309:
5301:
5298:
5297:
5272:
5271:
5270:
5261:
5260:
5252:
5249:
5248:
5221:
5220:
5219:
5210:
5209:
5193:
5192:
5191:
5182:
5181:
5168:
5167:
5166:
5161:
5158:
5157:
5134:
5133:
5124:
5123:
5111:
5110:
5101:
5100:
5091:
5090:
5085:
5082:
5081:
5074:
5052:
5051:
5049:
5046:
5045:
5025:
5024:
5010:
5008:
5005:
5004:
4980:
4979:
4970:
4969:
4967:
4964:
4963:
4940:
4939:
4930:
4929:
4927:
4924:
4923:
4898:
4892:
4891:
4890:
4888:
4885:
4884:
4864:
4858:
4857:
4856:
4847:
4846:
4844:
4841:
4840:
4817:
4811:
4810:
4809:
4800:
4799:
4790:
4784:
4783:
4782:
4777:
4774:
4773:
4750:
4744:
4743:
4742:
4733:
4732:
4720:
4714:
4713:
4712:
4703:
4702:
4693:
4687:
4686:
4685:
4680:
4677:
4676:
4653:
4652:
4643:
4642:
4630:
4629:
4620:
4619:
4610:
4609:
4604:
4601:
4600:
4576:
4575:
4566:
4565:
4563:
4560:
4559:
4537:
4536:
4527:
4526:
4524:
4521:
4520:
4501:
4500:
4498:
4495:
4494:
4467:
4466:
4457:
4456:
4444:
4443:
4434:
4433:
4424:
4423:
4418:
4415:
4414:
4387:
4381:
4380:
4379:
4370:
4369:
4367:
4364:
4363:
4339:
4336:
4335:
4315:
4309:
4308:
4307:
4298:
4297:
4295:
4292:
4291:
4271:
4265:
4264:
4263:
4254:
4253:
4251:
4248:
4247:
4230:
4224:
4223:
4222:
4220:
4217:
4216:
4196:
4193:
4192:
4160:
4156:
4147:
4146:
4144:
4141:
4140:
4117:
4111:
4110:
4109:
4100:
4099:
4090:
4084:
4083:
4082:
4077:
4074:
4073:
4066:
4040:
4034:
4033:
4032:
4030:
4027:
4026:
4010:
4007:
4006:
3988:
3981:
3977:
3971:
3965:
3964:
3963:
3958:
3955:
3954:
3923:
3917:
3916:
3915:
3906:
3900:
3899:
3898:
3896:
3893:
3892:
3875:
3869:
3868:
3867:
3865:
3862:
3861:
3816:
3812:
3810:
3807:
3806:
3789:
3788:
3786:
3783:
3782:
3760:
3757:
3756:
3739:
3733:
3732:
3731:
3729:
3726:
3725:
3684:
3680:
3671:
3667:
3665:
3662:
3661:
3623:
3619:
3617:
3614:
3613:
3587:
3586:
3577:
3573:
3571:
3568:
3567:
3529:
3525:
3495:
3491:
3473:
3472:
3463:
3459:
3450:
3444:
3443:
3442:
3440:
3437:
3436:
3420:
3417:
3416:
3400:
3397:
3396:
3364:
3360:
3351:
3350:
3348:
3345:
3344:
3341:
3319:
3318:
3316:
3313:
3312:
3289:
3288:
3279:
3278:
3269:
3268:
3263:
3260:
3259:
3239:
3238:
3224:
3212:
3211:
3202:
3198:
3189:
3183:
3182:
3181:
3166:
3165:
3156:
3155:
3153:
3150:
3149:
3117:
3113:
3104:
3103:
3101:
3098:
3097:
3072:
3069:
3068:
3051:
3045:
3044:
3043:
3031:
3027:
3025:
3022:
3021:
3004:
3003:
3001:
2998:
2997:
2976:
2975:
2967:
2964:
2963:
2927:
2924:
2923:
2907:
2904:
2903:
2886:
2885:
2876:
2872:
2870:
2867:
2866:
2849:
2848:
2840:
2837:
2836:
2820:
2817:
2816:
2780:
2776:
2767:
2766:
2764:
2761:
2760:
2744:
2741:
2740:
2724:
2721:
2720:
2713:
2675:
2671:
2663:
2660:
2659:
2642:
2639:
2638:
2621:
2615:
2614:
2613:
2611:
2608:
2607:
2572:
2568:
2559:
2555:
2546:
2540:
2539:
2538:
2507:
2503:
2476:
2472:
2463:
2462:
2460:
2457:
2456:
2421:
2417:
2402:
2401:
2399:
2396:
2395:
2359:
2355:
2353:
2350:
2349:
2328:
2325:
2324:
2306:
2303:
2302:
2300:, is called an
2269:
2265:
2257:
2254:
2253:
2237:
2234:
2233:
2217:
2214:
2213:
2193:
2189:
2180:
2176:
2174:
2171:
2170:
2167:
2146:
2143:
2142:
2110:
2106:
2104:
2101:
2100:
2084:
2081:
2080:
2047:
2043:
2035:
2026:
2022:
2020:
2017:
2016:
1986:
1982:
1980:
1977:
1976:
1957:
1952:
1911:
1907:
1883:
1879:
1877:
1874:
1873:
1849:
1846:
1845:
1829:
1826:
1825:
1822:
1785:
1781:
1779:
1776:
1775:
1741:
1737:
1735:
1732:
1731:
1696:
1693:
1692:
1668:
1665:
1664:
1648:
1645:
1644:
1628:
1625:
1624:
1621:fibred manifold
1604:
1601:
1600:
1567:
1563:
1545:
1541:
1539:
1536:
1535:
1516:
1513:
1512:
1496:
1493:
1492:
1466:
1463:
1462:
1446:
1443:
1442:
1420:
1417:
1416:
1389:
1387:
1384:
1383:
1367:
1364:
1363:
1341:
1338:
1337:
1318:
1316:
1313:
1312:
1296:
1293:
1292:
1251:
1247:
1245:
1242:
1241:
1220:
1215:
1202:
1198:
1193:
1190:
1189:
1172:
1169:
1168:
1151:
1148:
1147:
1131:
1128:
1127:
1110:
1105:
1093:
1090:
1089:
1069:
1034:
1011:
1002:
997:
960:
956:
954:
951:
950:
933:
930:
929:
913:
910:
909:
893:
890:
889:
873:
870:
869:
836:
832:
830:
827:
826:
805:
802:
801:
785:
782:
781:
764:
761:
760:
735:
730:
724:
721:
720:
703:
698:
686:
683:
682:
666:
663:
662:
644:
641:
640:
620:
618:
615:
614:
598:
595:
594:
578:
575:
574:
555:
552:
551:
534:
531:
530:
512:
509:
508:
482:
468:
465:
464:
447:
444:
443:
423:
420:
419:
400:
398:
395:
394:
378:
375:
374:
355:
352:
351:
348:smooth manifold
319:
316:
315:
299:
296:
295:
288:
279:
210:
206:
197:
196:
194:
191:
190:
173:
167:
166:
165:
163:
160:
159:
118:
47:
43:
32:
23:
22:
15:
12:
11:
5:
10235:
10225:
10224:
10219:
10214:
10209:
10192:
10191:
10189:
10188:
10183:
10178:
10173:
10168:
10167:
10166:
10156:
10151:
10146:
10141:
10136:
10131:
10125:
10123:
10119:
10118:
10116:
10115:
10110:
10105:
10100:
10095:
10090:
10084:
10082:
10078:
10077:
10074:
10073:
10071:
10070:
10065:
10060:
10055:
10050:
10045:
10040:
10035:
10030:
10025:
10019:
10017:
10011:
10010:
10008:
10007:
10002:
9997:
9992:
9987:
9982:
9977:
9967:
9962:
9957:
9947:
9942:
9937:
9932:
9927:
9922:
9916:
9914:
9908:
9907:
9905:
9904:
9899:
9894:
9893:
9892:
9882:
9877:
9876:
9875:
9865:
9860:
9855:
9850:
9849:
9848:
9838:
9833:
9832:
9831:
9821:
9816:
9810:
9808:
9804:
9803:
9801:
9800:
9795:
9790:
9785:
9784:
9783:
9773:
9768:
9763:
9757:
9755:
9748:
9742:
9741:
9739:
9738:
9733:
9723:
9718:
9704:
9699:
9694:
9689:
9684:
9682:Parallelizable
9679:
9674:
9669:
9668:
9667:
9657:
9652:
9647:
9642:
9637:
9632:
9627:
9622:
9617:
9612:
9602:
9592:
9586:
9584:
9578:
9577:
9575:
9574:
9569:
9564:
9562:Lie derivative
9559:
9557:Integral curve
9554:
9549:
9544:
9543:
9542:
9532:
9527:
9526:
9525:
9518:Diffeomorphism
9515:
9509:
9507:
9501:
9500:
9498:
9497:
9492:
9487:
9482:
9477:
9472:
9467:
9462:
9457:
9451:
9449:
9440:
9439:
9437:
9436:
9431:
9426:
9421:
9416:
9411:
9406:
9401:
9396:
9395:
9394:
9389:
9379:
9378:
9377:
9366:
9364:
9363:Basic concepts
9360:
9359:
9347:
9346:
9339:
9332:
9324:
9318:
9317:
9312:
9307:
9301:
9293:
9292:External links
9290:
9288:
9287:
9242:(4): 426–447.
9222:
9185:(6): 1450007.
9165:
9158:
9130:
9089:
9058:
9034:
9023:(2): 311–363.
9003:
8996:
8960:
8931:
8912:
8905:(in Russian).
8889:
8832:
8825:
8805:
8790:
8765:
8758:
8727:
8677:
8675:
8672:
8664:
8663:
8658:
8653:
8648:
8641:
8638:
8626:
8625:
8613:
8608:
8605:
8600:
8597:
8592:
8586:
8578:
8574:
8570:
8565:
8560:
8555:
8549:
8546:
8540:
8535:
8531:
8527:
8524:
8519:
8513:
8505:
8501:
8497:
8492:
8487:
8482:
8476:
8473:
8448:
8436:
8424:
8419:
8414:
8409:
8403:
8400:
8377:
8372:
8369:
8364:
8361:
8356:
8351:
8346:
8341:
8336:
8330:
8327:
8321:
8316:
8312:
8308:
8305:
8300:
8295:
8290:
8285:
8280:
8274:
8271:
8254:
8239:
8216:
8211:
8208:
8203:
8200:
8195:
8190:
8185:
8179:
8176:
8170:
8165:
8161:
8157:
8154:
8149:
8144:
8139:
8133:
8130:
8098:
8093:
8088:
8085:
8082:
8079:
8073:
8067:
8062:
8059:
8054:
8030:
8027:
8001:
7996:
7991:
7986:
7980:
7977:
7952:
7949:
7946:
7943:
7915:
7912:
7909:
7906:
7880:
7875:
7869:
7866:
7863:
7860:
7852:
7848:
7844:
7841:
7838:
7835:
7830:
7825:
7820:
7814:
7811:
7782:
7779:
7756:
7722:
7699:
7696:
7693:
7690:
7687:
7682:
7678:
7657:
7654:
7651:
7646:
7642:
7621:
7618:
7615:
7612:
7609:
7597:
7594:
7590:
7589:
7575:
7543:
7539:
7528:
7514:
7486:
7483:
7480:
7477:
7474:
7469:
7465:
7454:
7442:
7439:
7434:
7427:
7424:
7421:
7416:
7383:
7380:
7377:
7372:
7368:
7364:
7359:
7335:
7307:
7304:
7301:
7296:
7292:
7268:
7263:
7258:
7255:
7244:
7243:
7232:
7229:
7226:
7223:
7218:
7213:
7208:
7204:
7198:
7195:
7192:
7186:
7180:
7177:
7174:
7171:
7166:
7161:
7156:
7152:
7146:
7140:
7134:
7131:
7126:
7121:
7116:
7112:
7089:
7086:
7083:
7072:
7071:
7060:
7057:
7052:
7049:
7046:
7041:
7037:
7033:
7028:
7025:
7022:
7017:
7013:
7009:
7006:
7003:
6998:
6995:
6992:
6987:
6984:
6981:
6977:
6966:
6960:
6955:
6950:
6945:
6942:
6939:
6935:
6929:
6926:
6923:
6917:
6909:
6904:
6899:
6894:
6891:
6888:
6884:
6878:
6872:
6863:
6858:
6855:
6852:
6847:
6843:
6832:
6827:
6824:
6821:
6816:
6806:
6801:
6798:
6795:
6790:
6786:
6782:
6779:
6776:
6771:
6766:
6763:
6758:
6729:
6724:
6718:
6704:
6703:
6692:
6689:
6686:
6681:
6676:
6673:
6668:
6662:
6656:
6653:
6648:
6643:
6640:
6635:
6630:
6627:
6622:
6617:
6614:
6588:
6583:
6578:
6573:
6569:
6562:
6557:
6553:
6549:
6546:
6543:
6538:
6534:
6513:
6508:
6486:
6466:
6461:
6456:
6453:
6448:
6442:
6416:
6411:
6406:
6401:
6398:
6395:
6391:
6385:
6380:
6377:
6374:
6369:
6364:
6359:
6355:
6349:
6344:
6326:is actually a
6315:
6310:
6305:
6300:
6296:
6292:
6289:
6284:
6279:
6274:
6270:
6264:
6251:
6250:
6239:
6236:
6233:
6228:
6223:
6218:
6213:
6208:
6204:
6200:
6197:
6194:
6189:
6185:
6178:
6174:
6171:
6168:
6163:
6159:
6155:
6152:
6149:
6144:
6140:
6136:
6133:
6130:
6127:
6122:
6117:
6112:
6108:
6104:
6101:
6098:
6095:
6092:
6087:
6082:
6077:
6073:
6067:
6040:
6016:
5999:
5998:
5987:
5982:
5977:
5974:
5971:
5968:
5963:
5958:
5953:
5949:
5943:
5936:
5933:
5930:
5926:
5922:
5919:
5914:
5909:
5906:
5901:
5885:Poincaré Lemma
5872:
5867:
5864:
5861:
5851:
5842:
5838:
5833:
5823:
5815:
5812:
5807:
5802:
5797:
5793:
5778:
5777:
5766:
5763:
5760:
5755:
5750:
5745:
5741:
5737:
5734:
5729:
5724:
5719:
5715:
5711:
5708:
5703:
5698:
5693:
5689:
5670:
5669:
5658:
5653:
5648:
5643:
5639:
5633:
5630:
5627:
5623:
5619:
5616:
5611:
5606:
5603:
5579:
5553:
5550:
5545:
5540:
5535:
5530:
5527:
5522:
5517:
5512:
5507:
5502:
5497:
5472:
5438:
5422:
5419:
5417:
5414:
5389:
5386:
5383:
5380:
5374:
5368:
5365:
5360:
5354:
5348:
5345:
5340:
5336:
5313:
5308:
5305:
5280:
5275:
5269:
5264:
5259:
5256:
5236:
5233:
5227:
5224:
5218:
5213:
5208:
5205:
5201:
5196:
5190:
5185:
5180:
5176:
5171:
5165:
5145:
5142:
5137:
5132:
5127:
5122:
5119:
5114:
5109:
5104:
5099:
5094:
5089:
5073:
5070:
5055:
5033:
5028:
5023:
5019:
5016:
5013:
4988:
4983:
4978:
4973:
4948:
4943:
4938:
4933:
4901:
4895:
4872:
4867:
4861:
4855:
4850:
4828:
4825:
4820:
4814:
4808:
4803:
4798:
4793:
4787:
4781:
4761:
4758:
4753:
4747:
4741:
4736:
4731:
4728:
4723:
4717:
4711:
4706:
4701:
4696:
4690:
4684:
4664:
4661:
4656:
4651:
4646:
4641:
4638:
4633:
4628:
4623:
4618:
4613:
4608:
4597:
4596:
4584:
4579:
4574:
4569:
4556:
4545:
4540:
4535:
4530:
4517:
4504:
4478:
4475:
4470:
4465:
4460:
4455:
4452:
4447:
4442:
4437:
4432:
4427:
4422:
4395:
4390:
4384:
4378:
4373:
4343:
4323:
4318:
4312:
4306:
4301:
4279:
4274:
4268:
4262:
4257:
4233:
4227:
4200:
4180:
4177:
4174:
4171:
4168:
4163:
4159:
4155:
4150:
4128:
4125:
4120:
4114:
4108:
4103:
4098:
4093:
4087:
4081:
4065:
4062:
4043:
4037:
4014:
3991:
3987:
3984:
3980:
3974:
3968:
3962:
3932:
3929:
3926:
3920:
3914:
3909:
3903:
3878:
3872:
3842:
3839:
3836:
3833:
3830:
3825:
3822:
3819:
3815:
3792:
3770:
3767:
3764:
3742:
3736:
3713:
3710:
3707:
3704:
3701:
3698:
3695:
3692:
3687:
3683:
3679:
3674:
3670:
3649:
3646:
3643:
3640:
3637:
3632:
3629:
3626:
3622:
3601:
3598:
3595:
3590:
3585:
3580:
3576:
3555:
3552:
3549:
3546:
3543:
3538:
3535:
3532:
3528:
3524:
3521:
3518:
3515:
3512:
3509:
3504:
3501:
3498:
3494:
3490:
3487:
3484:
3481:
3476:
3471:
3466:
3462:
3458:
3453:
3447:
3424:
3404:
3384:
3381:
3378:
3375:
3372:
3367:
3363:
3359:
3354:
3340:
3337:
3322:
3300:
3297:
3292:
3287:
3282:
3277:
3272:
3267:
3247:
3242:
3237:
3234:
3227:
3220:
3215:
3210:
3205:
3201:
3197:
3192:
3186:
3180:
3177:
3174:
3169:
3164:
3159:
3137:
3134:
3131:
3128:
3125:
3120:
3116:
3112:
3107:
3082:
3079:
3076:
3054:
3048:
3042:
3039:
3034:
3030:
3007:
2979:
2974:
2971:
2931:
2911:
2889:
2884:
2879:
2875:
2852:
2847:
2844:
2824:
2800:
2797:
2794:
2791:
2788:
2783:
2779:
2775:
2770:
2748:
2728:
2712:
2709:
2695:
2692:
2689:
2686:
2683:
2678:
2674:
2670:
2667:
2646:
2624:
2618:
2595:
2592:
2589:
2586:
2583:
2580:
2575:
2571:
2567:
2562:
2558:
2554:
2549:
2543:
2537:
2534:
2530:
2527:
2524:
2521:
2518:
2515:
2510:
2506:
2502:
2499:
2496:
2493:
2490:
2487:
2484:
2479:
2475:
2471:
2466:
2444:
2441:
2438:
2435:
2432:
2429:
2424:
2420:
2416:
2413:
2410:
2405:
2379:
2376:
2373:
2370:
2367:
2362:
2358:
2332:
2310:
2289:
2286:
2283:
2280:
2277:
2272:
2268:
2264:
2261:
2241:
2221:
2201:
2196:
2192:
2188:
2183:
2179:
2166:
2163:
2150:
2130:
2127:
2124:
2121:
2118:
2113:
2109:
2088:
2064:
2061:
2058:
2055:
2050:
2046:
2042:
2034:
2029:
2025:
2015:and its image
2000:
1997:
1994:
1989:
1985:
1960:
1955:
1951:
1947:
1944:
1941:
1938:
1934:
1931:
1928:
1925:
1922:
1919:
1914:
1910:
1906:
1903:
1900:
1897:
1894:
1891:
1886:
1882:
1859:
1856:
1853:
1833:
1821:
1818:
1805:
1802:
1799:
1796:
1793:
1788:
1784:
1755:
1752:
1749:
1744:
1740:
1712:
1709:
1706:
1703:
1700:
1672:
1652:
1632:
1608:
1581:
1576:
1573:
1570:
1566:
1562:
1559:
1556:
1553:
1548:
1544:
1523:
1520:
1500:
1476:
1473:
1470:
1450:
1430:
1427:
1424:
1411:
1410:
1395:
1392:
1371:
1351:
1348:
1345:
1324:
1321:
1300:
1285:
1271:
1268:
1265:
1262:
1259:
1254:
1250:
1228:
1223:
1218:
1214:
1210:
1205:
1201:
1197:
1176:
1155:
1135:
1113:
1108:
1104:
1100:
1097:
1076:
1065:
1062:
1059:
1056:
1053:
1050:
1047:
1044:
1041:
1030:
1027:
1024:
1021:
1014:
1005:
1000:
996:
992:
989:
986:
983:
980:
977:
974:
971:
968:
963:
959:
937:
917:
897:
877:
856:
853:
850:
847:
844:
839:
835:
809:
789:
780:-jet space of
768:
743:
738:
733:
729:
706:
701:
697:
693:
690:
670:
648:
626:
623:
602:
582:
559:
538:
516:
495:
492:
488:
485:
481:
478:
475:
472:
451:
427:
406:
403:
382:
359:
335:
332:
329:
326:
323:
303:
287:
284:
278:
275:
263:affine schemes
230:
227:
224:
221:
218:
213:
209:
205:
200:
176:
170:
117:
114:
9:
6:
4:
3:
2:
10234:
10223:
10220:
10218:
10215:
10213:
10210:
10208:
10205:
10204:
10202:
10187:
10184:
10182:
10181:Supermanifold
10179:
10177:
10174:
10172:
10169:
10165:
10162:
10161:
10160:
10157:
10155:
10152:
10150:
10147:
10145:
10142:
10140:
10137:
10135:
10132:
10130:
10127:
10126:
10124:
10120:
10114:
10111:
10109:
10106:
10104:
10101:
10099:
10096:
10094:
10091:
10089:
10086:
10085:
10083:
10079:
10069:
10066:
10064:
10061:
10059:
10056:
10054:
10051:
10049:
10046:
10044:
10041:
10039:
10036:
10034:
10031:
10029:
10026:
10024:
10021:
10020:
10018:
10016:
10012:
10006:
10003:
10001:
9998:
9996:
9993:
9991:
9988:
9986:
9983:
9981:
9978:
9976:
9972:
9968:
9966:
9963:
9961:
9958:
9956:
9952:
9948:
9946:
9943:
9941:
9938:
9936:
9933:
9931:
9928:
9926:
9923:
9921:
9918:
9917:
9915:
9913:
9909:
9903:
9902:Wedge product
9900:
9898:
9895:
9891:
9888:
9887:
9886:
9883:
9881:
9878:
9874:
9871:
9870:
9869:
9866:
9864:
9861:
9859:
9856:
9854:
9851:
9847:
9846:Vector-valued
9844:
9843:
9842:
9839:
9837:
9834:
9830:
9827:
9826:
9825:
9822:
9820:
9817:
9815:
9812:
9811:
9809:
9805:
9799:
9796:
9794:
9791:
9789:
9786:
9782:
9779:
9778:
9777:
9776:Tangent space
9774:
9772:
9769:
9767:
9764:
9762:
9759:
9758:
9756:
9752:
9749:
9747:
9743:
9737:
9734:
9732:
9728:
9724:
9722:
9719:
9717:
9713:
9709:
9705:
9703:
9700:
9698:
9695:
9693:
9690:
9688:
9685:
9683:
9680:
9678:
9675:
9673:
9670:
9666:
9663:
9662:
9661:
9658:
9656:
9653:
9651:
9648:
9646:
9643:
9641:
9638:
9636:
9633:
9631:
9628:
9626:
9623:
9621:
9618:
9616:
9613:
9611:
9607:
9603:
9601:
9597:
9593:
9591:
9588:
9587:
9585:
9579:
9573:
9570:
9568:
9565:
9563:
9560:
9558:
9555:
9553:
9550:
9548:
9545:
9541:
9540:in Lie theory
9538:
9537:
9536:
9533:
9531:
9528:
9524:
9521:
9520:
9519:
9516:
9514:
9511:
9510:
9508:
9506:
9502:
9496:
9493:
9491:
9488:
9486:
9483:
9481:
9478:
9476:
9473:
9471:
9468:
9466:
9463:
9461:
9458:
9456:
9453:
9452:
9450:
9447:
9443:Main results
9441:
9435:
9432:
9430:
9427:
9425:
9424:Tangent space
9422:
9420:
9417:
9415:
9412:
9410:
9407:
9405:
9402:
9400:
9397:
9393:
9390:
9388:
9385:
9384:
9383:
9380:
9376:
9373:
9372:
9371:
9368:
9367:
9365:
9361:
9356:
9352:
9345:
9340:
9338:
9333:
9331:
9326:
9325:
9322:
9316:
9313:
9311:
9308:
9305:
9302:
9299:
9296:
9295:
9283:
9279:
9275:
9271:
9267:
9263:
9259:
9255:
9250:
9245:
9241:
9237:
9233:
9226:
9218:
9214:
9210:
9206:
9202:
9198:
9193:
9188:
9184:
9180:
9176:
9169:
9161:
9155:
9151:
9147:
9143:
9142:
9134:
9126:
9122:
9118:
9114:
9111:(1): 41–129.
9110:
9106:
9101:
9093:
9085:
9081:
9077:
9073:
9069:
9062:
9048:
9044:
9038:
9030:
9026:
9022:
9018:
9014:
9007:
8999:
8993:
8989:
8985:
8981:
8977:
8972:
8964:
8956:
8950:
8942:
8938:
8934:
8928:
8924:
8923:
8916:
8908:
8904:
8900:
8893:
8885:
8881:
8877:
8873:
8869:
8865:
8860:
8855:
8851:
8847:
8843:
8836:
8828:
8822:
8818:
8817:
8809:
8801:
8797:
8793:
8791:0-8218-2922-X
8787:
8783:
8782:
8774:
8772:
8770:
8761:
8755:
8751:
8747:
8743:
8742:
8734:
8732:
8723:
8719:
8715:
8711:
8707:
8703:
8699:
8695:
8690:
8682:
8678:
8671:
8669:
8662:
8659:
8657:
8654:
8652:
8649:
8647:
8644:
8643:
8637:
8635:
8631:
8603:
8598:
8590:
8576:
8572:
8568:
8553:
8533:
8529:
8525:
8517:
8503:
8499:
8495:
8480:
8471:
8460:
8446:
8437:
8407:
8398:
8367:
8362:
8349:
8334:
8314:
8310:
8306:
8293:
8278:
8269:
8258:
8255:
8252:
8206:
8201:
8183:
8163:
8159:
8155:
8137:
8128:
8117:
8114:
8113:
8112:
8071:
8060:
8057:
8025:
8015:
7984:
7931:
7894:
7873:
7850:
7846:
7836:
7818:
7799:
7794:
7790:
7788:
7778:
7776:
7771:
7769:
7743:
7739:
7735:
7694:
7691:
7688:
7680:
7676:
7652:
7644:
7640:
7619:
7613:
7610:
7607:
7593:
7563:
7559:
7541:
7537:
7529:
7502:
7484:
7481:
7478:
7475:
7472:
7467:
7463:
7455:
7440:
7437:
7425:
7422:
7419:
7414:
7399:
7381:
7378:
7375:
7370:
7366:
7362:
7323:
7305:
7302:
7299:
7294:
7290:
7282:
7281:
7280:
7253:
7230:
7227:
7224:
7206:
7196:
7193:
7190:
7178:
7175:
7172:
7154:
7144:
7132:
7114:
7102:
7101:
7100:
7087:
7084:
7081:
7058:
7050:
7047:
7044:
7039:
7035:
7031:
7026:
7023:
7020:
7015:
7011:
7004:
7001:
6996:
6993:
6990:
6985:
6982:
6979:
6975:
6964:
6943:
6940:
6937:
6927:
6924:
6921:
6892:
6889:
6886:
6876:
6861:
6856:
6853:
6850:
6845:
6841:
6825:
6822:
6819:
6814:
6804:
6799:
6796:
6793:
6788:
6784:
6777:
6761:
6747:
6746:
6745:
6743:
6722:
6690:
6687:
6666:
6654:
6628:
6605:
6604:
6603:
6602:
6576:
6571:
6567:
6560:
6555:
6551:
6547:
6544:
6541:
6536:
6532:
6524:generated by
6484:
6446:
6428:
6399:
6396:
6393:
6378:
6357:
6329:
6298:
6290:
6272:
6237:
6211:
6206:
6202:
6198:
6195:
6192:
6187:
6183:
6172:
6169:
6161:
6157:
6153:
6150:
6147:
6142:
6138:
6131:
6128:
6110:
6102:
6099:
6093:
6075:
6056:
6055:
6054:
6004:
5969:
5951:
5934:
5931:
5928:
5924:
5920:
5890:
5889:
5888:
5886:
5865:
5862:
5859:
5840:
5831:
5813:
5795:
5791:
5783:
5764:
5743:
5717:
5687:
5679:
5678:
5677:
5675:
5641:
5631:
5628:
5625:
5621:
5617:
5594:
5593:
5592:
5567:
5528:
5505:
5486:
5460:
5456:
5452:
5413:
5411:
5409:
5403:
5384:
5366:
5358:
5338:
5334:
5306:
5303:
5295:
5278:
5257:
5225:
5206:
5199:
5178:
5174:
5120:
5097:
5079:
5069:
5002:
4960:
4921:
4917:
4796:
4729:
4699:
4639:
4616:
4557:
4518:
4492:
4491:
4490:
4453:
4430:
4412:
4407:
4361:
4357:
4341:
4214:
4198:
4175:
4172:
4169:
4161:
4157:
4153:
4096:
4071:
4061:
4059:
4012:
3985:
3982:
3972:
3952:
3951:inverse limit
3948:
3930:
3927:
3924:
3907:
3876:
3859:
3854:
3837:
3834:
3831:
3823:
3820:
3817:
3813:
3768:
3765:
3762:
3740:
3708:
3705:
3699:
3696:
3693:
3685:
3681:
3672:
3668:
3644:
3641:
3638:
3630:
3627:
3624:
3620:
3596:
3593:
3578:
3574:
3550:
3547:
3544:
3536:
3533:
3530:
3526:
3522:
3516:
3513:
3510:
3502:
3499:
3496:
3492:
3488:
3482:
3479:
3464:
3460:
3456:
3451:
3422:
3402:
3379:
3376:
3373:
3365:
3361:
3357:
3336:
3275:
3235:
3232:
3203:
3199:
3195:
3190:
3175:
3132:
3129:
3126:
3118:
3114:
3110:
3094:
3080:
3077:
3074:
3052:
3040:
3037:
3032:
3028:
2995:
2972:
2969:
2959:
2957:
2953:
2949:
2945:
2929:
2909:
2882:
2877:
2873:
2845:
2842:
2822:
2814:
2795:
2792:
2789:
2781:
2777:
2773:
2746:
2726:
2718:
2708:
2706:
2690:
2687:
2684:
2676:
2672:
2668:
2665:
2644:
2622:
2587:
2584:
2581:
2573:
2569:
2560:
2556:
2552:
2547:
2532:
2528:
2522:
2519:
2516:
2508:
2504:
2500:
2491:
2488:
2485:
2477:
2473:
2469:
2436:
2433:
2430:
2422:
2418:
2411:
2408:
2394:
2390:
2374:
2371:
2368:
2360:
2356:
2346:
2330:
2322:
2308:
2284:
2281:
2278:
2270:
2266:
2262:
2259:
2239:
2219:
2194:
2190:
2181:
2177:
2162:
2148:
2125:
2122:
2119:
2111:
2107:
2086:
2078:
2075:, called the
2056:
2048:
2044:
2032:
2027:
2023:
2014:
1995:
1987:
1983:
1973:
1958:
1953:
1945:
1936:
1932:
1926:
1923:
1920:
1912:
1908:
1901:
1898:
1892:
1884:
1880:
1871:
1857:
1854:
1851:
1831:
1817:
1800:
1797:
1794:
1786:
1782:
1773:
1769:
1750:
1742:
1738:
1730:
1726:
1710:
1704:
1701:
1698:
1690:
1686:
1683:given by the
1670:
1650:
1630:
1622:
1606:
1597:
1595:
1594:fibre bundles
1579:
1574:
1571:
1568:
1564:
1554:
1546:
1542:
1521:
1518:
1498:
1490:
1474:
1471:
1468:
1448:
1428:
1425:
1422:
1393:
1390:
1369:
1349:
1346:
1343:
1322:
1319:
1298:
1290:
1286:
1284:
1282:
1266:
1263:
1260:
1252:
1248:
1221:
1216:
1212:
1208:
1203:
1199:
1187:
1174:
1153:
1133:
1111:
1106:
1098:
1087:
1063:
1060:
1057:
1054:
1051:
1048:
1042:
1028:
1025:
1022:
1019:
1003:
998:
990:
981:
975:
972:
969:
961:
957:
935:
915:
895:
875:
867:
851:
848:
845:
837:
833:
823:
820:
807:
787:
779:
766:
755:
741:
736:
731:
727:
704:
699:
691:
668:
660:
659:-th order jet
646:
624:
621:
600:
580:
571:
557:
549:
536:
527:
514:
493:
490:
486:
483:
479:
476:
473:
470:
463:at the point
462:
449:
441:
425:
404:
401:
380:
373:
370:-dimensional
357:
349:
346:-dimensional
330:
327:
324:
301:
292:
291:
283:
274:
272:
268:
264:
260:
256:
252:
248:
244:
225:
222:
219:
211:
207:
203:
158:
154:
149:
147:
142:
141:differentiate
137:
135:
131:
127:
123:
113:
111:
107:
103:
99:
95:
91:
87:
83:
77:
41:
37:
30:
19:
10138:
10108:Moving frame
10103:Morse theory
10093:Gauge theory
9885:Tensor field
9814:Closed/Exact
9793:Vector field
9761:Distribution
9702:Hypercomplex
9697:Quaternionic
9434:Vector field
9392:Smooth atlas
9239:
9235:
9225:
9182:
9178:
9168:
9140:
9133:
9108:
9104:
9092:
9075:
9071:
9061:
9050:. Retrieved
9046:
9037:
9020:
9016:
9006:
8979:
8975:
8963:
8921:
8915:
8906:
8902:
8892:
8849:
8845:
8835:
8815:
8808:
8780:
8740:
8700:(1): 21–78.
8697:
8693:
8681:
8665:
8627:
8438:
8256:
8228:
8115:
7797:
7795:
7791:
7784:
7772:
7745:
7711:
7599:
7591:
7245:
7073:
6705:
6429:
6252:
6000:
5779:
5671:
5487:
5454:
5426:
5424:
5416:Applications
5406:
5404:
5077:
5075:
5000:
4961:
4915:
4598:
4413:is a triple
4410:
4408:
4069:
4067:
4057:
3857:
3855:
3342:
3095:
2960:
2812:
2716:
2714:
2658:
2393:distribution
2348:
2344:
2301:
2168:
2077:prolongation
2076:
2011:is a smooth
1974:
1872:
1823:
1598:
1489:Grassmannian
1414:
1240:
1239:and provide
1167:
949:
825:
821:
759:
758:
756:
639:
572:
529:
507:
442:
439:
372:submanifolds
293:
280:
270:
266:
243:distribution
156:
152:
150:
138:
119:
109:
108:erential var
105:
39:
33:
10053:Levi-Civita
10043:Generalized
10015:Connections
9965:Lie algebra
9897:Volume form
9798:Vector flow
9771:Pushforward
9766:Lie bracket
9665:Lie algebra
9630:G-structure
9419:Pushforward
9399:Submanifold
9078:(1): 1–40.
9047:ncatlab.org
8630:Phase Space
5427:Vinogradov
5294:pushforward
3566:where both
2657:-planes at
824:denoted by
130:polynomials
102:portmanteau
82:geometrical
36:mathematics
10201:Categories
10176:Stratifold
10134:Diffeology
9930:Associated
9731:Symplectic
9716:Riemannian
9645:Hyperbolic
9572:Submersion
9480:Hopf–Rinow
9414:Submersion
9409:Smooth map
9052:2021-12-11
8941:1031947580
8674:References
8014:DG algebra
6601:filtration
6253:Note that
5326:, one has
4999:is called
4599:such that
4356:involutive
4211:-th order
4072:is a pair
4060:manifold.
2865:such that
2455:defined by
1774:subset of
1729:jet bundle
638:have same
29:Diffeology
10222:Manifolds
10058:Principal
10033:Ehresmann
9990:Subbundle
9980:Principal
9955:Fibration
9935:Cotangent
9807:Covectors
9660:Lie group
9640:Hermitian
9583:manifolds
9552:Immersion
9547:Foliation
9485:Noether's
9470:Frobenius
9465:De Rham's
9460:Darboux's
9351:Manifolds
9274:0393-0440
9249:0809.4164
9217:119704524
9209:0219-1997
9192:1204.2467
9125:0022-247X
9029:0030-6126
8949:cite book
8859:1308.1005
8722:121860845
8714:0167-8019
8607:¯
8591:∗
8573:∧
8569:⊗
8554:∙
8548:¯
8545:Ω
8534:∙
8518:∗
8500:∧
8481:∙
8475:¯
8408:∙
8402:¯
8371:¯
8350:⊗
8335:∙
8329:¯
8326:Ω
8315:∙
8279:∙
8273:¯
8210:¯
8184:∙
8178:¯
8175:Ω
8164:∙
8138:∙
8132:¯
8092:→
8029:¯
7985:∙
7979:¯
7976:Ω
7874:∗
7851:∙
7847:∧
7840:Γ
7819:∙
7813:¯
7810:Ω
7742:bicomplex
7653:π
7617:→
7608:π
7482:−
7363:∈
7203:Ω
7179:⊃
7176:⋯
7173:⊃
7151:Ω
7133:⊃
7111:Ω
7085:≥
6934:Ω
6883:Ω
6728:Ω
6691:⋯
6688:⊃
6672:Ω
6655:⊃
6639:Ω
6629:⊃
6613:Ω
6587:Ω
6577:∈
6548:∧
6545:⋯
6542:∧
6512:Ω
6452:Ω
6390:Ω
6379:⊂
6354:Ω
6295:Ω
6291:⊆
6269:Ω
6212:∈
6196:…
6177:∀
6151:⋯
6129:∣
6107:Ω
6103:∈
6072:Ω
6003:submodule
5973:Ω
5970:⊆
5948:Ω
5932:≥
5925:∑
5905:Ω
5863:−
5765:⋯
5762:⟶
5740:Ω
5736:⟶
5714:Ω
5710:⟶
5692:∞
5638:Ω
5629:≥
5622:∑
5602:Ω
5385:θ
5379:Φ
5367:⊆
5359:θ
5344:Φ
5339:θ
5307:∈
5304:θ
5268:→
5255:Φ
4900:∞
4866:∞
4819:∞
4792:∞
4752:∞
4722:∞
4695:∞
4389:∞
4317:∞
4273:∞
4232:∞
4154:⊂
4119:∞
4092:∞
4042:∞
3986:∈
3928:−
3913:→
3523:⊆
3489:∩
3395:of order
3358:⊂
3236:∈
3233:θ
3204:θ
3196:∩
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3111:⊂
3078:∈
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3053:θ
3041:⊂
3033:θ
2973:⊂
2883:⊆
2846:⊂
2774:⊂
2719:of order
2669:∈
2666:θ
2623:θ
2561:θ
2553:⊂
2548:θ
2536:↦
2533:θ
2498:→
2409:⊆
2331:θ
2263:∈
2260:θ
2182:θ
2013:embedding
1940:↦
1905:→
1855:⊆
1751:π
1708:→
1699:π
1687:of local
1572:−
1561:→
1347:∈
1217:σ
1061:⊂
1023:∈
888:-jets of
491:⊂
480:∩
474:∈
251:varieties
204:⊂
175:∞
126:varieties
92:play for
10154:Orbifold
10149:K-theory
10139:Diffiety
9863:Pullback
9677:Oriented
9655:Kenmotsu
9635:Hadamard
9581:Types of
9530:Geodesic
9355:Glossary
9282:21787052
8884:15871902
8800:47296188
8640:See also
7562:bordisms
6430:Now let
5408:category
5279:′
5226:′
5200:′
5175:′
5078:morphism
4772:, where
4411:diffiety
3067:for all
2813:solution
2212:, where
1975:The map
1689:sections
1623:over an
1461:and for
1394:″
1323:′
625:′
487:′
405:′
267:diffiety
40:diffiety
10098:History
10081:Related
9995:Tangent
9973:)
9953:)
9920:Adjoint
9912:Bundles
9890:density
9788:Torsion
9754:Vectors
9746:Tensors
9729:)
9714:)
9710:,
9708:Pseudo−
9687:Poisson
9620:Finsler
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9608:)
9600:Complex
9598:)
9567:Section
9254:Bibcode
8864:Bibcode
8852:: 003.
6477:be its
6001:be the
5457:) is a
4916:locally
4914:. Here
3020:, i.e.
2902:. When
2391:is the
1166:around
255:schemes
80:) is a
10063:Vector
10048:Koszul
10028:Cartan
10023:Affine
10005:Vector
10000:Tensor
9985:Spinor
9975:Normal
9971:Stable
9925:Affine
9829:bundle
9781:bundle
9727:Almost
9650:Kähler
9606:Almost
9596:Almost
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9490:Sard's
9446:(list)
9280:
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2606:where
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2321:-plane
1685:graphs
1362:while
1067:
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314:be an
10171:Sheaf
9945:Fiber
9721:Rizza
9692:Prime
9523:Local
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9375:Atlas
9278:S2CID
9244:arXiv
9213:S2CID
9187:arXiv
8880:S2CID
8854:arXiv
8718:S2CID
6836:where
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134:ideal
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88:that
10038:Form
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9712:Sub−
9625:Flat
9505:Maps
9270:ISSN
9205:ISSN
9154:ISBN
9121:ISSN
9025:ISSN
8992:ISBN
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8821:ISBN
8796:OCLC
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38:, a
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199:E
169:E
76:/
70:t
67:ˌ
64:ə
58:f
55:ˈ
52:ə
49:d
46:/
42:(
31:.
20:)
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