65:
7819:
While it is easy to find homeomorphisms that are not diffeomorphisms, it is more difficult to find a pair of homeomorphic manifolds that are not diffeomorphic. In dimensions 1, 2 and 3, any pair of homeomorphic smooth manifolds are diffeomorphic. In dimension 4 or greater, examples of homeomorphic
5289:
1358:
4324:
5734:-compact and not compact the full diffeomorphism group is not locally contractible for any of the two topologies. One has to restrict the group by controlling the deviation from the identity near infinity to obtain a diffeomorphism group which is a manifold; see (
3609:
3833:
5568:
4696:
6501:
7643:
The homotopy-type of the diffeomorphism groups of 3-manifolds are fairly well understood via the work of Ivanov, Hatcher, Gabai and
Rubinstein, although there are a few outstanding open cases (primarily 3-manifolds with finite
7071:; this space is convex and hence path-connected. A smooth, eventually constant path to the identity gives a second more elementary way of extending a diffeomorphism from the circle to the open unit disc (a special case of the
3248:
5099:
1184:
5964:
4091:
4102:
7828:) that is homeomorphic to the standard 7-sphere but not diffeomorphic to it. There are, in fact, 28 oriented diffeomorphism classes of manifolds homeomorphic to the 7-sphere (each of them is the total space of a
7606:
7363:
6134:
5884:
3144:
1078:
6753:
2968:
3940:
6296:
1462:
1862:
7405:
204:
6375:
1673:
808:
2895:
154:
7797:
7733:
6635:
6539:
4953:
1596:
6989:
The (orientation-preserving) diffeomorphism group of the circle is pathwise connected. This can be seen by noting that any such diffeomorphism can be lifted to a diffeomorphism
6171:
4772:
8005:
7972:
7943:
7914:
7881:
6570:
6242:
6206:
6054:
5004:
3034:
3005:
955:
3424:
4602:
3440:
2490:
7183:
6789:
6671:
4786:) is also interpreted as that type of complex number, the action is of complex multiplication in the appropriate complex number plane. As such, there is a type of angle (
2468:
690:
5700:
3702:
1770:
344:
4385:
2639:
1011:
742:
621:
7141:
5732:
5372:
3679:
3646:
7759:
7695:
7638:
6928:
6878:
2395:
2318:
2182:
2108:
2075:
2045:
2015:
1989:
1959:
1744:
1710:
1495:
7505:
7222:
5598:
5399:
4895:
2847:
2827:
2727:
2707:
458:
431:
271:
7534:
7105:
3285:
5999:
5431:
4542:
4451:
2365:
2288:
7669:
7453:
7007:
6948:
6898:
6845:
6818:
6691:
6590:
6406:
6074:
6023:
5816:
5792:
5768:
5666:
5642:
5622:
5423:
5352:
5332:
5312:
5091:
5024:
4973:
4915:
4864:
4832:
4519:
4495:
4475:
4428:
4408:
3365:
3345:
3325:
3305:
2807:
2787:
2767:
2747:
2687:
2663:
2601:
2577:
2557:
2537:
2517:
2415:
2338:
2265:
2245:
2222:
2202:
2148:
2128:
1933:
1913:
1893:
1814:
1794:
1515:
1381:
1172:
1148:
1122:
1102:
975:
919:
899:
868:
848:
828:
710:
661:
641:
589:
569:
549:
529:
498:
478:
404:
384:
364:
314:
294:
244:
224:
119:
99:
7069:
4620:
1174:
to be globally invertible (under the sole condition that its derivative be a bijective map at each point). For example, consider the "realification" of the
6414:
5284:{\displaystyle d_{K}(f,g)=\sup \nolimits _{x\in K}d(f(x),g(x))+\sum \nolimits _{1\leq p\leq r}\sup \nolimits _{x\in K}\left\|D^{p}f(x)-D^{p}g(x)\right\|}
5062:. When the manifold is not compact, the strong topology captures the behavior of functions "at infinity" and is not metrizable. It is, however, still
3155:
8240:
1353:{\displaystyle {\begin{cases}f:\mathbb {R} ^{2}\setminus \{(0,0)\}\to \mathbb {R} ^{2}\setminus \{(0,0)\}\\(x,y)\mapsto (x^{2}-y^{2},2xy).\end{cases}}}
878:
Testing whether a differentiable map is a diffeomorphism can be made locally under some mild restrictions. This is the
Hadamard-Caccioppoli theorem:
4802:
being invertible, the type of complex number is uniform over the surface. Consequently, a surface deformation or diffeomorphism of surfaces has the
9328:
7468:
122:
8519:
8176:
4319:{\displaystyle J_{h}={\begin{pmatrix}2x\cos(x^{2}+y^{2})&2y\cos(x^{2}+y^{2})\\-2x\sin(x^{2}+y^{2})&-2y\sin(x^{2}+y^{2})\end{pmatrix}}.}
5895:
3972:
9323:
7546:
7296:
6951:
8610:
6079:
7812:
Since every diffeomorphism is a homeomorphism, given a pair of manifolds which are diffeomorphic to each other they are in particular
5824:
8634:
8829:
3049:
1023:
6696:
8699:
8442:
8321:
8160:
8098:
6244:
consists of two components, consisting of the orientation-preserving and orientation-reversing diffeomorphisms. In fact, the
5645:
2900:
8925:
5672:. Moreover, the transition maps from one chart of this atlas to another are smooth, making the diffeomorphism group into a
2492:
to itself because its derivative vanishes at 0 (and hence its inverse is not differentiable at 0). This is an example of a
3852:
2580:
8978:
8506:
7416:
9262:
7412:
6255:
1393:
7408:
7286:
diffeomorphism; those equivalent to a diffeomorphism leaving a simple closed curve invariant; and those equivalent to
8460:
8339:
8281:
8263:
5711:
9027:
1819:
7419:
matrices. Thurston accomplished his classification by observing that the mapping class group acted naturally on a
9010:
8619:
7371:
7279:
167:
17:
6301:
1607:
7420:
751:
9376:
9222:
8629:
8300:
7261:
6848:
2852:
8185:
6824:
127:
9207:
8930:
8704:
5795:
7945:, and also there are uncountably many pairwise non-diffeomorphic differentiable manifolds homeomorphic to
7764:
7700:
6595:
6506:
6377:. In particular, the general linear group is also a deformation retract of the full diffeomorphism group.
4920:
1524:
9252:
8295:
7428:
58:
7456:
6139:
9257:
9227:
8935:
8891:
8872:
8639:
8583:
6821:
4709:
1018:
664:
7981:
7948:
7919:
7890:
7857:
7283:
6544:
6218:
6180:
6028:
4978:
3010:
2981:
931:
8794:
8659:
3604:{\displaystyle g(x,y)=\left(a_{0}+a_{1,0}x+a_{0,1}y+\cdots ,\ b_{0}+b_{1,0}x+b_{0,1}y+\cdots \right)}
745:
8290:
7232:
For manifolds, the diffeomorphism group is usually not connected. Its component group is called the
3370:
1193:
9371:
9179:
9044:
8736:
8578:
7265:
7241:
6900:
is at least two-dimensional, the diffeomorphism group acts transitively on the configuration space
4778:, fixing the origin, and expressible as the action of a complex number of a particular type. When (
4551:
2341:
3828:{\displaystyle J_{g}(0,0)={\begin{pmatrix}a_{1,0}&a_{0,1}\\b_{1,0}&b_{0,1}\end{pmatrix}}.}
2473:
8876:
8846:
8770:
8760:
8716:
8546:
8499:
7735:
has more than two components. Via Milnor, Kahn and
Antonelli, however, it is known that provided
7161:
6758:
6640:
5573:
The diffeomorphism group equipped with its weak topology is locally homeomorphic to the space of
4498:
4352:
2418:
57:
that maps one differentiable manifold to another such that both the function and its inverse are
50:
7197:
of the diffeomorphism group by the subgroup of classes extending to diffeomorphisms of the ball
2431:
669:
9217:
8836:
8731:
8644:
8551:
5679:
1749:
323:
6978:
yields a diffeomorphism on the open disc. An elegant proof was provided shortly afterwards by
4358:
2612:
984:
715:
594:
8866:
8861:
8029:
8016:
7186:
7113:
5717:
5357:
4775:
3651:
3618:
7738:
7674:
7640:
have the homotopy-type of their mapping class groups (i.e. the components are contractible).
7617:
6903:
6853:
2666:
2642:
2607:. Every diffeomorphism is a homeomorphism, but not every homeomorphism is a diffeomorphism.
2370:
2293:
2157:
2083:
2050:
2020:
1994:
1964:
1938:
1719:
1685:
1470:
9197:
9135:
8983:
8687:
8677:
8649:
8624:
8534:
8381:
8200:
8064:
8059:
7975:
7483:
7200:
6245:
5576:
5563:{\displaystyle d(f,g)=\sum \nolimits _{n}2^{-n}{\frac {d_{K_{n}}(f,g)}{1+d_{K_{n}}(f,g)}}.}
5377:
4873:
3951:
2832:
2812:
2712:
2692:
2151:
1713:
436:
409:
249:
7510:
7424:
7081:
3261:
8:
9335:
9017:
8895:
8880:
8809:
8568:
8455:, Translations of Mathematical Monographs, vol. 158, American Mathematical Society,
8020:
7612:
7233:
6249:
5978:
4867:
2604:
69:
54:
9308:
8204:
5707:
5676:
with smooth right translations; left translations and inversion are only continuous. If
4691:{\displaystyle du={\frac {\partial u}{\partial x}}dx+{\frac {\partial u}{\partial y}}dy}
4524:
4433:
2978:
Since any manifold can be locally parametrised, we can consider some explicit maps from
2347:
2270:
9277:
9232:
9129:
9000:
8804:
8492:
8419:
8401:
8350:
8232:
8170:
7825:
7654:
7460:
7438:
7253:
7237:
6992:
6933:
6883:
6830:
6803:
6676:
6575:
6391:
6059:
6008:
5801:
5777:
5753:
5651:
5627:
5607:
5408:
5402:
5337:
5317:
5297:
5076:
5009:
4958:
4900:
4849:
4817:
4607:
4504:
4480:
4460:
4413:
4393:
4388:
3350:
3330:
3310:
3290:
2792:
2772:
2752:
2732:
2672:
2648:
2586:
2562:
2542:
2522:
2502:
2400:
2323:
2250:
2230:
2207:
2187:
2133:
2113:
1918:
1898:
1878:
1799:
1779:
1773:
1500:
1366:
1157:
1133:
1107:
1087:
1081:
960:
904:
884:
853:
833:
813:
695:
646:
626:
574:
554:
534:
514:
483:
463:
389:
369:
349:
299:
279:
229:
209:
104:
84:
8814:
8348:
Leslie, J. A. (1967), "On a differential structure for the group of diffeomorphisms",
8132:
8115:
7110:
The corresponding extension problem for diffeomorphisms of higher-dimensional spheres
7012:
4330:
The
Jacobian matrix has zero determinant everywhere! In fact we see that the image of
9212:
9192:
9187:
9094:
9005:
8819:
8799:
8654:
8593:
8456:
8438:
8423:
8369:
8364:
8335:
8317:
8277:
8259:
8224:
8216:
8156:
8094:
8048:
8037:
8033:
7645:
7287:
7269:
6496:{\displaystyle 0\to {\text{Diff}}_{0}(M)\to {\text{Diff}}(M)\to \Sigma (\pi _{0}(M))}
6381:
5070:
5039:
4545:
4454:
870:
is said to be a diffeomorphism if it is bijective, smooth and its inverse is smooth.
72:
of a rectangular grid on a square under a diffeomorphism from the square onto itself.
8334:, Mathematical Surveys and Monographs, vol. 53, American Mathematical Society,
8236:
8054:
9350:
9144:
9099:
9022:
8993:
8851:
8784:
8779:
8774:
8764:
8556:
8539:
8411:
8359:
8251:
8208:
8127:
7848:
7275:
7076:
6971:
6209:
5706:. Moreover, the transition maps are smooth, making the diffeomorphism group into a
5059:
4835:
4791:
1151:
978:
480:
times continuously differentiable bijective map between them whose inverse is also
31:
5703:
3243:{\displaystyle J_{f}={\begin{pmatrix}2x&3y^{2}\\2x&-3y^{2}\end{pmatrix}}.}
9293:
9202:
9032:
8988:
8754:
8469:
8377:
8313:
7884:
7844:
7194:
7075:). Moreover, the diffeomorphism group of the circle has the homotopy-type of the
7072:
6983:
6979:
6409:
6385:
5673:
5027:
4839:
1865:
922:
9159:
9084:
9054:
8952:
8945:
8885:
8856:
8726:
8721:
8025:
7190:
1175:
8415:
5798:. Somewhat formally, this is seen by making a small change to the coordinate
9365:
9345:
9169:
9164:
9149:
9139:
9089:
9066:
8940:
8900:
8841:
8789:
8588:
8373:
8220:
8212:
8069:
7820:
but not diffeomorphic pairs exist. The first such example was constructed by
7366:
7257:
7156:
7152:
5055:
2493:
64:
8388:
Michor, Peter W.; Mumford, David (2013), "A zoo of diffeomorphism groups on
8276:, Cambridge Mathematical Tracts, vol. 156, Cambridge University Press,
6986:, apparently unaware of this result, produced a completely different proof.
6967:
9272:
9267:
9109:
9076:
9049:
8957:
8598:
7829:
7824:
in dimension 7. He constructed a smooth 7-dimensional manifold (called now
7813:
7540:
7464:
5771:
5669:
8228:
7539:
The diffeomorphism group of the torus has the homotopy-type of its linear
7144:
9115:
9104:
9061:
8962:
8563:
8430:
7821:
7697:
are poorly understood. For example, it is an open problem whether or not
7148:
7143:
was much studied in the 1950s and 1960s, with notable contributions from
5747:
5063:
4335:
3682:
3255:
46:
38:
5959:{\displaystyle L_{h}=h^{\mu }(x){\frac {\partial }{\partial x^{\mu }}}.}
4086:{\displaystyle h(x,y)=\left(\sin(x^{2}+y^{2}),\cos(x^{2}+y^{2})\right).}
9340:
9298:
9124:
9037:
8669:
8573:
8484:
8042:
7840:
7800:
7432:
7245:
5668:
is finite and the manifold is compact, the space of vector fields is a
3955:
2645:, it satisfies the definition above. More precisely: Pick any cover of
1679:
1384:
1014:
8045:, smooth parameterizations on a set, which makes a diffeological space
9154:
9119:
8824:
8711:
7852:
6975:
6002:
5093:, the weak topology is the topology induced by the family of metrics
4348:
1872:
1518:
161:
8258:, Mathematics and its Applications, vol. 400, Kluwer Academic,
6847:. More generally, the diffeomorphism group acts transitively on the
9318:
9313:
9303:
8694:
8515:
8184:
Chaudhuri, Shyamoli; Kawai, Hikaru; Tye, S.-H. Henry (1987-08-15).
7833:
7601:{\displaystyle S^{1}\times S^{1}\times {\text{GL}}(2,\mathbb {Z} )}
7358:{\displaystyle S^{1}\times S^{1}=\mathbb {R} ^{2}/\mathbb {Z} ^{2}}
7249:
6174:
1601:
Since the differential at a point (for a differentiable function)
925:
8406:
6974:
of any homeomorphism or diffeomorphism of the unit circle to the
6129:{\displaystyle {\text{Diff}}(G)\simeq G\times {\text{Diff}}(G,e)}
8153:
The implicit function theorem: history, theory, and applications
8091:
The implicit function theorem: history, theory, and applications
7427:; as this enlarged space was homeomorphic to a closed ball, the
5879:{\displaystyle x^{\mu }\mapsto x^{\mu }+\varepsilon h^{\mu }(x)}
8910:
8088:
509:
2897:
is then a diffeomorphism as in the definition above, whenever
1871:
Diffeomorphisms are necessarily between manifolds of the same
7291:
4795:
4787:
3139:{\displaystyle f(x,y)=\left(x^{2}+y^{3},x^{2}-y^{3}\right).}
7155:. An obstruction to such extensions is given by the finite
5058:, these two topologies agree. The weak topology is always
4355:
and may be described by a diffeomorphism. A diffeomorphism
2519:
is a map between differentiable manifolds, a diffeomorphic
1346:
1073:{\displaystyle Df_{x}:\mathbb {R} ^{n}\to \mathbb {R} ^{n}}
7463:
of the group of orientation-preserving diffeomorphisms is
6025:
in its own diffeomorphism group via left-translation. Let
503:
8051:, metric study of shape and form in computational anatomy
7467:. This had first been proved for a product of circles by
6748:{\displaystyle {\text{Diff}}(M)\to \Sigma (\pi _{0}(M))}
2963:{\displaystyle f(\phi ^{-1}(U))\subseteq \psi ^{-1}(V)}
7843:. In the early 1980s, a combination of results due to
5889:
so the infinitesimal generators are the vector fields
5006:. This is a "large" group, in the sense thatâprovided
4124:
3739:
3685:, and the omitted terms are of degree at least two in
3177:
7984:
7951:
7922:
7893:
7860:
7816:
to each other. The converse is not true in general.
7767:
7741:
7703:
7677:
7657:
7620:
7549:
7513:
7486:
7441:
7407:
and the classification becomes classical in terms of
7374:
7299:
7203:
7164:
7116:
7084:
7015:
6995:
6936:
6906:
6886:
6856:
6833:
6806:
6761:
6699:
6679:
6643:
6598:
6578:
6547:
6509:
6417:
6394:
6304:
6258:
6221:
6183:
6142:
6082:
6062:
6031:
6011:
5981:
5898:
5827:
5804:
5780:
5756:
5720:
5682:
5654:
5630:
5610:
5579:
5434:
5411:
5380:
5360:
5340:
5320:
5300:
5102:
5079:
5012:
4981:
4961:
4923:
4903:
4876:
4852:
4820:
4798:) that is preserved in such a multiplication. Due to
4712:
4623:
4554:
4527:
4507:
4483:
4463:
4436:
4416:
4396:
4361:
4105:
3975:
3855:
3705:
3654:
3621:
3443:
3373:
3353:
3333:
3313:
3293:
3264:
3158:
3052:
3013:
2984:
2903:
2855:
2835:
2815:
2795:
2775:
2755:
2735:
2715:
2695:
2675:
2651:
2615:
2589:
2565:
2545:
2525:
2505:
2476:
2434:
2403:
2373:
2350:
2326:
2296:
2273:
2253:
2233:
2210:
2190:
2160:
2136:
2116:
2086:
2053:
2023:
1997:
1967:
1941:
1921:
1901:
1881:
1822:
1802:
1782:
1752:
1722:
1688:
1610:
1527:
1503:
1473:
1396:
1369:
1187:
1160:
1136:
1110:
1090:
1026:
987:
963:
934:
907:
887:
856:
836:
816:
754:
718:
698:
672:
649:
629:
597:
577:
557:
537:
517:
486:
466:
439:
412:
392:
372:
352:
326:
302:
282:
252:
232:
212:
170:
130:
107:
87:
8474:
Jahresbericht der
Deutschen Mathematiker-Vereinigung
7611:
The diffeomorphism groups of orientable surfaces of
3935:{\displaystyle a_{1,0}b_{0,1}-a_{0,1}b_{1,0}\neq 0,}
2047:
could never be injective. In both cases, therefore,
7807:
6298:of diffeomorphisms fixing the origin under the map
7999:
7966:
7937:
7908:
7875:
7791:
7753:
7727:
7689:
7663:
7632:
7600:
7528:
7499:
7447:
7399:
7357:
7216:
7177:
7135:
7099:
7063:
7001:
6942:
6922:
6892:
6872:
6839:
6812:
6783:
6747:
6685:
6665:
6629:
6584:
6564:
6533:
6495:
6400:
6384:of points, the diffeomorphism group is simply the
6369:
6290:
6236:
6200:
6165:
6128:
6068:
6048:
6017:
5993:
5958:
5878:
5810:
5786:
5762:
5726:
5694:
5660:
5636:
5616:
5592:
5562:
5417:
5393:
5366:
5346:
5326:
5306:
5283:
5085:
5018:
4998:
4967:
4947:
4909:
4889:
4858:
4826:
4766:
4690:
4596:
4536:
4513:
4489:
4469:
4445:
4422:
4402:
4379:
4318:
4085:
3934:
3827:
3673:
3640:
3603:
3418:
3359:
3339:
3319:
3299:
3279:
3242:
3138:
3028:
2999:
2962:
2889:
2841:
2821:
2801:
2781:
2761:
2741:
2721:
2701:
2681:
2657:
2633:
2595:
2571:
2551:
2531:
2511:
2484:
2462:
2409:
2389:
2359:
2332:
2312:
2282:
2259:
2239:
2216:
2196:
2176:
2142:
2122:
2102:
2069:
2039:
2009:
1983:
1953:
1927:
1907:
1887:
1856:
1808:
1788:
1764:
1738:
1704:
1667:
1590:
1509:
1489:
1456:
1375:
1352:
1166:
1142:
1116:
1096:
1072:
1005:
969:
949:
913:
893:
862:
842:
822:
802:
736:
704:
684:
655:
635:
615:
583:
563:
543:
523:
492:
472:
452:
425:
398:
378:
358:
338:
308:
288:
265:
238:
218:
206:is differentiable as well. If these functions are
198:
148:
113:
93:
8183:
6291:{\displaystyle {\text{Diff}}(\mathbb {R} ^{n},0)}
5374:-compact, there is a sequence of compact subsets
1457:{\displaystyle \det Df_{x}=4(x^{2}+y^{2})\neq 0.}
9363:
8256:The structure of classical diffeomorphism groups
7471:; it was proved in full generality by Thurston.
6961:
5624:, this follows by fixing a Riemannian metric on
4806:of preserving (the appropriate type of) angles.
1397:
8435:Collected Works Vol. III, Differential Topology
7280:classifying elements of the mapping class group
1682:, it has a well-defined inverse if and only if
7651:The homotopy-type of diffeomorphism groups of
4351:, a stress-induced transformation is called a
30:"Diffeo" redirects here. For the company, see
8500:
8387:
8186:"Path-integral formulation of closed strings"
5735:
3307:could only be a diffeomorphism away from the
1857:{\displaystyle \partial f_{i}/\partial x_{j}}
8329:
8151:Krantz, Steven G.; Parks, Harold R. (2013).
7799:does not have the homotopy-type of a finite
6215:The diffeomorphism group of Euclidean space
2539:is a stronger condition than a homeomorphic
2470:, for example, is not a diffeomorphism from
2344:(or, locally, a "local submersion"); and if
1274:
1256:
1235:
1217:
8150:
7887:pairwise non-diffeomorphic open subsets of
7400:{\displaystyle {\text{SL}}(2,\mathbb {Z} )}
3946:i.e. the linear terms in the components of
199:{\displaystyle f^{-1}\colon N\rightarrow M}
8507:
8493:
8175:: CS1 maint: location missing publisher (
8089:Steven G. Krantz; Harold R. Parks (2013).
7264:showed that it can be identified with the
6370:{\displaystyle f(x)\to f(tx)/t,t\in (0,1]}
4548:. Suppose that in a chart of the surface,
3693:. We can calculate the Jacobian matrix at
1668:{\displaystyle Df_{x}:T_{x}U\to T_{f(x)}V}
8405:
8363:
8332:The convenient setting of global analysis
8131:
7987:
7954:
7925:
7896:
7863:
7591:
7390:
7345:
7328:
6269:
6224:
5038:The diffeomorphism group has two natural
3426:, and thus it cannot be a diffeomorphism.
3016:
2987:
2478:
1868:is often used for explicit computations.
1517:is not invertible because it fails to be
1243:
1204:
1060:
1045:
937:
803:{\displaystyle g_{|U\cap X}=f_{|U\cap X}}
8514:
7365:, the mapping class group is simply the
7282:into three types: those equivalent to a
63:
8330:Kriegl, Andreas; Michor, Peter (1997),
8250:
6955:
4809:
4342:
2890:{\displaystyle \psi f\phi ^{-1}:U\to V}
504:Diffeomorphisms of subsets of manifolds
27:Isomorphism of differentiable manifolds
14:
9364:
8468:
8429:
8394:Annals of Global Analysis and Geometry
8347:
8307:
8155:. Modern BirkhÀuser classics. Boston.
7507:has the homotopy-type of the subgroup
5601:
5051:
4097:We can calculate the Jacobian matrix:
3150:We can calculate the Jacobian matrix:
149:{\displaystyle f\colon M\rightarrow N}
8488:
8450:
8271:
8113:
6791:that preserve diffeomorphism classes.
6572:that preserves all the components of
4834:be a differentiable manifold that is
7792:{\displaystyle {\text{Diff}}(S^{n})}
7728:{\displaystyle {\text{Diff}}(S^{4})}
6637:is the permutation group of the set
6630:{\displaystyle \Sigma (\pi _{0}(M))}
6534:{\displaystyle {\text{Diff}}_{0}(M)}
4948:{\displaystyle {\text{Diff}}^{r}(M)}
2421:(or, locally, a "local immersion").
1591:{\displaystyle f(1,0)=(1,0)=f(-1,0)}
873:
8093:. Springer. p. Theorem 6.2.4.
6056:denote the diffeomorphism group of
5702:, the space of vector fields is a
5457:
5187:
4457:. In fact, it is required that for
500:times continuously differentiable.
226:times continuously differentiable,
81:Given two differentiable manifolds
24:
8472:(1926), "Lösung der Aufgabe 41.",
7166:
6717:
6693:). Moreover, the image of the map
6599:
6465:
6166:{\displaystyle {\text{Diff}}(G,e)}
6005:, there is a natural inclusion of
5937:
5933:
5689:
5026:is not zero-dimensionalâit is not
4673:
4665:
4644:
4636:
2227:Given a smooth map from dimension
1991:could never be surjective, and if
1841:
1823:
25:
9388:
8437:, American Mathematical Society,
8133:10.1090/s0002-9939-1959-0112149-8
8116:"Diffeomorphisms of the 2-sphere"
7916:each of which is homeomorphic to
7839:More unusual phenomena occur for
7536:. This was proven by Steve Smale.
7474:
4767:{\displaystyle (du,dv)=(dx,dy)Df}
1253:
1214:
8246:from the original on 2018-07-21.
8000:{\displaystyle \mathbb {R} ^{4}}
7967:{\displaystyle \mathbb {R} ^{4}}
7938:{\displaystyle \mathbb {R} ^{4}}
7909:{\displaystyle \mathbb {R} ^{4}}
7876:{\displaystyle \mathbb {R} ^{4}}
7808:Homeomorphism and diffeomorphism
7240:), the mapping class group is a
7227:
6565:{\displaystyle {\text{Diff}}(M)}
6237:{\displaystyle \mathbb {R} ^{n}}
6201:{\displaystyle {\text{Diff}}(G)}
6049:{\displaystyle {\text{Diff}}(G)}
4999:{\displaystyle {\text{Diff}}(M)}
3029:{\displaystyle \mathbb {R} ^{2}}
3000:{\displaystyle \mathbb {R} ^{2}}
2809:as, respectively, the images of
950:{\displaystyle \mathbb {R} ^{n}}
623:is said to be smooth if for all
8453:Infinite-dimensional Lie groups
6795:
5750:of the diffeomorphism group of
5314:varies over compact subsets of
2184:will also be bijective for all
346:) if there is a diffeomorphism
8547:Differentiable/Smooth manifold
8274:Harmonic Mappings in the Plane
8107:
8082:
7786:
7773:
7722:
7709:
7595:
7581:
7523:
7517:
7394:
7380:
7094:
7088:
7058:
7049:
7043:
7034:
7022:
7016:
6778:
6772:
6742:
6739:
6733:
6720:
6714:
6711:
6705:
6660:
6654:
6624:
6621:
6615:
6602:
6559:
6553:
6528:
6522:
6490:
6487:
6481:
6468:
6462:
6459:
6453:
6445:
6442:
6436:
6421:
6364:
6352:
6332:
6323:
6317:
6314:
6308:
6285:
6264:
6195:
6189:
6160:
6148:
6123:
6111:
6094:
6088:
6043:
6037:
5928:
5922:
5873:
5867:
5838:
5741:
5551:
5539:
5511:
5499:
5450:
5438:
5277:
5273:
5267:
5248:
5242:
5225:
5180:
5177:
5171:
5162:
5156:
5150:
5125:
5113:
4993:
4987:
4942:
4936:
4755:
4737:
4731:
4713:
4588:
4576:
4570:
4558:
4371:
4302:
4276:
4256:
4230:
4208:
4182:
4165:
4139:
4072:
4046:
4034:
4008:
3991:
3979:
3728:
3716:
3459:
3447:
3419:{\displaystyle f(x,y)=f(-x,y)}
3413:
3398:
3389:
3377:
3068:
3056:
2957:
2951:
2932:
2929:
2923:
2907:
2881:
2625:
2496:that is not a diffeomorphism.
2444:
2438:
2428:necessarily a diffeomorphism.
2424:A differentiable bijection is
1657:
1651:
1640:
1585:
1570:
1561:
1549:
1543:
1531:
1445:
1419:
1337:
1299:
1296:
1293:
1281:
1271:
1259:
1238:
1232:
1220:
1055:
997:
785:
761:
728:
607:
460:-diffeomorphic if there is an
433:-differentiable manifolds are
190:
140:
13:
1:
8144:
7288:pseudo-Anosov diffeomorphisms
6962:Extensions of diffeomorphisms
5969:
4597:{\displaystyle f(x,y)=(u,v).}
3843:is a local diffeomorphism at
3254:The Jacobian matrix has zero
2603:and its inverse need only be
1013:is a diffeomorphism if it is
76:
8365:10.1016/0040-9383(67)90038-9
7480:The diffeomorphism group of
7459:smooth closed manifold, the
6076:, then there is a splitting
5796:Lie bracket of vector fields
5604:). Over a compact subset of
2729:be charts on, respectively,
2485:{\displaystyle \mathbb {R} }
1497:is bijective at each point,
7:
9253:Classification of manifolds
8296:Encyclopedia of Mathematics
8010:
7832:over the 4-sphere with the
7429:Brouwer fixed-point theorem
7178:{\displaystyle \Gamma _{n}}
6820:, the diffeomorphism group
6784:{\displaystyle \pi _{0}(M)}
6666:{\displaystyle \pi _{0}(M)}
6408:is any manifold there is a
5209:
5132:
5033:
2973:
2641:is a diffeomorphism if, in
2579:and its inverse need to be
59:continuously differentiable
10:
9393:
7248:; this has been proved by
2463:{\displaystyle f(x)=x^{3}}
1080:is bijective (and hence a
685:{\displaystyle U\subset M}
29:
9329:over commutative algebras
9286:
9245:
9178:
9075:
8971:
8918:
8909:
8745:
8668:
8607:
8527:
8416:10.1007/s10455-013-9380-2
7431:became applicable. Smale
7278:refined this analysis by
6800:For a connected manifold
5736:Michor & Mumford 2013
5712:regular Fréchet Lie group
5695:{\displaystyle r=\infty }
2077:fails to be a bijection.
1765:{\displaystyle n\times n}
339:{\displaystyle M\simeq N}
9045:Riemann curvature tensor
8272:Duren, Peter L. (2004),
8213:10.1103/physrevd.36.1148
8075:
7851:led to the discovery of
7266:outer automorphism group
7242:finitely presented group
7187:group of twisted spheres
7009:of the reals satisfying
5818:at each point in space:
5054:). When the manifold is
4380:{\displaystyle f:U\to V}
2634:{\displaystyle f:M\to N}
2559:. For a diffeomorphism,
1006:{\displaystyle f:U\to V}
737:{\displaystyle g:U\to N}
616:{\displaystyle f:X\to Y}
51:differentiable manifolds
8308:Hirsch, Morris (1997),
7236:. In dimension 2 (i.e.
7136:{\displaystyle S^{n-1}}
5727:{\displaystyle \sigma }
5367:{\displaystyle \sigma }
3674:{\displaystyle b_{i,j}}
3641:{\displaystyle a_{i,j}}
3367:is not bijective since
2583:; for a homeomorphism,
2154:(since, by continuity,
981:, a differentiable map
8837:Manifold with boundary
8552:Differential structure
8451:Omori, Hideki (1997),
8001:
7968:
7939:
7910:
7877:
7793:
7755:
7754:{\displaystyle n>6}
7729:
7691:
7690:{\displaystyle n>3}
7665:
7634:
7633:{\displaystyle g>1}
7602:
7530:
7501:
7449:
7401:
7359:
7218:
7179:
7137:
7101:
7065:
7003:
6944:
6924:
6923:{\displaystyle F_{k}M}
6894:
6874:
6873:{\displaystyle C_{k}M}
6841:
6814:
6785:
6749:
6687:
6667:
6631:
6586:
6566:
6535:
6497:
6402:
6371:
6292:
6238:
6202:
6167:
6130:
6070:
6050:
6019:
5995:
5960:
5880:
5812:
5788:
5764:
5728:
5696:
5662:
5638:
5618:
5594:
5564:
5419:
5395:
5368:
5348:
5328:
5308:
5285:
5087:
5020:
5000:
4969:
4949:
4917:to itself, denoted by
4911:
4891:
4860:
4828:
4768:
4692:
4598:
4538:
4521:in which the Jacobian
4515:
4491:
4471:
4447:
4430:has a Jacobian matrix
4424:
4404:
4381:
4320:
4087:
3936:
3829:
3675:
3642:
3605:
3420:
3361:
3341:
3321:
3301:
3281:
3244:
3140:
3030:
3001:
2964:
2891:
2843:
2823:
2803:
2783:
2763:
2743:
2723:
2703:
2683:
2659:
2635:
2597:
2573:
2553:
2533:
2513:
2486:
2464:
2411:
2391:
2390:{\displaystyle Df_{x}}
2361:
2334:
2314:
2313:{\displaystyle Df_{x}}
2284:
2261:
2241:
2218:
2204:sufficiently close to
2198:
2178:
2177:{\displaystyle Df_{y}}
2144:
2124:
2104:
2103:{\displaystyle Df_{x}}
2071:
2070:{\displaystyle Df_{x}}
2041:
2040:{\displaystyle Df_{x}}
2011:
2010:{\displaystyle n>k}
1985:
1984:{\displaystyle Df_{x}}
1955:
1954:{\displaystyle n<k}
1929:
1909:
1889:
1858:
1810:
1790:
1772:matrix of first-order
1766:
1740:
1739:{\displaystyle Df_{x}}
1706:
1705:{\displaystyle Df_{x}}
1669:
1592:
1511:
1491:
1490:{\displaystyle Df_{x}}
1458:
1377:
1354:
1168:
1144:
1118:
1098:
1074:
1007:
971:
951:
915:
895:
864:
844:
824:
804:
738:
712:and a smooth function
706:
686:
657:
637:
617:
585:
565:
545:
525:
494:
474:
454:
427:
400:
380:
360:
340:
310:
290:
267:
240:
220:
200:
150:
115:
95:
73:
8310:Differential Topology
8120:Proc. Amer. Math. Soc
8030:gravitational anomaly
8017:Anosov diffeomorphism
8002:
7969:
7940:
7911:
7878:
7794:
7756:
7730:
7692:
7666:
7635:
7603:
7531:
7502:
7500:{\displaystyle S^{2}}
7450:
7402:
7360:
7290:. In the case of the
7219:
7217:{\displaystyle B^{n}}
7180:
7138:
7102:
7066:
7004:
6945:
6925:
6895:
6875:
6842:
6815:
6786:
6755:is the bijections of
6750:
6688:
6668:
6632:
6587:
6567:
6536:
6498:
6403:
6372:
6293:
6239:
6203:
6168:
6131:
6071:
6051:
6020:
5996:
5961:
5881:
5813:
5789:
5765:
5729:
5714:. If the manifold is
5697:
5663:
5639:
5619:
5595:
5593:{\displaystyle C^{r}}
5565:
5420:
5396:
5394:{\displaystyle K_{n}}
5369:
5349:
5329:
5309:
5286:
5088:
5021:
5001:
4970:
4950:
4912:
4892:
4890:{\displaystyle C^{r}}
4861:
4829:
4776:linear transformation
4769:
4693:
4599:
4539:
4516:
4492:
4472:
4448:
4425:
4405:
4382:
4321:
4088:
3937:
3830:
3676:
3643:
3606:
3421:
3362:
3342:
3322:
3302:
3282:
3245:
3141:
3031:
3002:
2965:
2892:
2844:
2842:{\displaystyle \psi }
2824:
2822:{\displaystyle \phi }
2804:
2784:
2764:
2744:
2724:
2722:{\displaystyle \psi }
2704:
2702:{\displaystyle \phi }
2684:
2660:
2636:
2598:
2574:
2554:
2534:
2514:
2487:
2465:
2412:
2392:
2362:
2335:
2315:
2285:
2262:
2242:
2219:
2199:
2179:
2145:
2125:
2105:
2072:
2042:
2012:
1986:
1956:
1930:
1910:
1895:going from dimension
1890:
1859:
1811:
1791:
1767:
1741:
1707:
1670:
1593:
1512:
1492:
1459:
1378:
1355:
1169:
1145:
1119:
1099:
1075:
1008:
972:
952:
916:
896:
865:
845:
825:
805:
739:
707:
687:
658:
638:
618:
586:
566:
546:
526:
495:
475:
455:
453:{\displaystyle C^{r}}
428:
426:{\displaystyle C^{r}}
401:
381:
361:
341:
311:
291:
268:
266:{\displaystyle C^{r}}
241:
221:
201:
151:
116:
96:
67:
9377:Mathematical physics
8984:Covariant derivative
8535:Topological manifold
8312:, Berlin, New York:
8065:Local diffeomorphism
8060:Large diffeomorphism
7982:
7949:
7920:
7891:
7858:
7765:
7739:
7701:
7675:
7655:
7618:
7547:
7529:{\displaystyle O(3)}
7511:
7484:
7439:
7372:
7297:
7201:
7162:
7114:
7100:{\displaystyle O(2)}
7082:
7013:
6993:
6934:
6904:
6884:
6854:
6831:
6804:
6759:
6697:
6677:
6641:
6596:
6576:
6545:
6507:
6415:
6392:
6302:
6256:
6246:general linear group
6219:
6181:
6140:
6080:
6060:
6029:
6009:
5979:
5896:
5825:
5802:
5778:
5754:
5718:
5680:
5652:
5648:for that metric. If
5628:
5608:
5577:
5432:
5409:
5378:
5358:
5338:
5318:
5298:
5100:
5077:
5010:
4979:
4959:
4921:
4901:
4874:
4850:
4844:diffeomorphism group
4818:
4810:Diffeomorphism group
4710:
4698:, and similarly for
4621:
4552:
4525:
4505:
4481:
4461:
4434:
4414:
4394:
4359:
4343:Surface deformations
4103:
3973:
3952:linearly independent
3853:
3703:
3652:
3619:
3441:
3371:
3351:
3331:
3311:
3291:
3280:{\displaystyle xy=0}
3262:
3156:
3050:
3011:
2982:
2901:
2853:
2833:
2813:
2793:
2773:
2753:
2733:
2713:
2693:
2673:
2669:and do the same for
2649:
2613:
2587:
2563:
2543:
2523:
2503:
2474:
2432:
2401:
2371:
2348:
2324:
2294:
2271:
2251:
2231:
2208:
2188:
2158:
2152:local diffeomorphism
2134:
2114:
2084:
2051:
2021:
1995:
1965:
1939:
1919:
1899:
1879:
1820:
1800:
1780:
1750:
1720:
1712:is a bijection. The
1686:
1608:
1525:
1501:
1471:
1394:
1367:
1185:
1158:
1134:
1130:It is essential for
1108:
1088:
1024:
985:
961:
932:
905:
885:
854:
834:
814:
752:
716:
696:
670:
647:
627:
595:
575:
555:
535:
515:
484:
464:
437:
410:
390:
370:
350:
324:
300:
280:
250:
230:
210:
168:
128:
105:
85:
9018:Exterior derivative
8620:AtiyahâSinger index
8569:Riemannian manifold
8205:1987PhRvD..36.1148C
8070:Superdiffeomorphism
7234:mapping class group
6952:multiply transitive
6849:configuration space
6673:(the components of
6541:is the subgroup of
6250:deformation retract
5994:{\displaystyle M=G}
4897:diffeomorphisms of
1776:whose entry in the
1774:partial derivatives
830:is an extension of
55:invertible function
9324:Secondary calculus
9278:Singularity theory
9233:Parallel transport
9001:De Rham cohomology
8640:Generalized Stokes
7997:
7964:
7935:
7906:
7873:
7789:
7751:
7725:
7687:
7661:
7646:fundamental groups
7630:
7598:
7526:
7497:
7461:identity component
7445:
7397:
7355:
7254:W. B. R. Lickorish
7214:
7189:", defined as the
7175:
7133:
7097:
7061:
6999:
6972:harmonic extension
6970:asked whether the
6940:
6930:and the action on
6920:
6890:
6870:
6837:
6810:
6781:
6745:
6683:
6663:
6627:
6582:
6562:
6531:
6493:
6398:
6367:
6288:
6234:
6198:
6163:
6126:
6066:
6046:
6015:
5991:
5956:
5876:
5808:
5794:equipped with the
5784:
5760:
5724:
5692:
5658:
5634:
5614:
5590:
5560:
5415:
5391:
5364:
5344:
5324:
5304:
5281:
5083:
5016:
4996:
4965:
4945:
4907:
4887:
4856:
4824:
4804:conformal property
4764:
4688:
4608:total differential
4594:
4537:{\displaystyle Df}
4534:
4511:
4487:
4467:
4446:{\displaystyle Df}
4443:
4420:
4400:
4377:
4316:
4307:
4083:
3932:
3825:
3816:
3671:
3638:
3601:
3416:
3357:
3337:
3317:
3297:
3277:
3240:
3231:
3136:
3026:
2997:
2960:
2887:
2839:
2819:
2799:
2779:
2759:
2739:
2719:
2699:
2679:
2655:
2631:
2593:
2569:
2549:
2529:
2509:
2482:
2460:
2407:
2387:
2360:{\displaystyle Df}
2357:
2330:
2310:
2283:{\displaystyle Df}
2280:
2257:
2237:
2214:
2194:
2174:
2140:
2120:
2110:is a bijection at
2100:
2067:
2037:
2007:
1981:
1951:
1925:
1905:
1885:
1854:
1806:
1786:
1762:
1736:
1716:representation of
1702:
1665:
1588:
1507:
1487:
1454:
1387:and it satisfies
1373:
1350:
1345:
1164:
1140:
1114:
1094:
1082:linear isomorphism
1070:
1003:
967:
947:
911:
891:
860:
840:
820:
800:
734:
702:
682:
653:
633:
613:
581:
561:
541:
521:
490:
470:
450:
423:
396:
376:
356:
336:
306:
286:
263:
236:
216:
196:
146:
123:differentiable map
111:
91:
74:
9359:
9358:
9241:
9240:
9006:Differential form
8660:Whitney embedding
8594:Differential form
8444:978-0-8218-4230-0
8323:978-0-387-90148-0
8252:Banyaga, Augustin
8193:Physical Review D
8162:978-1-4614-5980-4
8100:978-1-4614-5980-4
8049:Diffeomorphometry
8038:quantum mechanics
7771:
7707:
7664:{\displaystyle n}
7579:
7448:{\displaystyle M}
7425:TeichmĂŒller space
7378:
7270:fundamental group
7002:{\displaystyle f}
6943:{\displaystyle M}
6893:{\displaystyle M}
6840:{\displaystyle M}
6813:{\displaystyle M}
6703:
6686:{\displaystyle M}
6585:{\displaystyle M}
6551:
6514:
6451:
6428:
6401:{\displaystyle M}
6262:
6187:
6146:
6109:
6086:
6069:{\displaystyle G}
6035:
6018:{\displaystyle G}
5951:
5811:{\displaystyle x}
5787:{\displaystyle M}
5763:{\displaystyle M}
5661:{\displaystyle r}
5637:{\displaystyle M}
5617:{\displaystyle M}
5555:
5418:{\displaystyle M}
5347:{\displaystyle M}
5334:. Indeed, since
5327:{\displaystyle M}
5307:{\displaystyle K}
5086:{\displaystyle M}
5071:Riemannian metric
5019:{\displaystyle M}
4985:
4968:{\displaystyle r}
4928:
4910:{\displaystyle M}
4859:{\displaystyle M}
4827:{\displaystyle M}
4680:
4651:
4514:{\displaystyle p}
4490:{\displaystyle U}
4470:{\displaystyle p}
4455:invertible matrix
4423:{\displaystyle V}
4403:{\displaystyle U}
3847:if, and only if,
3535:
3360:{\displaystyle f}
3340:{\displaystyle y}
3320:{\displaystyle x}
3300:{\displaystyle f}
2802:{\displaystyle V}
2782:{\displaystyle U}
2762:{\displaystyle N}
2742:{\displaystyle M}
2682:{\displaystyle N}
2667:coordinate charts
2658:{\displaystyle M}
2643:coordinate charts
2596:{\displaystyle f}
2572:{\displaystyle f}
2552:{\displaystyle f}
2532:{\displaystyle f}
2512:{\displaystyle f}
2417:is said to be an
2410:{\displaystyle f}
2333:{\displaystyle f}
2320:) is surjective,
2260:{\displaystyle k}
2240:{\displaystyle n}
2217:{\displaystyle x}
2197:{\displaystyle y}
2143:{\displaystyle f}
2123:{\displaystyle x}
1928:{\displaystyle k}
1908:{\displaystyle n}
1888:{\displaystyle f}
1864:. This so-called
1809:{\displaystyle j}
1789:{\displaystyle i}
1510:{\displaystyle f}
1376:{\displaystyle f}
1178:square function
1167:{\displaystyle f}
1154:for the function
1143:{\displaystyle V}
1117:{\displaystyle U}
1097:{\displaystyle x}
970:{\displaystyle V}
914:{\displaystyle V}
894:{\displaystyle U}
874:Local description
863:{\displaystyle f}
843:{\displaystyle f}
823:{\displaystyle g}
705:{\displaystyle p}
656:{\displaystyle X}
636:{\displaystyle p}
584:{\displaystyle N}
564:{\displaystyle Y}
544:{\displaystyle M}
524:{\displaystyle X}
493:{\displaystyle r}
473:{\displaystyle r}
399:{\displaystyle N}
379:{\displaystyle M}
359:{\displaystyle f}
320:(usually denoted
309:{\displaystyle N}
289:{\displaystyle M}
273:-diffeomorphism.
239:{\displaystyle f}
219:{\displaystyle r}
114:{\displaystyle N}
94:{\displaystyle M}
16:(Redirected from
9384:
9351:Stratified space
9309:Fréchet manifold
9023:Interior product
8916:
8915:
8613:
8509:
8502:
8495:
8486:
8485:
8481:
8470:Kneser, Hellmuth
8465:
8447:
8426:
8409:
8384:
8367:
8344:
8326:
8304:
8291:"Diffeomorphism"
8286:
8268:
8247:
8245:
8199:(4): 1148â1168.
8190:
8180:
8174:
8166:
8138:
8137:
8135:
8111:
8105:
8104:
8086:
8028:also known as a
8021:Arnold's cat map
8006:
8004:
8003:
7998:
7996:
7995:
7990:
7973:
7971:
7970:
7965:
7963:
7962:
7957:
7944:
7942:
7941:
7936:
7934:
7933:
7928:
7915:
7913:
7912:
7907:
7905:
7904:
7899:
7885:uncountably many
7882:
7880:
7879:
7874:
7872:
7871:
7866:
7849:Michael Freedman
7798:
7796:
7795:
7790:
7785:
7784:
7772:
7769:
7760:
7758:
7757:
7752:
7734:
7732:
7731:
7726:
7721:
7720:
7708:
7705:
7696:
7694:
7693:
7688:
7670:
7668:
7667:
7662:
7639:
7637:
7636:
7631:
7607:
7605:
7604:
7599:
7594:
7580:
7577:
7572:
7571:
7559:
7558:
7535:
7533:
7532:
7527:
7506:
7504:
7503:
7498:
7496:
7495:
7454:
7452:
7451:
7446:
7421:compactification
7406:
7404:
7403:
7398:
7393:
7379:
7376:
7364:
7362:
7361:
7356:
7354:
7353:
7348:
7342:
7337:
7336:
7331:
7322:
7321:
7309:
7308:
7276:William Thurston
7272:of the surface.
7260:). Max Dehn and
7223:
7221:
7220:
7215:
7213:
7212:
7184:
7182:
7181:
7176:
7174:
7173:
7142:
7140:
7139:
7134:
7132:
7131:
7106:
7104:
7103:
7098:
7077:orthogonal group
7070:
7068:
7067:
7064:{\displaystyle }
7062:
7008:
7006:
7005:
7000:
6949:
6947:
6946:
6941:
6929:
6927:
6926:
6921:
6916:
6915:
6899:
6897:
6896:
6891:
6879:
6877:
6876:
6871:
6866:
6865:
6846:
6844:
6843:
6838:
6819:
6817:
6816:
6811:
6790:
6788:
6787:
6782:
6771:
6770:
6754:
6752:
6751:
6746:
6732:
6731:
6704:
6701:
6692:
6690:
6689:
6684:
6672:
6670:
6669:
6664:
6653:
6652:
6636:
6634:
6633:
6628:
6614:
6613:
6591:
6589:
6588:
6583:
6571:
6569:
6568:
6563:
6552:
6549:
6540:
6538:
6537:
6532:
6521:
6520:
6515:
6512:
6502:
6500:
6499:
6494:
6480:
6479:
6452:
6449:
6435:
6434:
6429:
6426:
6407:
6405:
6404:
6399:
6388:. Similarly, if
6376:
6374:
6373:
6368:
6339:
6297:
6295:
6294:
6289:
6278:
6277:
6272:
6263:
6260:
6252:of the subgroup
6243:
6241:
6240:
6235:
6233:
6232:
6227:
6210:identity element
6207:
6205:
6204:
6199:
6188:
6185:
6172:
6170:
6169:
6164:
6147:
6144:
6135:
6133:
6132:
6127:
6110:
6107:
6087:
6084:
6075:
6073:
6072:
6067:
6055:
6053:
6052:
6047:
6036:
6033:
6024:
6022:
6021:
6016:
6000:
5998:
5997:
5992:
5965:
5963:
5962:
5957:
5952:
5950:
5949:
5948:
5932:
5921:
5920:
5908:
5907:
5885:
5883:
5882:
5877:
5866:
5865:
5850:
5849:
5837:
5836:
5817:
5815:
5814:
5809:
5793:
5791:
5790:
5785:
5770:consists of all
5769:
5767:
5766:
5761:
5733:
5731:
5730:
5725:
5710:and even into a
5708:Fréchet manifold
5701:
5699:
5698:
5693:
5667:
5665:
5664:
5659:
5643:
5641:
5640:
5635:
5623:
5621:
5620:
5615:
5599:
5597:
5596:
5591:
5589:
5588:
5569:
5567:
5566:
5561:
5556:
5554:
5538:
5537:
5536:
5535:
5514:
5498:
5497:
5496:
5495:
5480:
5478:
5477:
5465:
5464:
5424:
5422:
5421:
5416:
5400:
5398:
5397:
5392:
5390:
5389:
5373:
5371:
5370:
5365:
5353:
5351:
5350:
5345:
5333:
5331:
5330:
5325:
5313:
5311:
5310:
5305:
5290:
5288:
5287:
5282:
5280:
5276:
5263:
5262:
5238:
5237:
5223:
5222:
5207:
5206:
5146:
5145:
5112:
5111:
5092:
5090:
5089:
5084:
5025:
5023:
5022:
5017:
5005:
5003:
5002:
4997:
4986:
4983:
4974:
4972:
4971:
4966:
4954:
4952:
4951:
4946:
4935:
4934:
4929:
4926:
4916:
4914:
4913:
4908:
4896:
4894:
4893:
4888:
4886:
4885:
4865:
4863:
4862:
4857:
4836:second-countable
4833:
4831:
4830:
4825:
4773:
4771:
4770:
4765:
4697:
4695:
4694:
4689:
4681:
4679:
4671:
4663:
4652:
4650:
4642:
4634:
4603:
4601:
4600:
4595:
4543:
4541:
4540:
4535:
4520:
4518:
4517:
4512:
4496:
4494:
4493:
4488:
4476:
4474:
4473:
4468:
4452:
4450:
4449:
4444:
4429:
4427:
4426:
4421:
4409:
4407:
4406:
4401:
4386:
4384:
4383:
4378:
4325:
4323:
4322:
4317:
4312:
4311:
4301:
4300:
4288:
4287:
4255:
4254:
4242:
4241:
4207:
4206:
4194:
4193:
4164:
4163:
4151:
4150:
4115:
4114:
4092:
4090:
4089:
4084:
4079:
4075:
4071:
4070:
4058:
4057:
4033:
4032:
4020:
4019:
3941:
3939:
3938:
3933:
3922:
3921:
3906:
3905:
3887:
3886:
3871:
3870:
3834:
3832:
3831:
3826:
3821:
3820:
3813:
3812:
3795:
3794:
3775:
3774:
3757:
3756:
3715:
3714:
3680:
3678:
3677:
3672:
3670:
3669:
3647:
3645:
3644:
3639:
3637:
3636:
3610:
3608:
3607:
3602:
3600:
3596:
3586:
3585:
3564:
3563:
3545:
3544:
3533:
3520:
3519:
3498:
3497:
3479:
3478:
3425:
3423:
3422:
3417:
3366:
3364:
3363:
3358:
3347:-axis. However,
3346:
3344:
3343:
3338:
3326:
3324:
3323:
3318:
3306:
3304:
3303:
3298:
3286:
3284:
3283:
3278:
3249:
3247:
3246:
3241:
3236:
3235:
3228:
3227:
3200:
3199:
3168:
3167:
3145:
3143:
3142:
3137:
3132:
3128:
3127:
3126:
3114:
3113:
3101:
3100:
3088:
3087:
3035:
3033:
3032:
3027:
3025:
3024:
3019:
3006:
3004:
3003:
2998:
2996:
2995:
2990:
2969:
2967:
2966:
2961:
2950:
2949:
2922:
2921:
2896:
2894:
2893:
2888:
2874:
2873:
2848:
2846:
2845:
2840:
2828:
2826:
2825:
2820:
2808:
2806:
2805:
2800:
2788:
2786:
2785:
2780:
2768:
2766:
2765:
2760:
2748:
2746:
2745:
2740:
2728:
2726:
2725:
2720:
2708:
2706:
2705:
2700:
2688:
2686:
2685:
2680:
2664:
2662:
2661:
2656:
2640:
2638:
2637:
2632:
2602:
2600:
2599:
2594:
2578:
2576:
2575:
2570:
2558:
2556:
2555:
2550:
2538:
2536:
2535:
2530:
2518:
2516:
2515:
2510:
2491:
2489:
2488:
2483:
2481:
2469:
2467:
2466:
2461:
2459:
2458:
2416:
2414:
2413:
2408:
2397:) is injective,
2396:
2394:
2393:
2388:
2386:
2385:
2366:
2364:
2363:
2358:
2340:is said to be a
2339:
2337:
2336:
2331:
2319:
2317:
2316:
2311:
2309:
2308:
2289:
2287:
2286:
2281:
2266:
2264:
2263:
2258:
2246:
2244:
2243:
2238:
2223:
2221:
2220:
2215:
2203:
2201:
2200:
2195:
2183:
2181:
2180:
2175:
2173:
2172:
2150:is said to be a
2149:
2147:
2146:
2141:
2129:
2127:
2126:
2121:
2109:
2107:
2106:
2101:
2099:
2098:
2076:
2074:
2073:
2068:
2066:
2065:
2046:
2044:
2043:
2038:
2036:
2035:
2016:
2014:
2013:
2008:
1990:
1988:
1987:
1982:
1980:
1979:
1960:
1958:
1957:
1952:
1934:
1932:
1931:
1926:
1914:
1912:
1911:
1906:
1894:
1892:
1891:
1886:
1863:
1861:
1860:
1855:
1853:
1852:
1840:
1835:
1834:
1815:
1813:
1812:
1807:
1795:
1793:
1792:
1787:
1771:
1769:
1768:
1763:
1745:
1743:
1742:
1737:
1735:
1734:
1711:
1709:
1708:
1703:
1701:
1700:
1674:
1672:
1671:
1666:
1661:
1660:
1636:
1635:
1623:
1622:
1597:
1595:
1594:
1589:
1516:
1514:
1513:
1508:
1496:
1494:
1493:
1488:
1486:
1485:
1463:
1461:
1460:
1455:
1444:
1443:
1431:
1430:
1412:
1411:
1382:
1380:
1379:
1374:
1359:
1357:
1356:
1351:
1349:
1348:
1324:
1323:
1311:
1310:
1252:
1251:
1246:
1213:
1212:
1207:
1173:
1171:
1170:
1165:
1152:simply connected
1149:
1147:
1146:
1141:
1123:
1121:
1120:
1115:
1103:
1101:
1100:
1095:
1084:) at each point
1079:
1077:
1076:
1071:
1069:
1068:
1063:
1054:
1053:
1048:
1039:
1038:
1012:
1010:
1009:
1004:
979:simply connected
976:
974:
973:
968:
956:
954:
953:
948:
946:
945:
940:
920:
918:
917:
912:
900:
898:
897:
892:
869:
867:
866:
861:
850:). The function
849:
847:
846:
841:
829:
827:
826:
821:
809:
807:
806:
801:
799:
798:
788:
775:
774:
764:
743:
741:
740:
735:
711:
709:
708:
703:
691:
689:
688:
683:
662:
660:
659:
654:
642:
640:
639:
634:
622:
620:
619:
614:
590:
588:
587:
582:
570:
568:
567:
562:
550:
548:
547:
542:
530:
528:
527:
522:
499:
497:
496:
491:
479:
477:
476:
471:
459:
457:
456:
451:
449:
448:
432:
430:
429:
424:
422:
421:
405:
403:
402:
397:
385:
383:
382:
377:
365:
363:
362:
357:
345:
343:
342:
337:
315:
313:
312:
307:
295:
293:
292:
287:
272:
270:
269:
264:
262:
261:
245:
243:
242:
237:
225:
223:
222:
217:
205:
203:
202:
197:
183:
182:
164:and its inverse
155:
153:
152:
147:
120:
118:
117:
112:
100:
98:
97:
92:
32:Diffeo (company)
21:
9392:
9391:
9387:
9386:
9385:
9383:
9382:
9381:
9372:Diffeomorphisms
9362:
9361:
9360:
9355:
9294:Banach manifold
9287:Generalizations
9282:
9237:
9174:
9071:
9033:Ricci curvature
8989:Cotangent space
8967:
8905:
8747:
8741:
8700:Exponential map
8664:
8609:
8603:
8523:
8513:
8463:
8445:
8431:Milnor, John W.
8342:
8324:
8314:Springer-Verlag
8289:
8284:
8266:
8243:
8188:
8168:
8167:
8163:
8147:
8142:
8141:
8112:
8108:
8101:
8087:
8083:
8078:
8013:
7991:
7986:
7985:
7983:
7980:
7979:
7958:
7953:
7952:
7950:
7947:
7946:
7929:
7924:
7923:
7921:
7918:
7917:
7900:
7895:
7894:
7892:
7889:
7888:
7867:
7862:
7861:
7859:
7856:
7855:
7845:Simon Donaldson
7836:as the fiber).
7826:Milnor's sphere
7810:
7780:
7776:
7768:
7766:
7763:
7762:
7740:
7737:
7736:
7716:
7712:
7704:
7702:
7699:
7698:
7676:
7673:
7672:
7671:-manifolds for
7656:
7653:
7652:
7619:
7616:
7615:
7590:
7576:
7567:
7563:
7554:
7550:
7548:
7545:
7544:
7512:
7509:
7508:
7491:
7487:
7485:
7482:
7481:
7477:
7440:
7437:
7436:
7389:
7375:
7373:
7370:
7369:
7349:
7344:
7343:
7338:
7332:
7327:
7326:
7317:
7313:
7304:
7300:
7298:
7295:
7294:
7230:
7208:
7204:
7202:
7199:
7198:
7195:component group
7193:of the abelian
7169:
7165:
7163:
7160:
7159:
7121:
7117:
7115:
7112:
7111:
7083:
7080:
7079:
7073:Alexander trick
7014:
7011:
7010:
6994:
6991:
6990:
6984:Gustave Choquet
6980:Hellmuth Kneser
6964:
6958:, p. 29).
6935:
6932:
6931:
6911:
6907:
6905:
6902:
6901:
6885:
6882:
6881:
6861:
6857:
6855:
6852:
6851:
6832:
6829:
6828:
6805:
6802:
6801:
6798:
6766:
6762:
6760:
6757:
6756:
6727:
6723:
6700:
6698:
6695:
6694:
6678:
6675:
6674:
6648:
6644:
6642:
6639:
6638:
6609:
6605:
6597:
6594:
6593:
6577:
6574:
6573:
6548:
6546:
6543:
6542:
6516:
6511:
6510:
6508:
6505:
6504:
6475:
6471:
6448:
6430:
6425:
6424:
6416:
6413:
6412:
6410:group extension
6393:
6390:
6389:
6386:symmetric group
6335:
6303:
6300:
6299:
6273:
6268:
6267:
6259:
6257:
6254:
6253:
6228:
6223:
6222:
6220:
6217:
6216:
6208:that fixes the
6184:
6182:
6179:
6178:
6143:
6141:
6138:
6137:
6106:
6083:
6081:
6078:
6077:
6061:
6058:
6057:
6032:
6030:
6027:
6026:
6010:
6007:
6006:
5980:
5977:
5976:
5972:
5944:
5940:
5936:
5931:
5916:
5912:
5903:
5899:
5897:
5894:
5893:
5861:
5857:
5845:
5841:
5832:
5828:
5826:
5823:
5822:
5803:
5800:
5799:
5779:
5776:
5775:
5755:
5752:
5751:
5744:
5719:
5716:
5715:
5681:
5678:
5677:
5674:Banach manifold
5653:
5650:
5649:
5646:exponential map
5629:
5626:
5625:
5609:
5606:
5605:
5600:vector fields (
5584:
5580:
5578:
5575:
5574:
5531:
5527:
5526:
5522:
5515:
5491:
5487:
5486:
5482:
5481:
5479:
5470:
5466:
5460:
5456:
5433:
5430:
5429:
5410:
5407:
5406:
5385:
5381:
5379:
5376:
5375:
5359:
5356:
5355:
5339:
5336:
5335:
5319:
5316:
5315:
5299:
5296:
5295:
5258:
5254:
5233:
5229:
5228:
5224:
5212:
5208:
5190:
5186:
5135:
5131:
5107:
5103:
5101:
5098:
5097:
5078:
5075:
5074:
5036:
5028:locally compact
5011:
5008:
5007:
4982:
4980:
4977:
4976:
4975:is understood,
4960:
4957:
4956:
4930:
4925:
4924:
4922:
4919:
4918:
4902:
4899:
4898:
4881:
4877:
4875:
4872:
4871:
4851:
4848:
4847:
4819:
4816:
4815:
4812:
4711:
4708:
4707:
4706:Then the image
4672:
4664:
4662:
4643:
4635:
4633:
4622:
4619:
4618:
4553:
4550:
4549:
4526:
4523:
4522:
4506:
4503:
4502:
4482:
4479:
4478:
4462:
4459:
4458:
4435:
4432:
4431:
4415:
4412:
4411:
4395:
4392:
4391:
4360:
4357:
4356:
4345:
4306:
4305:
4296:
4292:
4283:
4279:
4259:
4250:
4246:
4237:
4233:
4212:
4211:
4202:
4198:
4189:
4185:
4168:
4159:
4155:
4146:
4142:
4120:
4119:
4110:
4106:
4104:
4101:
4100:
4066:
4062:
4053:
4049:
4028:
4024:
4015:
4011:
4001:
3997:
3974:
3971:
3970:
3911:
3907:
3895:
3891:
3876:
3872:
3860:
3856:
3854:
3851:
3850:
3815:
3814:
3802:
3798:
3796:
3784:
3780:
3777:
3776:
3764:
3760:
3758:
3746:
3742:
3735:
3734:
3710:
3706:
3704:
3701:
3700:
3659:
3655:
3653:
3650:
3649:
3626:
3622:
3620:
3617:
3616:
3575:
3571:
3553:
3549:
3540:
3536:
3509:
3505:
3487:
3483:
3474:
3470:
3469:
3465:
3442:
3439:
3438:
3372:
3369:
3368:
3352:
3349:
3348:
3332:
3329:
3328:
3312:
3309:
3308:
3292:
3289:
3288:
3263:
3260:
3259:
3258:if and only if
3230:
3229:
3223:
3219:
3211:
3202:
3201:
3195:
3191:
3186:
3173:
3172:
3163:
3159:
3157:
3154:
3153:
3122:
3118:
3109:
3105:
3096:
3092:
3083:
3079:
3078:
3074:
3051:
3048:
3047:
3020:
3015:
3014:
3012:
3009:
3008:
2991:
2986:
2985:
2983:
2980:
2979:
2976:
2942:
2938:
2914:
2910:
2902:
2899:
2898:
2866:
2862:
2854:
2851:
2850:
2834:
2831:
2830:
2814:
2811:
2810:
2794:
2791:
2790:
2774:
2771:
2770:
2754:
2751:
2750:
2734:
2731:
2730:
2714:
2711:
2710:
2694:
2691:
2690:
2674:
2671:
2670:
2650:
2647:
2646:
2614:
2611:
2610:
2588:
2585:
2584:
2564:
2561:
2560:
2544:
2541:
2540:
2524:
2521:
2520:
2504:
2501:
2500:
2477:
2475:
2472:
2471:
2454:
2450:
2433:
2430:
2429:
2402:
2399:
2398:
2381:
2377:
2372:
2369:
2368:
2349:
2346:
2345:
2325:
2322:
2321:
2304:
2300:
2295:
2292:
2291:
2272:
2269:
2268:
2252:
2249:
2248:
2232:
2229:
2228:
2209:
2206:
2205:
2189:
2186:
2185:
2168:
2164:
2159:
2156:
2155:
2135:
2132:
2131:
2115:
2112:
2111:
2094:
2090:
2085:
2082:
2081:
2061:
2057:
2052:
2049:
2048:
2031:
2027:
2022:
2019:
2018:
1996:
1993:
1992:
1975:
1971:
1966:
1963:
1962:
1940:
1937:
1936:
1920:
1917:
1916:
1900:
1897:
1896:
1880:
1877:
1876:
1866:Jacobian matrix
1848:
1844:
1836:
1830:
1826:
1821:
1818:
1817:
1801:
1798:
1797:
1781:
1778:
1777:
1751:
1748:
1747:
1730:
1726:
1721:
1718:
1717:
1696:
1692:
1687:
1684:
1683:
1647:
1643:
1631:
1627:
1618:
1614:
1609:
1606:
1605:
1526:
1523:
1522:
1502:
1499:
1498:
1481:
1477:
1472:
1469:
1468:
1439:
1435:
1426:
1422:
1407:
1403:
1395:
1392:
1391:
1368:
1365:
1364:
1344:
1343:
1319:
1315:
1306:
1302:
1278:
1277:
1247:
1242:
1241:
1208:
1203:
1202:
1189:
1188:
1186:
1183:
1182:
1159:
1156:
1155:
1135:
1132:
1131:
1109:
1106:
1105:
1089:
1086:
1085:
1064:
1059:
1058:
1049:
1044:
1043:
1034:
1030:
1025:
1022:
1021:
986:
983:
982:
962:
959:
958:
941:
936:
935:
933:
930:
929:
906:
903:
902:
886:
883:
882:
876:
855:
852:
851:
835:
832:
831:
815:
812:
811:
784:
783:
779:
760:
759:
755:
753:
750:
749:
717:
714:
713:
697:
694:
693:
671:
668:
667:
648:
645:
644:
628:
625:
624:
596:
593:
592:
576:
573:
572:
556:
553:
552:
536:
533:
532:
516:
513:
512:
506:
485:
482:
481:
465:
462:
461:
444:
440:
438:
435:
434:
417:
413:
411:
408:
407:
391:
388:
387:
371:
368:
367:
351:
348:
347:
325:
322:
321:
301:
298:
297:
281:
278:
277:
257:
253:
251:
248:
247:
231:
228:
227:
211:
208:
207:
175:
171:
169:
166:
165:
129:
126:
125:
106:
103:
102:
86:
83:
82:
79:
35:
28:
23:
22:
18:Diffeomorphisms
15:
12:
11:
5:
9390:
9380:
9379:
9374:
9357:
9356:
9354:
9353:
9348:
9343:
9338:
9333:
9332:
9331:
9321:
9316:
9311:
9306:
9301:
9296:
9290:
9288:
9284:
9283:
9281:
9280:
9275:
9270:
9265:
9260:
9255:
9249:
9247:
9243:
9242:
9239:
9238:
9236:
9235:
9230:
9225:
9220:
9215:
9210:
9205:
9200:
9195:
9190:
9184:
9182:
9176:
9175:
9173:
9172:
9167:
9162:
9157:
9152:
9147:
9142:
9132:
9127:
9122:
9112:
9107:
9102:
9097:
9092:
9087:
9081:
9079:
9073:
9072:
9070:
9069:
9064:
9059:
9058:
9057:
9047:
9042:
9041:
9040:
9030:
9025:
9020:
9015:
9014:
9013:
9003:
8998:
8997:
8996:
8986:
8981:
8975:
8973:
8969:
8968:
8966:
8965:
8960:
8955:
8950:
8949:
8948:
8938:
8933:
8928:
8922:
8920:
8913:
8907:
8906:
8904:
8903:
8898:
8888:
8883:
8869:
8864:
8859:
8854:
8849:
8847:Parallelizable
8844:
8839:
8834:
8833:
8832:
8822:
8817:
8812:
8807:
8802:
8797:
8792:
8787:
8782:
8777:
8767:
8757:
8751:
8749:
8743:
8742:
8740:
8739:
8734:
8729:
8727:Lie derivative
8724:
8722:Integral curve
8719:
8714:
8709:
8708:
8707:
8697:
8692:
8691:
8690:
8683:Diffeomorphism
8680:
8674:
8672:
8666:
8665:
8663:
8662:
8657:
8652:
8647:
8642:
8637:
8632:
8627:
8622:
8616:
8614:
8605:
8604:
8602:
8601:
8596:
8591:
8586:
8581:
8576:
8571:
8566:
8561:
8560:
8559:
8554:
8544:
8543:
8542:
8531:
8529:
8528:Basic concepts
8525:
8524:
8512:
8511:
8504:
8497:
8489:
8483:
8482:
8466:
8461:
8448:
8443:
8427:
8400:(4): 529â540,
8385:
8358:(2): 263â271,
8345:
8340:
8327:
8322:
8305:
8287:
8282:
8269:
8264:
8248:
8181:
8161:
8146:
8143:
8140:
8139:
8126:(4): 621â626.
8114:Smale (1959).
8106:
8099:
8080:
8079:
8077:
8074:
8073:
8072:
8067:
8062:
8057:
8055:Ătale morphism
8052:
8046:
8040:
8026:Diffeo anomaly
8023:
8012:
8009:
7994:
7989:
7976:embed smoothly
7961:
7956:
7932:
7927:
7903:
7898:
7870:
7865:
7809:
7806:
7805:
7804:
7788:
7783:
7779:
7775:
7750:
7747:
7744:
7724:
7719:
7715:
7711:
7686:
7683:
7680:
7660:
7649:
7641:
7629:
7626:
7623:
7609:
7597:
7593:
7589:
7586:
7583:
7575:
7570:
7566:
7562:
7557:
7553:
7537:
7525:
7522:
7519:
7516:
7494:
7490:
7476:
7475:Homotopy types
7473:
7444:
7396:
7392:
7388:
7385:
7382:
7352:
7347:
7341:
7335:
7330:
7325:
7320:
7316:
7312:
7307:
7303:
7229:
7226:
7211:
7207:
7172:
7168:
7130:
7127:
7124:
7120:
7096:
7093:
7090:
7087:
7060:
7057:
7054:
7051:
7048:
7045:
7042:
7039:
7036:
7033:
7030:
7027:
7024:
7021:
7018:
6998:
6963:
6960:
6939:
6919:
6914:
6910:
6889:
6869:
6864:
6860:
6836:
6809:
6797:
6794:
6793:
6792:
6780:
6777:
6774:
6769:
6765:
6744:
6741:
6738:
6735:
6730:
6726:
6722:
6719:
6716:
6713:
6710:
6707:
6682:
6662:
6659:
6656:
6651:
6647:
6626:
6623:
6620:
6617:
6612:
6608:
6604:
6601:
6581:
6561:
6558:
6555:
6530:
6527:
6524:
6519:
6492:
6489:
6486:
6483:
6478:
6474:
6470:
6467:
6464:
6461:
6458:
6455:
6447:
6444:
6441:
6438:
6433:
6423:
6420:
6397:
6378:
6366:
6363:
6360:
6357:
6354:
6351:
6348:
6345:
6342:
6338:
6334:
6331:
6328:
6325:
6322:
6319:
6316:
6313:
6310:
6307:
6287:
6284:
6281:
6276:
6271:
6266:
6231:
6226:
6213:
6197:
6194:
6191:
6162:
6159:
6156:
6153:
6150:
6125:
6122:
6119:
6116:
6113:
6105:
6102:
6099:
6096:
6093:
6090:
6065:
6045:
6042:
6039:
6014:
5990:
5987:
5984:
5971:
5968:
5967:
5966:
5955:
5947:
5943:
5939:
5935:
5930:
5927:
5924:
5919:
5915:
5911:
5906:
5902:
5887:
5886:
5875:
5872:
5869:
5864:
5860:
5856:
5853:
5848:
5844:
5840:
5835:
5831:
5807:
5783:
5759:
5743:
5740:
5723:
5691:
5688:
5685:
5657:
5644:and using the
5633:
5613:
5587:
5583:
5571:
5570:
5559:
5553:
5550:
5547:
5544:
5541:
5534:
5530:
5525:
5521:
5518:
5513:
5510:
5507:
5504:
5501:
5494:
5490:
5485:
5476:
5473:
5469:
5463:
5459:
5455:
5452:
5449:
5446:
5443:
5440:
5437:
5414:
5388:
5384:
5363:
5343:
5323:
5303:
5292:
5291:
5279:
5275:
5272:
5269:
5266:
5261:
5257:
5253:
5250:
5247:
5244:
5241:
5236:
5232:
5227:
5221:
5218:
5215:
5211:
5205:
5202:
5199:
5196:
5193:
5189:
5185:
5182:
5179:
5176:
5173:
5170:
5167:
5164:
5161:
5158:
5155:
5152:
5149:
5144:
5141:
5138:
5134:
5130:
5127:
5124:
5121:
5118:
5115:
5110:
5106:
5082:
5035:
5032:
5015:
4995:
4992:
4989:
4964:
4944:
4941:
4938:
4933:
4906:
4884:
4880:
4855:
4823:
4811:
4808:
4763:
4760:
4757:
4754:
4751:
4748:
4745:
4742:
4739:
4736:
4733:
4730:
4727:
4724:
4721:
4718:
4715:
4704:
4703:
4687:
4684:
4678:
4675:
4670:
4667:
4661:
4658:
4655:
4649:
4646:
4641:
4638:
4632:
4629:
4626:
4593:
4590:
4587:
4584:
4581:
4578:
4575:
4572:
4569:
4566:
4563:
4560:
4557:
4533:
4530:
4510:
4486:
4466:
4442:
4439:
4419:
4399:
4376:
4373:
4370:
4367:
4364:
4344:
4341:
4340:
4339:
4328:
4327:
4326:
4315:
4310:
4304:
4299:
4295:
4291:
4286:
4282:
4278:
4275:
4272:
4269:
4266:
4263:
4260:
4258:
4253:
4249:
4245:
4240:
4236:
4232:
4229:
4226:
4223:
4220:
4217:
4214:
4213:
4210:
4205:
4201:
4197:
4192:
4188:
4184:
4181:
4178:
4175:
4172:
4169:
4167:
4162:
4158:
4154:
4149:
4145:
4141:
4138:
4135:
4132:
4129:
4126:
4125:
4123:
4118:
4113:
4109:
4095:
4094:
4093:
4082:
4078:
4074:
4069:
4065:
4061:
4056:
4052:
4048:
4045:
4042:
4039:
4036:
4031:
4027:
4023:
4018:
4014:
4010:
4007:
4004:
4000:
3996:
3993:
3990:
3987:
3984:
3981:
3978:
3965:
3964:
3960:
3959:
3944:
3943:
3942:
3931:
3928:
3925:
3920:
3917:
3914:
3910:
3904:
3901:
3898:
3894:
3890:
3885:
3882:
3879:
3875:
3869:
3866:
3863:
3859:
3837:
3836:
3835:
3824:
3819:
3811:
3808:
3805:
3801:
3797:
3793:
3790:
3787:
3783:
3779:
3778:
3773:
3770:
3767:
3763:
3759:
3755:
3752:
3749:
3745:
3741:
3740:
3738:
3733:
3730:
3727:
3724:
3721:
3718:
3713:
3709:
3681:are arbitrary
3668:
3665:
3662:
3658:
3635:
3632:
3629:
3625:
3613:
3612:
3611:
3599:
3595:
3592:
3589:
3584:
3581:
3578:
3574:
3570:
3567:
3562:
3559:
3556:
3552:
3548:
3543:
3539:
3532:
3529:
3526:
3523:
3518:
3515:
3512:
3508:
3504:
3501:
3496:
3493:
3490:
3486:
3482:
3477:
3473:
3468:
3464:
3461:
3458:
3455:
3452:
3449:
3446:
3433:
3432:
3428:
3427:
3415:
3412:
3409:
3406:
3403:
3400:
3397:
3394:
3391:
3388:
3385:
3382:
3379:
3376:
3356:
3336:
3327:-axis and the
3316:
3296:
3287:. We see that
3276:
3273:
3270:
3267:
3252:
3251:
3250:
3239:
3234:
3226:
3222:
3218:
3215:
3212:
3210:
3207:
3204:
3203:
3198:
3194:
3190:
3187:
3185:
3182:
3179:
3178:
3176:
3171:
3166:
3162:
3148:
3147:
3146:
3135:
3131:
3125:
3121:
3117:
3112:
3108:
3104:
3099:
3095:
3091:
3086:
3082:
3077:
3073:
3070:
3067:
3064:
3061:
3058:
3055:
3042:
3041:
3023:
3018:
2994:
2989:
2975:
2972:
2959:
2956:
2953:
2948:
2945:
2941:
2937:
2934:
2931:
2928:
2925:
2920:
2917:
2913:
2909:
2906:
2886:
2883:
2880:
2877:
2872:
2869:
2865:
2861:
2858:
2838:
2818:
2798:
2778:
2758:
2738:
2718:
2698:
2678:
2665:by compatible
2654:
2630:
2627:
2624:
2621:
2618:
2592:
2581:differentiable
2568:
2548:
2528:
2508:
2480:
2457:
2453:
2449:
2446:
2443:
2440:
2437:
2406:
2384:
2380:
2376:
2367:(or, locally,
2356:
2353:
2329:
2307:
2303:
2299:
2290:(or, locally,
2279:
2276:
2256:
2236:
2213:
2193:
2171:
2167:
2163:
2139:
2119:
2097:
2093:
2089:
2064:
2060:
2056:
2034:
2030:
2026:
2006:
2003:
2000:
1978:
1974:
1970:
1950:
1947:
1944:
1924:
1904:
1884:
1851:
1847:
1843:
1839:
1833:
1829:
1825:
1816:-th column is
1805:
1785:
1761:
1758:
1755:
1733:
1729:
1725:
1699:
1695:
1691:
1676:
1675:
1664:
1659:
1656:
1653:
1650:
1646:
1642:
1639:
1634:
1630:
1626:
1621:
1617:
1613:
1587:
1584:
1581:
1578:
1575:
1572:
1569:
1566:
1563:
1560:
1557:
1554:
1551:
1548:
1545:
1542:
1539:
1536:
1533:
1530:
1506:
1484:
1480:
1476:
1465:
1464:
1453:
1450:
1447:
1442:
1438:
1434:
1429:
1425:
1421:
1418:
1415:
1410:
1406:
1402:
1399:
1372:
1361:
1360:
1347:
1342:
1339:
1336:
1333:
1330:
1327:
1322:
1318:
1314:
1309:
1305:
1301:
1298:
1295:
1292:
1289:
1286:
1283:
1280:
1279:
1276:
1273:
1270:
1267:
1264:
1261:
1258:
1255:
1250:
1245:
1240:
1237:
1234:
1231:
1228:
1225:
1222:
1219:
1216:
1211:
1206:
1201:
1198:
1195:
1194:
1192:
1163:
1139:
1127:Some remarks:
1113:
1093:
1067:
1062:
1057:
1052:
1047:
1042:
1037:
1033:
1029:
1002:
999:
996:
993:
990:
966:
944:
939:
910:
890:
875:
872:
859:
839:
819:
797:
794:
791:
787:
782:
778:
773:
770:
767:
763:
758:
744:such that the
733:
730:
727:
724:
721:
701:
681:
678:
675:
652:
632:
612:
609:
606:
603:
600:
580:
571:of a manifold
560:
540:
531:of a manifold
520:
505:
502:
489:
469:
447:
443:
420:
416:
395:
375:
355:
335:
332:
329:
305:
285:
276:Two manifolds
260:
256:
235:
215:
195:
192:
189:
186:
181:
178:
174:
158:diffeomorphism
145:
142:
139:
136:
133:
110:
90:
78:
75:
43:diffeomorphism
26:
9:
6:
4:
3:
2:
9389:
9378:
9375:
9373:
9370:
9369:
9367:
9352:
9349:
9347:
9346:Supermanifold
9344:
9342:
9339:
9337:
9334:
9330:
9327:
9326:
9325:
9322:
9320:
9317:
9315:
9312:
9310:
9307:
9305:
9302:
9300:
9297:
9295:
9292:
9291:
9289:
9285:
9279:
9276:
9274:
9271:
9269:
9266:
9264:
9261:
9259:
9256:
9254:
9251:
9250:
9248:
9244:
9234:
9231:
9229:
9226:
9224:
9221:
9219:
9216:
9214:
9211:
9209:
9206:
9204:
9201:
9199:
9196:
9194:
9191:
9189:
9186:
9185:
9183:
9181:
9177:
9171:
9168:
9166:
9163:
9161:
9158:
9156:
9153:
9151:
9148:
9146:
9143:
9141:
9137:
9133:
9131:
9128:
9126:
9123:
9121:
9117:
9113:
9111:
9108:
9106:
9103:
9101:
9098:
9096:
9093:
9091:
9088:
9086:
9083:
9082:
9080:
9078:
9074:
9068:
9067:Wedge product
9065:
9063:
9060:
9056:
9053:
9052:
9051:
9048:
9046:
9043:
9039:
9036:
9035:
9034:
9031:
9029:
9026:
9024:
9021:
9019:
9016:
9012:
9011:Vector-valued
9009:
9008:
9007:
9004:
9002:
8999:
8995:
8992:
8991:
8990:
8987:
8985:
8982:
8980:
8977:
8976:
8974:
8970:
8964:
8961:
8959:
8956:
8954:
8951:
8947:
8944:
8943:
8942:
8941:Tangent space
8939:
8937:
8934:
8932:
8929:
8927:
8924:
8923:
8921:
8917:
8914:
8912:
8908:
8902:
8899:
8897:
8893:
8889:
8887:
8884:
8882:
8878:
8874:
8870:
8868:
8865:
8863:
8860:
8858:
8855:
8853:
8850:
8848:
8845:
8843:
8840:
8838:
8835:
8831:
8828:
8827:
8826:
8823:
8821:
8818:
8816:
8813:
8811:
8808:
8806:
8803:
8801:
8798:
8796:
8793:
8791:
8788:
8786:
8783:
8781:
8778:
8776:
8772:
8768:
8766:
8762:
8758:
8756:
8753:
8752:
8750:
8744:
8738:
8735:
8733:
8730:
8728:
8725:
8723:
8720:
8718:
8715:
8713:
8710:
8706:
8705:in Lie theory
8703:
8702:
8701:
8698:
8696:
8693:
8689:
8686:
8685:
8684:
8681:
8679:
8676:
8675:
8673:
8671:
8667:
8661:
8658:
8656:
8653:
8651:
8648:
8646:
8643:
8641:
8638:
8636:
8633:
8631:
8628:
8626:
8623:
8621:
8618:
8617:
8615:
8612:
8608:Main results
8606:
8600:
8597:
8595:
8592:
8590:
8589:Tangent space
8587:
8585:
8582:
8580:
8577:
8575:
8572:
8570:
8567:
8565:
8562:
8558:
8555:
8553:
8550:
8549:
8548:
8545:
8541:
8538:
8537:
8536:
8533:
8532:
8530:
8526:
8521:
8517:
8510:
8505:
8503:
8498:
8496:
8491:
8490:
8487:
8479:
8476:(in German),
8475:
8471:
8467:
8464:
8462:0-8218-4575-6
8458:
8454:
8449:
8446:
8440:
8436:
8432:
8428:
8425:
8421:
8417:
8413:
8408:
8403:
8399:
8395:
8391:
8386:
8383:
8379:
8375:
8371:
8366:
8361:
8357:
8353:
8352:
8346:
8343:
8341:0-8218-0780-3
8337:
8333:
8328:
8325:
8319:
8315:
8311:
8306:
8302:
8298:
8297:
8292:
8288:
8285:
8283:0-521-64121-7
8279:
8275:
8270:
8267:
8265:0-7923-4475-8
8261:
8257:
8253:
8249:
8242:
8238:
8234:
8230:
8226:
8222:
8218:
8214:
8210:
8206:
8202:
8198:
8194:
8187:
8182:
8178:
8172:
8164:
8158:
8154:
8149:
8148:
8134:
8129:
8125:
8121:
8117:
8110:
8102:
8096:
8092:
8085:
8081:
8071:
8068:
8066:
8063:
8061:
8058:
8056:
8053:
8050:
8047:
8044:
8041:
8039:
8035:
8031:
8027:
8024:
8022:
8018:
8015:
8014:
8008:
7992:
7977:
7959:
7930:
7901:
7886:
7868:
7854:
7850:
7846:
7842:
7837:
7835:
7831:
7827:
7823:
7817:
7815:
7802:
7781:
7777:
7748:
7745:
7742:
7717:
7713:
7684:
7681:
7678:
7658:
7650:
7647:
7642:
7627:
7624:
7621:
7614:
7610:
7587:
7584:
7573:
7568:
7564:
7560:
7555:
7551:
7542:
7541:automorphisms
7538:
7520:
7514:
7492:
7488:
7479:
7478:
7472:
7470:
7469:Michel Herman
7466:
7462:
7458:
7442:
7434:
7430:
7426:
7422:
7418:
7414:
7410:
7386:
7383:
7368:
7367:modular group
7350:
7339:
7333:
7323:
7318:
7314:
7310:
7305:
7301:
7293:
7289:
7285:
7281:
7277:
7273:
7271:
7267:
7263:
7262:Jakob Nielsen
7259:
7258:Allen Hatcher
7255:
7251:
7247:
7244:generated by
7243:
7239:
7235:
7228:Connectedness
7225:
7209:
7205:
7196:
7192:
7188:
7170:
7158:
7157:abelian group
7154:
7153:Stephen Smale
7150:
7146:
7128:
7125:
7122:
7118:
7108:
7091:
7085:
7078:
7074:
7055:
7052:
7046:
7040:
7037:
7031:
7028:
7025:
7019:
6996:
6987:
6985:
6981:
6977:
6973:
6969:
6959:
6957:
6953:
6937:
6917:
6912:
6908:
6887:
6867:
6862:
6858:
6850:
6834:
6826:
6823:
6807:
6775:
6767:
6763:
6736:
6728:
6724:
6708:
6680:
6657:
6649:
6645:
6618:
6610:
6606:
6579:
6556:
6525:
6517:
6484:
6476:
6472:
6456:
6439:
6431:
6418:
6411:
6395:
6387:
6383:
6380:For a finite
6379:
6361:
6358:
6355:
6349:
6346:
6343:
6340:
6336:
6329:
6326:
6320:
6311:
6305:
6282:
6279:
6274:
6251:
6247:
6229:
6214:
6212:of the group.
6211:
6192:
6176:
6157:
6154:
6151:
6120:
6117:
6114:
6103:
6100:
6097:
6091:
6063:
6040:
6012:
6004:
5988:
5985:
5982:
5974:
5973:
5953:
5945:
5941:
5925:
5917:
5913:
5909:
5904:
5900:
5892:
5891:
5890:
5870:
5862:
5858:
5854:
5851:
5846:
5842:
5833:
5829:
5821:
5820:
5819:
5805:
5797:
5781:
5773:
5772:vector fields
5757:
5749:
5739:
5737:
5721:
5713:
5709:
5705:
5704:Fréchet space
5686:
5683:
5675:
5671:
5655:
5647:
5631:
5611:
5603:
5585:
5581:
5557:
5548:
5545:
5542:
5532:
5528:
5523:
5519:
5516:
5508:
5505:
5502:
5492:
5488:
5483:
5474:
5471:
5467:
5461:
5453:
5447:
5444:
5441:
5435:
5428:
5427:
5426:
5412:
5404:
5386:
5382:
5361:
5341:
5321:
5301:
5270:
5264:
5259:
5255:
5251:
5245:
5239:
5234:
5230:
5219:
5216:
5213:
5203:
5200:
5197:
5194:
5191:
5183:
5174:
5168:
5165:
5159:
5153:
5147:
5142:
5139:
5136:
5128:
5122:
5119:
5116:
5108:
5104:
5096:
5095:
5094:
5080:
5072:
5067:
5065:
5061:
5057:
5053:
5049:
5045:
5041:
5031:
5029:
5013:
4990:
4962:
4939:
4931:
4904:
4882:
4878:
4869:
4853:
4845:
4841:
4837:
4821:
4807:
4805:
4801:
4797:
4793:
4789:
4785:
4781:
4777:
4761:
4758:
4752:
4749:
4746:
4743:
4740:
4734:
4728:
4725:
4722:
4719:
4716:
4701:
4685:
4682:
4676:
4668:
4659:
4656:
4653:
4647:
4639:
4630:
4627:
4624:
4617:
4616:
4615:
4613:
4609:
4604:
4591:
4585:
4582:
4579:
4573:
4567:
4564:
4561:
4555:
4547:
4531:
4528:
4508:
4500:
4497:, there is a
4484:
4464:
4456:
4440:
4437:
4417:
4397:
4390:
4374:
4368:
4365:
4362:
4354:
4350:
4337:
4333:
4329:
4313:
4308:
4297:
4293:
4289:
4284:
4280:
4273:
4270:
4267:
4264:
4261:
4251:
4247:
4243:
4238:
4234:
4227:
4224:
4221:
4218:
4215:
4203:
4199:
4195:
4190:
4186:
4179:
4176:
4173:
4170:
4160:
4156:
4152:
4147:
4143:
4136:
4133:
4130:
4127:
4121:
4116:
4111:
4107:
4099:
4098:
4096:
4080:
4076:
4067:
4063:
4059:
4054:
4050:
4043:
4040:
4037:
4029:
4025:
4021:
4016:
4012:
4005:
4002:
3998:
3994:
3988:
3985:
3982:
3976:
3969:
3968:
3967:
3966:
3962:
3961:
3957:
3953:
3949:
3945:
3929:
3926:
3923:
3918:
3915:
3912:
3908:
3902:
3899:
3896:
3892:
3888:
3883:
3880:
3877:
3873:
3867:
3864:
3861:
3857:
3849:
3848:
3846:
3842:
3838:
3822:
3817:
3809:
3806:
3803:
3799:
3791:
3788:
3785:
3781:
3771:
3768:
3765:
3761:
3753:
3750:
3747:
3743:
3736:
3731:
3725:
3722:
3719:
3711:
3707:
3699:
3698:
3696:
3692:
3688:
3684:
3666:
3663:
3660:
3656:
3633:
3630:
3627:
3623:
3614:
3597:
3593:
3590:
3587:
3582:
3579:
3576:
3572:
3568:
3565:
3560:
3557:
3554:
3550:
3546:
3541:
3537:
3530:
3527:
3524:
3521:
3516:
3513:
3510:
3506:
3502:
3499:
3494:
3491:
3488:
3484:
3480:
3475:
3471:
3466:
3462:
3456:
3453:
3450:
3444:
3437:
3436:
3435:
3434:
3430:
3429:
3410:
3407:
3404:
3401:
3395:
3392:
3386:
3383:
3380:
3374:
3354:
3334:
3314:
3294:
3274:
3271:
3268:
3265:
3257:
3253:
3237:
3232:
3224:
3220:
3216:
3213:
3208:
3205:
3196:
3192:
3188:
3183:
3180:
3174:
3169:
3164:
3160:
3152:
3151:
3149:
3133:
3129:
3123:
3119:
3115:
3110:
3106:
3102:
3097:
3093:
3089:
3084:
3080:
3075:
3071:
3065:
3062:
3059:
3053:
3046:
3045:
3044:
3043:
3039:
3038:
3037:
3021:
2992:
2971:
2954:
2946:
2943:
2939:
2935:
2926:
2918:
2915:
2911:
2904:
2884:
2878:
2875:
2870:
2867:
2863:
2859:
2856:
2836:
2816:
2796:
2776:
2756:
2736:
2716:
2696:
2676:
2668:
2652:
2644:
2628:
2622:
2619:
2616:
2608:
2606:
2590:
2582:
2566:
2546:
2526:
2506:
2497:
2495:
2494:homeomorphism
2455:
2451:
2447:
2441:
2435:
2427:
2422:
2420:
2404:
2382:
2378:
2374:
2354:
2351:
2343:
2327:
2305:
2301:
2297:
2277:
2274:
2254:
2247:to dimension
2234:
2225:
2211:
2191:
2169:
2165:
2161:
2153:
2137:
2117:
2095:
2091:
2087:
2078:
2062:
2058:
2054:
2032:
2028:
2024:
2004:
2001:
1998:
1976:
1972:
1968:
1948:
1945:
1942:
1922:
1915:to dimension
1902:
1882:
1874:
1869:
1867:
1849:
1845:
1837:
1831:
1827:
1803:
1783:
1775:
1759:
1756:
1753:
1731:
1727:
1723:
1715:
1697:
1693:
1689:
1681:
1662:
1654:
1648:
1644:
1637:
1632:
1628:
1624:
1619:
1615:
1611:
1604:
1603:
1602:
1599:
1582:
1579:
1576:
1573:
1567:
1564:
1558:
1555:
1552:
1546:
1540:
1537:
1534:
1528:
1520:
1504:
1482:
1478:
1474:
1467:Thus, though
1451:
1448:
1440:
1436:
1432:
1427:
1423:
1416:
1413:
1408:
1404:
1400:
1390:
1389:
1388:
1386:
1370:
1340:
1334:
1331:
1328:
1325:
1320:
1316:
1312:
1307:
1303:
1290:
1287:
1284:
1268:
1265:
1262:
1248:
1229:
1226:
1223:
1209:
1199:
1196:
1190:
1181:
1180:
1179:
1177:
1161:
1153:
1137:
1128:
1125:
1111:
1091:
1083:
1065:
1050:
1040:
1035:
1031:
1027:
1020:
1016:
1000:
994:
991:
988:
980:
964:
942:
927:
924:
908:
888:
879:
871:
857:
837:
817:
795:
792:
789:
780:
776:
771:
768:
765:
756:
747:
731:
725:
722:
719:
699:
679:
676:
673:
666:
650:
630:
610:
604:
601:
598:
591:, a function
578:
558:
551:and a subset
538:
518:
511:
501:
487:
467:
445:
441:
418:
414:
393:
373:
353:
333:
330:
327:
319:
318:diffeomorphic
303:
283:
274:
258:
254:
233:
213:
193:
187:
184:
179:
176:
172:
163:
159:
143:
137:
134:
131:
124:
108:
88:
71:
66:
62:
60:
56:
52:
48:
44:
40:
33:
19:
9273:Moving frame
9268:Morse theory
9258:Gauge theory
9050:Tensor field
8979:Closed/Exact
8958:Vector field
8926:Distribution
8867:Hypercomplex
8862:Quaternionic
8682:
8599:Vector field
8557:Smooth atlas
8477:
8473:
8452:
8434:
8397:
8393:
8389:
8355:
8349:
8331:
8309:
8294:
8273:
8255:
8196:
8192:
8152:
8123:
8119:
8109:
8090:
8084:
7974:that do not
7883:: there are
7838:
7830:fiber bundle
7818:
7814:homeomorphic
7811:
7274:
7231:
7109:
6988:
6965:
6956:Banyaga 1997
6825:transitively
6799:
6796:Transitivity
5888:
5745:
5670:Banach space
5572:
5293:
5068:
5047:
5043:
5037:
4843:
4813:
4803:
4799:
4783:
4779:
4705:
4699:
4611:
4605:
4546:non-singular
4499:neighborhood
4387:between two
4346:
4331:
3947:
3844:
3840:
3839:We see that
3694:
3690:
3686:
3683:real numbers
2977:
2609:
2498:
2425:
2423:
2226:
2079:
1870:
1796:-th row and
1677:
1600:
1466:
1362:
1129:
1126:
1019:differential
926:open subsets
880:
877:
746:restrictions
665:neighborhood
507:
317:
275:
246:is called a
157:
80:
42:
36:
9218:Levi-Civita
9208:Generalized
9180:Connections
9130:Lie algebra
9062:Volume form
8963:Vector flow
8936:Pushforward
8931:Lie bracket
8830:Lie algebra
8795:G-structure
8584:Pushforward
8564:Submanifold
7841:4-manifolds
7822:John Milnor
7433:conjectured
7246:Dehn twists
7149:John Milnor
6982:. In 1945,
5748:Lie algebra
5742:Lie algebra
5602:Leslie 1967
5052:Hirsch 1997
4453:that is an
4353:deformation
4336:unit circle
3956:polynomials
3256:determinant
1017:and if the
810:(note that
663:there is a
160:if it is a
53:. It is an
47:isomorphism
39:mathematics
9366:Categories
9341:Stratifold
9299:Diffeology
9095:Associated
8896:Symplectic
8881:Riemannian
8810:Hyperbolic
8737:Submersion
8645:HopfâRinow
8579:Submersion
8574:Smooth map
8145:References
8043:Diffeology
7801:CW-complex
7417:hyperbolic
6968:Tibor RadĂł
5060:metrizable
5040:topologies
4792:hyperbolic
3615:where the
2849:. The map
2605:continuous
2342:submersion
1875:. Imagine
1680:linear map
1385:surjective
957:such that
77:Definition
9223:Principal
9198:Ehresmann
9155:Subbundle
9145:Principal
9120:Fibration
9100:Cotangent
8972:Covectors
8825:Lie group
8805:Hermitian
8748:manifolds
8717:Immersion
8712:Foliation
8650:Noether's
8635:Frobenius
8630:De Rham's
8625:Darboux's
8516:Manifolds
8424:118624866
8407:1211.5704
8374:0040-9383
8301:EMS Press
8221:0556-2821
8171:cite book
8032:, a type
7574:×
7561:×
7413:parabolic
7311:×
7167:Γ
7145:René Thom
7126:−
6976:unit disc
6966:In 1926,
6764:π
6725:π
6718:Σ
6715:→
6646:π
6607:π
6600:Σ
6473:π
6466:Σ
6463:→
6446:→
6422:→
6350:∈
6318:→
6104:×
6098:≃
6003:Lie group
5946:μ
5938:∂
5934:∂
5918:μ
5863:μ
5855:ε
5847:μ
5839:↦
5834:μ
5722:σ
5690:∞
5472:−
5458:∑
5425:. Then:
5362:σ
5252:−
5217:∈
5201:≤
5195:≤
5188:∑
5140:∈
5069:Fixing a
4955:or, when
4840:Hausdorff
4788:Euclidean
4674:∂
4666:∂
4645:∂
4637:∂
4372:→
4349:mechanics
4274:
4262:−
4228:
4216:−
4180:
4137:
4044:
4006:
3924:≠
3889:−
3594:⋯
3528:⋯
3402:−
3214:−
3116:−
2944:−
2940:ψ
2936:⊆
2916:−
2912:ϕ
2882:→
2868:−
2864:ϕ
2857:ψ
2837:ψ
2817:ϕ
2717:ψ
2697:ϕ
2626:→
2419:immersion
1873:dimension
1842:∂
1824:∂
1757:×
1641:→
1574:−
1519:injective
1449:≠
1313:−
1297:↦
1254:∖
1239:→
1215:∖
1056:→
998:→
923:connected
793:∩
769:∩
729:→
677:⊂
608:→
331:≃
191:→
185::
177:−
162:bijection
141:→
135::
9319:Orbifold
9314:K-theory
9304:Diffiety
9028:Pullback
8842:Oriented
8820:Kenmotsu
8800:Hadamard
8746:Types of
8695:Geodesic
8520:Glossary
8480:(2): 123
8433:(2007),
8351:Topology
8254:(1997),
8241:Archived
8237:41709882
8019:such as
8011:See also
7834:3-sphere
7457:oriented
7435:that if
7409:elliptic
7284:periodic
7250:Max Dehn
7238:surfaces
7191:quotient
6175:subgroup
6136:, where
5970:Examples
5278:‖
5226:‖
5034:Topology
4782:,
4389:surfaces
2974:Examples
508:Given a
9263:History
9246:Related
9160:Tangent
9138:)
9118:)
9085:Adjoint
9077:Bundles
9055:density
8953:Torsion
8919:Vectors
8911:Tensors
8894:)
8879:)
8875:,
8873:Pseudoâ
8852:Poisson
8785:Finsler
8780:Fibered
8775:Contact
8773:)
8765:Complex
8763:)
8732:Section
8382:0210147
8303:, 2001
8229:9958280
8201:Bibcode
8034:anomaly
7268:of the
7185:, the "
6503:. Here
6173:is the
5056:compact
4870:of all
4866:is the
4334:is the
2769:, with
1746:is the
1176:complex
748:agree:
9228:Vector
9213:Koszul
9193:Cartan
9188:Affine
9170:Vector
9165:Tensor
9150:Spinor
9140:Normal
9136:Stable
9090:Affine
8994:bundle
8946:bundle
8892:Almost
8815:KĂ€hler
8771:Almost
8761:Almost
8755:Closed
8655:Sard's
8611:(list)
8459:
8441:
8422:
8380:
8372:
8338:
8320:
8280:
8262:
8235:
8227:
8219:
8159:
8097:
7853:exotic
7465:simple
7455:is an
7256:, and
6880:. If
6592:, and
5401:whose
5048:strong
4842:. The
4544:stays
3534:
2689:. Let
1714:matrix
1521:(e.g.
1150:to be
1015:proper
510:subset
406:. Two
45:is an
9336:Sheaf
9110:Fiber
8886:Rizza
8857:Prime
8688:Local
8678:Curve
8540:Atlas
8420:S2CID
8402:arXiv
8244:(PDF)
8233:S2CID
8189:(PDF)
8076:Notes
7613:genus
7292:torus
6248:is a
6001:is a
5975:When
5403:union
5064:Baire
4868:group
4796:slope
4794:, or
4774:is a
3007:into
2499:When
2267:, if
2130:then
2017:then
1961:then
1935:. If
1678:is a
1363:Then
366:from
156:is a
70:image
9203:Form
9105:Dual
9038:flow
8901:Tame
8877:Subâ
8790:Flat
8670:Maps
8457:ISBN
8439:ISBN
8392:.",
8370:ISSN
8336:ISBN
8318:ISBN
8278:ISBN
8260:ISBN
8225:PMID
8217:ISSN
8177:link
8157:ISBN
8095:ISBN
7847:and
7770:Diff
7746:>
7706:Diff
7682:>
7625:>
7415:and
7151:and
6822:acts
6702:Diff
6550:Diff
6513:Diff
6450:Diff
6427:Diff
6261:Diff
6186:Diff
6145:Diff
6108:Diff
6085:Diff
6034:Diff
5746:The
5046:and
5044:weak
4984:Diff
4927:Diff
4838:and
4814:Let
4606:The
4410:and
3950:are
3689:and
3648:and
2829:and
2789:and
2749:and
2709:and
2002:>
1946:<
921:are
316:are
296:and
121:, a
101:and
68:The
41:, a
9125:Jet
8412:doi
8360:doi
8209:doi
8128:doi
8036:in
7978:in
7423:of
6950:is
6827:on
6382:set
6177:of
5774:on
5738:).
5405:is
5354:is
5294:as
5210:sup
5133:sup
5073:on
4846:of
4614:is
4610:of
4501:of
4477:in
4347:In
4271:sin
4225:sin
4177:cos
4134:cos
4041:cos
4003:sin
3963:Let
3954:as
3431:Let
3040:Let
2426:not
2224:).
2080:If
1598:).
1398:det
1383:is
1104:in
977:is
928:of
881:If
692:of
643:in
386:to
49:of
37:In
9368::
9116:Co
8478:35
8418:,
8410:,
8398:44
8396:,
8378:MR
8376:,
8368:,
8354:,
8316:,
8299:,
8293:,
8239:.
8231:.
8223:.
8215:.
8207:.
8197:36
8195:.
8191:.
8173:}}
8169:{{
8124:10
8122:.
8118:.
8007:.
7761:,
7648:).
7578:GL
7543::
7411:,
7377:SL
7252:,
7224:.
7147:,
7107:.
5066:.
5042::
5030:.
4800:Df
4790:,
4784:dy
4780:dx
3697::
3036:.
2970:.
1452:0.
1124:.
901:,
61:.
9134:(
9114:(
8890:(
8871:(
8769:(
8759:(
8522:)
8518:(
8508:e
8501:t
8494:v
8414::
8404::
8390:R
8362::
8356:6
8211::
8203::
8179:)
8165:.
8136:.
8130::
8103:.
7993:4
7988:R
7960:4
7955:R
7931:4
7926:R
7902:4
7897:R
7869:4
7864:R
7803:.
7787:)
7782:n
7778:S
7774:(
7749:6
7743:n
7723:)
7718:4
7714:S
7710:(
7685:3
7679:n
7659:n
7628:1
7622:g
7608:.
7596:)
7592:Z
7588:,
7585:2
7582:(
7569:1
7565:S
7556:1
7552:S
7524:)
7521:3
7518:(
7515:O
7493:2
7489:S
7443:M
7395:)
7391:Z
7387:,
7384:2
7381:(
7351:2
7346:Z
7340:/
7334:2
7329:R
7324:=
7319:1
7315:S
7306:1
7302:S
7210:n
7206:B
7171:n
7129:1
7123:n
7119:S
7095:)
7092:2
7089:(
7086:O
7059:]
7056:1
7053:+
7050:)
7047:x
7044:(
7041:f
7038:=
7035:)
7032:1
7029:+
7026:x
7023:(
7020:f
7017:[
6997:f
6954:(
6938:M
6918:M
6913:k
6909:F
6888:M
6868:M
6863:k
6859:C
6835:M
6808:M
6779:)
6776:M
6773:(
6768:0
6743:)
6740:)
6737:M
6734:(
6729:0
6721:(
6712:)
6709:M
6706:(
6681:M
6661:)
6658:M
6655:(
6650:0
6625:)
6622:)
6619:M
6616:(
6611:0
6603:(
6580:M
6560:)
6557:M
6554:(
6529:)
6526:M
6523:(
6518:0
6491:)
6488:)
6485:M
6482:(
6477:0
6469:(
6460:)
6457:M
6454:(
6443:)
6440:M
6437:(
6432:0
6419:0
6396:M
6365:]
6362:1
6359:,
6356:0
6353:(
6347:t
6344:,
6341:t
6337:/
6333:)
6330:x
6327:t
6324:(
6321:f
6315:)
6312:x
6309:(
6306:f
6286:)
6283:0
6280:,
6275:n
6270:R
6265:(
6230:n
6225:R
6196:)
6193:G
6190:(
6161:)
6158:e
6155:,
6152:G
6149:(
6124:)
6121:e
6118:,
6115:G
6112:(
6101:G
6095:)
6092:G
6089:(
6064:G
6044:)
6041:G
6038:(
6013:G
5989:G
5986:=
5983:M
5954:.
5942:x
5929:)
5926:x
5923:(
5914:h
5910:=
5905:h
5901:L
5874:)
5871:x
5868:(
5859:h
5852:+
5843:x
5830:x
5806:x
5782:M
5758:M
5687:=
5684:r
5656:r
5632:M
5612:M
5586:r
5582:C
5558:.
5552:)
5549:g
5546:,
5543:f
5540:(
5533:n
5529:K
5524:d
5520:+
5517:1
5512:)
5509:g
5506:,
5503:f
5500:(
5493:n
5489:K
5484:d
5475:n
5468:2
5462:n
5454:=
5451:)
5448:g
5445:,
5442:f
5439:(
5436:d
5413:M
5387:n
5383:K
5342:M
5322:M
5302:K
5274:)
5271:x
5268:(
5265:g
5260:p
5256:D
5249:)
5246:x
5243:(
5240:f
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5231:D
5220:K
5214:x
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5192:1
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5181:)
5178:)
5175:x
5172:(
5169:g
5166:,
5163:)
5160:x
5157:(
5154:f
5151:(
5148:d
5143:K
5137:x
5129:=
5126:)
5123:g
5120:,
5117:f
5114:(
5109:K
5105:d
5081:M
5050:(
5014:M
4994:)
4991:M
4988:(
4963:r
4943:)
4940:M
4937:(
4932:r
4905:M
4883:r
4879:C
4854:M
4822:M
4762:f
4759:D
4756:)
4753:y
4750:d
4747:,
4744:x
4741:d
4738:(
4735:=
4732:)
4729:v
4726:d
4723:,
4720:u
4717:d
4714:(
4702:.
4700:v
4686:y
4683:d
4677:y
4669:u
4660:+
4657:x
4654:d
4648:x
4640:u
4631:=
4628:u
4625:d
4612:u
4592:.
4589:)
4586:v
4583:,
4580:u
4577:(
4574:=
4571:)
4568:y
4565:,
4562:x
4559:(
4556:f
4532:f
4529:D
4509:p
4485:U
4465:p
4441:f
4438:D
4418:V
4398:U
4375:V
4369:U
4366::
4363:f
4338:.
4332:h
4314:.
4309:)
4303:)
4298:2
4294:y
4290:+
4285:2
4281:x
4277:(
4268:y
4265:2
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4252:2
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4244:+
4239:2
4235:x
4231:(
4222:x
4219:2
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4174:y
4171:2
4166:)
4161:2
4157:y
4153:+
4148:2
4144:x
4140:(
4131:x
4128:2
4122:(
4117:=
4112:h
4108:J
4081:.
4077:)
4073:)
4068:2
4064:y
4060:+
4055:2
4051:x
4047:(
4038:,
4035:)
4030:2
4026:y
4022:+
4017:2
4013:x
4009:(
3999:(
3995:=
3992:)
3989:y
3986:,
3983:x
3980:(
3977:h
3958:.
3948:g
3930:,
3927:0
3919:0
3916:,
3913:1
3909:b
3903:1
3900:,
3897:0
3893:a
3884:1
3881:,
3878:0
3874:b
3868:0
3865:,
3862:1
3858:a
3845:0
3841:g
3823:.
3818:)
3810:1
3807:,
3804:0
3800:b
3792:0
3789:,
3786:1
3782:b
3772:1
3769:,
3766:0
3762:a
3754:0
3751:,
3748:1
3744:a
3737:(
3732:=
3729:)
3726:0
3723:,
3720:0
3717:(
3712:g
3708:J
3695:0
3691:y
3687:x
3667:j
3664:,
3661:i
3657:b
3634:j
3631:,
3628:i
3624:a
3598:)
3591:+
3588:y
3583:1
3580:,
3577:0
3573:b
3569:+
3566:x
3561:0
3558:,
3555:1
3551:b
3547:+
3542:0
3538:b
3531:,
3525:+
3522:y
3517:1
3514:,
3511:0
3507:a
3503:+
3500:x
3495:0
3492:,
3489:1
3485:a
3481:+
3476:0
3472:a
3467:(
3463:=
3460:)
3457:y
3454:,
3451:x
3448:(
3445:g
3414:)
3411:y
3408:,
3405:x
3399:(
3396:f
3393:=
3390:)
3387:y
3384:,
3381:x
3378:(
3375:f
3355:f
3335:y
3315:x
3295:f
3275:0
3272:=
3269:y
3266:x
3238:.
3233:)
3225:2
3221:y
3217:3
3209:x
3206:2
3197:2
3193:y
3189:3
3184:x
3181:2
3175:(
3170:=
3165:f
3161:J
3134:.
3130:)
3124:3
3120:y
3111:2
3107:x
3103:,
3098:3
3094:y
3090:+
3085:2
3081:x
3076:(
3072:=
3069:)
3066:y
3063:,
3060:x
3057:(
3054:f
3022:2
3017:R
2993:2
2988:R
2958:)
2955:V
2952:(
2947:1
2933:)
2930:)
2927:U
2924:(
2919:1
2908:(
2905:f
2885:V
2879:U
2876::
2871:1
2860:f
2797:V
2777:U
2757:N
2737:M
2677:N
2653:M
2629:N
2623:M
2620::
2617:f
2591:f
2567:f
2547:f
2527:f
2507:f
2479:R
2456:3
2452:x
2448:=
2445:)
2442:x
2439:(
2436:f
2405:f
2383:x
2379:f
2375:D
2355:f
2352:D
2328:f
2306:x
2302:f
2298:D
2278:f
2275:D
2255:k
2235:n
2212:x
2192:y
2170:y
2166:f
2162:D
2138:f
2118:x
2096:x
2092:f
2088:D
2063:x
2059:f
2055:D
2033:x
2029:f
2025:D
2005:k
1999:n
1977:x
1973:f
1969:D
1949:k
1943:n
1923:k
1903:n
1883:f
1850:j
1846:x
1838:/
1832:i
1828:f
1804:j
1784:i
1760:n
1754:n
1732:x
1728:f
1724:D
1698:x
1694:f
1690:D
1663:V
1658:)
1655:x
1652:(
1649:f
1645:T
1638:U
1633:x
1629:T
1625::
1620:x
1616:f
1612:D
1586:)
1583:0
1580:,
1577:1
1571:(
1568:f
1565:=
1562:)
1559:0
1556:,
1553:1
1550:(
1547:=
1544:)
1541:0
1538:,
1535:1
1532:(
1529:f
1505:f
1483:x
1479:f
1475:D
1446:)
1441:2
1437:y
1433:+
1428:2
1424:x
1420:(
1417:4
1414:=
1409:x
1405:f
1401:D
1371:f
1341:.
1338:)
1335:y
1332:x
1329:2
1326:,
1321:2
1317:y
1308:2
1304:x
1300:(
1294:)
1291:y
1288:,
1285:x
1282:(
1275:}
1272:)
1269:0
1266:,
1263:0
1260:(
1257:{
1249:2
1244:R
1236:}
1233:)
1230:0
1227:,
1224:0
1221:(
1218:{
1210:2
1205:R
1200::
1197:f
1191:{
1162:f
1138:V
1112:U
1092:x
1066:n
1061:R
1051:n
1046:R
1041::
1036:x
1032:f
1028:D
1001:V
995:U
992::
989:f
965:V
943:n
938:R
909:V
889:U
858:f
838:f
818:g
796:X
790:U
786:|
781:f
777:=
772:X
766:U
762:|
757:g
732:N
726:U
723::
720:g
700:p
680:M
674:U
651:X
631:p
611:Y
605:X
602::
599:f
579:N
559:Y
539:M
519:X
488:r
468:r
446:r
442:C
419:r
415:C
394:N
374:M
354:f
334:N
328:M
304:N
284:M
259:r
255:C
234:f
214:r
194:M
188:N
180:1
173:f
144:N
138:M
132:f
109:N
89:M
34:.
20:)
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