3627:'s 1982 breakthrough on the Ricci flow was his "pinching estimate" which, informally stated, says that for a Riemannian metric which appears in a 3-manifold Ricci flow with positive Ricci curvature, the eigenvalues of the Ricci tensor are close to one another relative to the size of their sum. If one normalizes the sum, then, the eigenvalues are close to one another in an absolute sense. In this sense, each of the metrics appearing in a 3-manifold Ricci flow of positive Ricci curvature "approximately" satisfies the conditions of the Schur lemma. The Schur lemma itself is not explicitly applied, but its proof is effectively carried out through Hamilton's calculations.
3634:'s extension of Hamilton's work to higher dimensions, where the main part of the work is that the Weyl tensor and the semi-traceless Riemann tensor become zero in the long-time limit. This extends to the more general Ricci flow convergence theorems, some expositions of which directly use the Schur lemma. This includes the proof of the
4822:
1122:
5095:
4628:
966:
338:
27:
is a result that says, heuristically, whenever certain curvatures are pointwise constant then they are forced to be globally constant. The proof is essentially a one-step calculation, which has only one input: the second
Bianchi identity.
3544:
3417:
2466:
2662:
4949:
3266:
1748:
754:
1674:
411:
2287:
1573:
1226:
879:
832:
1358:
563:
1949:
1449:
2095:
4944:
4510:
4280:
3754:
3691:
233:
4171:
3893:
5191:
2838:
4446:
2588:
he a smooth symmetric (0,2)-tensor field whose covariant derivative, with respect to the Levi-Civita connection, is completely symmetric. The symmetry condition is an analogue of the
4336:
4050:
4221:
444:
3059:
2754:
4419:
4133:
3153:
4874:
4817:{\displaystyle \int _{M}(R-{\overline {R}})^{2}\,d\mu _{g}\leq {\frac {4n(n-1)}{(n-2)^{2}}}\int _{M}{\Big |}\operatorname {Ric} -{\frac {1}{n}}Rg{\Big |}_{g}^{2}\,d\mu _{g}.}
4623:
627:
4371:
4085:
3994:
244:
2595:
1151:
4596:
1174:
5214:
4845:
4475:
3926:
3295:
3182:
3110:
3088:
2987:
2884:
2682:
2366:
2174:
2050:
2028:
1875:
1789:
Note that the dimensional restriction is important, since every two-dimensional
Riemannian manifold which does not have constant curvature would be a counterexample.
1602:
1478:
1406:
1286:
647:
491:
194:
3952:
3855:
3570:
3449:
3443:
2864:
2492:
2313:
2121:
1975:
1384:
961:
589:
6495:
4570:
4507:
3829:
3605:
2943:
2539:
2206:
1831:
1783:
1510:
1262:
704:
66:
5686:
4884:
have shown that if the traceless second fundamental form of a compact surface is approximately zero then the mean curvature is approximately constant. Precisely
3322:
2907:
2797:
2566:
777:
93:
5390:
5155:
5135:
5115:
4906:
3794:
3774:
3315:
3007:
2963:
2774:
2702:
2586:
2333:
2141:
1995:
1895:
1851:
1306:
1117:{\displaystyle |B|_{g}^{2}=\left|B-{\frac {1}{n}}\left(\operatorname {tr} ^{g}B\right)g\right|_{g}^{2}+{\frac {1}{n}}\left(\operatorname {tr} ^{g}B\right)^{2}.}
926:
906:
687:
667:
511:
471:
161:
133:
113:
2372:
6490:
3796:
is the mean curvature. The Schur lemma implies that the mean curvature is constant, and the image of this embedding then must be a standard round sphere.
5777:
5553:
5403:
3188:
1680:
5801:
5996:
2965:
be a smooth symmetric (0,2)-tensor field whose covariant derivative is totally symmetric as a (0,3)-tensor field. Then the following are equivalent:
6548:
3616:
The Schur lemmas are frequently employed to prove roundness of geometric objects. A noteworthy example is to characterize the limits of convergent
1609:
349:
5558:
3630:
In the same way, the Schur lemma for the
Riemann tensor is employed to study convergence of Ricci flow in higher dimensions. This goes back to
5866:
5445:
5090:{\displaystyle \int _{\Sigma }(H-{\overline {H}})^{2}\,d\mu _{g}\leq C\int _{\Sigma }{\Big |}h-{\frac {1}{2}}Hg{\Big |}_{g}^{2}\,d\mu _{g},}
6092:
5347:
2213:
6145:
5673:
3645:, which was modeled on Hamilton's work. In the final two sentences of Huisken's paper, it is concluded that one has a smooth embedding
1518:
6429:
1785:
is a connected smooth pseudo-Riemannian manifold, then the first three conditions are equivalent, and they imply the fourth condition.
1179:
5418:
5383:
5363:
4536:
have shown that if the traceless Ricci tensor is approximately zero then the scalar curvature is approximately constant. Precisely:
3696:
6194:
4881:
837:
5786:
6177:
2802:
782:
1311:
516:
1900:
3607:
is a connected and smooth pseudo-Riemannian manifold, then the first three are equivalent, and imply the fourth and fifth.
1411:
6543:
6389:
2055:
6538:
6374:
6097:
5871:
5480:
5376:
4911:
6419:
4517:
must be the warped product of an interval and a constant-curvature
Riemannian manifold. See O'Neill (1983, page 341).
4528:
Consider the Schur lemma in the form "If the traceless Ricci tensor is zero then the scalar curvature is constant."
6424:
6394:
6102:
6058:
6039:
5806:
5750:
5501:
4525:
Recent research has investigated the case that the conditions of the Schur lemma are only approximately satisfied.
2542:
4226:
1264:
be a connected smooth
Riemannian manifold whose dimension is not equal to two. Then the following are equivalent:
5961:
5826:
5630:
3648:
199:
5563:
4138:
3860:
6346:
6211:
5903:
5745:
5423:
5160:
238:
4424:
4285:
3999:
6043:
6013:
5937:
5927:
5883:
5713:
5666:
5609:
4176:
422:
5811:
5568:
3641:
The Schur lemma for
Codazzi tensors is employed directly in Huisken's foundational paper on convergence of
3012:
2707:
6384:
6003:
5898:
5718:
5640:
5635:
5573:
5506:
5465:
5271:
Böhm, Christoph; Wilking, Burkhard (2008). "Manifolds with positive curvature operators are space forms".
4514:
4376:
4090:
6033:
6028:
3115:
5328:
De Lellis, Camillo; Müller, Stefan (2005). "Optimal rigidity estimates for nearly umbilical surfaces".
4850:
6364:
6302:
6150:
5854:
5844:
5816:
5791:
5701:
5470:
4601:
333:{\displaystyle \operatorname {Rm} _{p}:T_{p}M\times T_{p}M\times T_{p}M\times T_{p}M\to \mathbb {R} }
594:
6502:
6475:
6184:
6062:
6047:
5976:
5735:
5527:
5496:
5484:
5455:
5438:
5399:
4341:
4055:
3957:
3642:
3624:
140:
69:
20:
6444:
6399:
6296:
6167:
5971:
5796:
5659:
5625:
5522:
5491:
5981:
5532:
1127:
3539:{\displaystyle |h_{p}|_{g}^{2}\leq \textstyle {\frac {1}{n}}(\operatorname {tr} ^{g}h_{p})^{2}}
6379:
6359:
6354:
6261:
6172:
5986:
5966:
5821:
5760:
5537:
5359:
4877:
4575:
4529:
1156:
690:
5351:
Interscience
Publishers, a division of John Wiley & Sons, New York-London 1963 xi+329 pp.
5199:
4830:
4451:
3898:
3271:
3158:
3095:
3064:
2972:
2869:
2667:
2338:
2146:
2035:
2000:
1860:
1578:
1454:
1391:
1271:
632:
476:
166:
6517:
6311:
6266:
6189:
6160:
6018:
5951:
5946:
5941:
5931:
5723:
5706:
5604:
5599:
5475:
5428:
3931:
3834:
3549:
3422:
2843:
2589:
2471:
2292:
2100:
1954:
1363:
931:
698:
568:
416:
4543:
4480:
3802:
3578:
3412:{\displaystyle |h_{p}|_{g}^{2}=\textstyle {\frac {1}{n}}(\operatorname {tr} ^{g}h_{p})^{2}}
2916:
2512:
2179:
1804:
1756:
1483:
1235:
39:
6553:
6460:
6369:
6199:
6155:
5921:
5433:
3631:
1228:{ With these observations in mind, one can restate the Schur lemma in the following form:
343:
2461:{\displaystyle |\operatorname {Rm} _{p}|_{g}^{2}=\textstyle {\frac {2}{n(n-1)}}R_{p}^{2}}
2889:
2779:
2548:
759:
75:
6326:
6251:
6221:
6119:
6112:
6052:
6023:
5893:
5888:
5849:
5358:
Pure and
Applied Mathematics, 103. Academic Press, Inc. , New York, 1983. xiii+468 pp.
5140:
5120:
5100:
4891:
4876:
whose traceless second fundamental form is zero, then its mean curvature is constant."
3779:
3759:
3635:
3617:
3300:
2992:
2948:
2759:
2687:
2571:
2318:
2126:
1980:
1880:
1836:
1291:
911:
891:
672:
652:
496:
456:
146:
118:
98:
16:
Whenever certain curvatures are pointwise constant then they must be globally constant
6532:
6512:
6336:
6331:
6316:
6306:
6256:
6233:
6107:
6067:
6008:
5956:
5755:
5460:
5252:
Huisken, Gerhard (1985). "Ricci deformation of the metric on a
Riemannian manifold".
4533:
2657:{\displaystyle \operatorname {div} ^{g}h=d{\big (}\operatorname {tr} ^{g}h{\big )}.}
6439:
6434:
6276:
6243:
6216:
6124:
5765:
5290:
Huisken, Gerhard (1984). "Flow by mean curvature of convex surfaces into spheres".
2945:
be a connected smooth
Riemannian manifold whose dimension is not equal to one. Let
2496:
the sum of the Weyl curvature and semi-traceless part of the Riemann tensor is zero
3261:{\displaystyle h_{p}={\frac {1}{n}}\left(\operatorname {tr} ^{g}h_{p}\right)g_{p}}
2499:
both the Weyl curvature and the semi-traceless part of the Riemann tensor are zero
1743:{\displaystyle |\operatorname {Ric} |_{g}^{2}\leq \textstyle {\frac {1}{n}}R^{2}.}
5368:
1797:
The following is an immediate corollary of the Schur lemma for the Ricci tensor.
6282:
6271:
6228:
6129:
5730:
4572:
is a closed Riemannian manifold with nonnegative Ricci curvature and dimension
6507:
6465:
6291:
6204:
5836:
5740:
5578:
5233:
Hamilton, Richard S. (1982). "Three-manifolds with positive Ricci curvature".
749:{\displaystyle \operatorname {div} _{g}\operatorname {Ric} ={\frac {1}{2}}dR.}
1669:{\displaystyle |\operatorname {Ric} |_{g}^{2}=\textstyle {\frac {1}{n}}R^{2}}
6321:
6286:
5991:
5878:
3831:
is a connected thrice-differentiable Riemannian manifold, and that for each
406:{\displaystyle \operatorname {Ric} _{p}:T_{p}M\times T_{p}M\to \mathbb {R} }
4509:
has constant curvature. A particularly notable application of this is that
669:; this could also be phrased as asserting that each connected component of
6485:
6480:
6470:
5861:
5682:
5450:
697:
The Schur lemma is a simple consequence of the "twice-contracted second
5651:
3799:
Another application relates full isotropy and curvature. Suppose that
6077:
5309:
De Lellis, Camillo; Topping, Peter M. (2012). "Almost-Schur lemma".
2282:{\displaystyle \operatorname {sec} _{p}(V)={\frac {1}{n(n-1)}}R_{p}}
4421:
Since isometries preserve sectional curvature, this implies that
1568:{\displaystyle \operatorname {Ric} _{p}={\frac {1}{n}}R_{p}g_{p}}
5655:
5372:
1221:{\displaystyle \kappa ={\frac {1}{n}}\operatorname {tr} ^{g}B.}
2909:
As above, one can then state the Schur lemma in this context:
4908:
such that, for any smooth compact connected embedded surface
4625:
denotes the average value of the scalar curvature, one has
874:{\displaystyle \textstyle d\kappa ={\frac {n}{2}}d\kappa .}
5356:
Semi-Riemannian geometry. With applications to relativity.
1833:
be a connected smooth Riemannian manifold whose dimension
2592:; continuing the analogy, one takes a trace to find that
827:{\displaystyle \operatorname {Ric} _{p}=\kappa (p)g_{p},}
1853:
is not equal to two. Then the following are equivalent:
1353:{\displaystyle \operatorname {Ric} _{p}=\kappa (p)g_{p}}
558:{\displaystyle \operatorname {Ric} _{p}=\kappa (p)g_{p}}
4827:
Next, consider the Schur lemma in the special form "If
143:, which assigns to every 2-dimensional linear subspace
3488:
3361:
2414:
1944:{\displaystyle \operatorname {sec} _{p}(V)=\kappa (p)}
1715:
1644:
841:
5202:
5163:
5143:
5123:
5103:
4952:
4914:
4894:
4853:
4833:
4631:
4604:
4578:
4546:
4483:
4454:
4427:
4379:
4344:
4288:
4229:
4179:
4141:
4093:
4058:
4002:
3960:
3934:
3901:
3863:
3837:
3805:
3782:
3762:
3699:
3651:
3581:
3552:
3452:
3425:
3325:
3303:
3274:
3191:
3161:
3118:
3098:
3067:
3015:
2995:
2975:
2951:
2919:
2892:
2872:
2846:
2805:
2782:
2762:
2710:
2690:
2670:
2598:
2574:
2551:
2515:
2474:
2375:
2341:
2321:
2295:
2216:
2182:
2149:
2129:
2103:
2058:
2038:
2003:
1983:
1957:
1903:
1883:
1863:
1839:
1807:
1759:
1683:
1612:
1581:
1521:
1486:
1457:
1444:{\displaystyle \operatorname {Ric} _{p}=\kappa g_{p}}
1414:
1394:
1366:
1314:
1294:
1274:
1238:
1182:
1159:
1130:
969:
934:
914:
894:
840:
785:
762:
707:
675:
655:
635:
597:
571:
519:
499:
479:
459:
425:
352:
247:
202:
169:
149:
121:
101:
78:
42:
6453:
6412:
6345:
6242:
6138:
6085:
6076:
5912:
5835:
5774:
5694:
5618:
5592:
5546:
5515:
5411:
2090:{\displaystyle \operatorname {sec} _{p}(V)=\kappa }
5208:
5185:
5149:
5129:
5109:
5089:
4939:{\displaystyle \Sigma \subseteq \mathbb {R} ^{3},}
4938:
4900:
4868:
4839:
4816:
4617:
4590:
4564:
4501:
4469:
4440:
4413:
4365:
4330:
4274:
4215:
4165:
4127:
4079:
4044:
3988:
3946:
3920:
3887:
3849:
3823:
3788:
3768:
3748:
3685:
3599:
3564:
3538:
3437:
3411:
3309:
3289:
3260:
3176:
3147:
3104:
3082:
3053:
3001:
2981:
2957:
2937:
2901:
2878:
2858:
2832:
2791:
2768:
2748:
2696:
2676:
2656:
2580:
2560:
2533:
2486:
2460:
2360:
2327:
2307:
2281:
2200:
2168:
2135:
2115:
2089:
2044:
2022:
1989:
1969:
1943:
1889:
1869:
1845:
1825:
1777:
1742:
1668:
1596:
1567:
1504:
1472:
1443:
1400:
1378:
1352:
1300:
1280:
1256:
1220:
1168:
1145:
1116:
955:
920:
900:
873:
826:
771:
748:
681:
661:
641:
621:
583:
557:
505:
485:
465:
438:
405:
332:
227:
188:
155:
127:
107:
87:
60:
5054:
5024:
4781:
4751:
2911:
1799:
1230:
756:understood as an equality of smooth 1-forms on
451:
5667:
5384:
5348:Foundations of differential geometry. Vol. I.
2646:
2623:
8:
4275:{\displaystyle P,Q\in {\text{Gr}}(2,T_{p}M)}
3749:{\displaystyle |h|^{2}={\frac {1}{n}}H^{2},}
473:is not equal to two. If there is a function
3686:{\displaystyle S^{n}\to \mathbb {R} ^{n+1}}
2886:is constant on each connected component of
1604:that is, the traceless Ricci tensor is zero
884:Alternative formulations of the assumptions
649:is constant on each connected component of
228:{\displaystyle \operatorname {sec} _{p}(V)}
6082:
5674:
5660:
5652:
5554:Fundamental theorem of Riemannian geometry
5391:
5377:
5369:
4166:{\displaystyle \operatorname {Isom} (M,g)}
3888:{\displaystyle \operatorname {Isom} (M,g)}
95:Recall that this defines for each element
5311:Calc. Var. Partial Differential Equations
5201:
5196:As an application, one can conclude that
5186:{\displaystyle \operatorname {tr} _{g}h.}
5168:
5162:
5142:
5122:
5102:
5078:
5070:
5064:
5059:
5053:
5052:
5035:
5023:
5022:
5016:
5000:
4992:
4986:
4972:
4957:
4951:
4927:
4923:
4922:
4913:
4893:
4860:
4856:
4855:
4852:
4832:
4805:
4797:
4791:
4786:
4780:
4779:
4762:
4750:
4749:
4743:
4730:
4688:
4679:
4671:
4665:
4651:
4636:
4630:
4605:
4603:
4577:
4545:
4482:
4453:
4432:
4426:
4387:
4378:
4343:
4287:
4260:
4242:
4228:
4198:
4180:
4178:
4140:
4101:
4092:
4057:
4001:
3977:
3959:
3933:
3906:
3900:
3862:
3836:
3804:
3781:
3761:
3737:
3723:
3714:
3709:
3700:
3698:
3671:
3667:
3666:
3656:
3650:
3580:
3551:
3529:
3519:
3506:
3489:
3479:
3474:
3469:
3462:
3453:
3451:
3424:
3402:
3392:
3379:
3362:
3352:
3347:
3342:
3335:
3326:
3324:
3302:
3273:
3252:
3237:
3224:
3205:
3196:
3190:
3160:
3139:
3123:
3117:
3097:
3066:
3045:
3020:
3014:
2994:
2974:
2950:
2918:
2891:
2871:
2845:
2833:{\displaystyle d\kappa =n\cdot d\kappa .}
2804:
2781:
2761:
2740:
2715:
2709:
2689:
2669:
2645:
2644:
2632:
2622:
2621:
2603:
2597:
2573:
2550:
2514:
2473:
2451:
2446:
2415:
2405:
2400:
2395:
2385:
2376:
2374:
2346:
2340:
2320:
2315:and all two-dimensional linear subspaces
2294:
2273:
2242:
2221:
2215:
2181:
2154:
2148:
2128:
2123:and all two-dimensional linear subspaces
2102:
2063:
2057:
2037:
2008:
2002:
1982:
1977:and all two-dimensional linear subspaces
1956:
1908:
1902:
1882:
1862:
1838:
1806:
1758:
1730:
1716:
1706:
1701:
1696:
1684:
1682:
1659:
1645:
1635:
1630:
1625:
1613:
1611:
1580:
1559:
1549:
1535:
1526:
1520:
1485:
1456:
1435:
1419:
1413:
1393:
1365:
1344:
1319:
1313:
1293:
1273:
1237:
1203:
1189:
1181:
1158:
1129:
1105:
1088:
1068:
1059:
1054:
1029:
1010:
989:
984:
979:
970:
968:
933:
913:
893:
851:
839:
815:
790:
784:
761:
727:
712:
706:
674:
654:
634:
596:
570:
549:
524:
518:
498:
478:
458:
430:
424:
399:
398:
386:
370:
357:
351:
326:
325:
313:
297:
281:
265:
252:
246:
207:
201:
174:
168:
148:
120:
100:
77:
41:
5345:Shoshichi Kobayashi and Katsumi Nomizu.
4441:{\displaystyle \operatorname {sec} _{p}}
5225:
4331:{\displaystyle \varphi :(M,g)\to (M,g)}
4045:{\displaystyle \varphi :(M,g)\to (M,g)}
4216:{\displaystyle {\text{Gr}}(2,T_{p}M),}
1793:The Schur lemma for the Riemann tensor
449:The Schur lemma states the following:
439:{\displaystyle \operatorname {R} _{p}}
5216:itself is 'close' to a round sphere.
3054:{\displaystyle h_{p}=\kappa (p)g_{p}}
2749:{\displaystyle h_{p}=\kappa (p)g_{p}}
7:
779:Substituting in the given condition
346:, which is a symmetric bilinear map
32:The Schur lemma for the Ricci tensor
4847:is a connected embedded surface in
4414:{\displaystyle d\varphi _{p}(P)=Q.}
4128:{\displaystyle d\varphi _{p}(v)=w.}
3776:is the second fundamental form and
2505:The Schur lemma for Codazzi tensors
908:be a symmetric bilinear form on an
5203:
5017:
4958:
4915:
4834:
3148:{\displaystyle h_{p}=\kappa g_{p}}
427:
14:
2799:then upon substitution one finds
928:-dimensional inner product space
5235:Journal of Differential Geometry
5117:is the second fundamental form,
4869:{\displaystyle \mathbb {R} ^{3}}
6549:Theorems in Riemannian geometry
4618:{\displaystyle {\overline {R}}}
3297:that is, the traceless form of
5714:Differentiable/Smooth manifold
4983:
4963:
4727:
4714:
4709:
4697:
4662:
4642:
4559:
4547:
4496:
4484:
4399:
4393:
4354:
4348:
4325:
4313:
4310:
4307:
4295:
4269:
4247:
4207:
4185:
4160:
4148:
4113:
4107:
4068:
4062:
4039:
4027:
4024:
4021:
4009:
3882:
3870:
3818:
3806:
3710:
3701:
3662:
3594:
3582:
3526:
3499:
3470:
3454:
3399:
3372:
3343:
3327:
3038:
3032:
2932:
2920:
2733:
2727:
2528:
2516:
2436:
2424:
2396:
2377:
2263:
2251:
2236:
2230:
2195:
2183:
2078:
2072:
1938:
1932:
1923:
1917:
1820:
1808:
1772:
1760:
1697:
1685:
1626:
1614:
1499:
1487:
1337:
1331:
1251:
1239:
980:
971:
947:
935:
808:
802:
622:{\displaystyle d\kappa (p)=0.}
610:
604:
542:
536:
395:
322:
222:
216:
55:
43:
1:
4477:The Schur lemma implies that
4366:{\displaystyle \varphi (p)=p}
4080:{\displaystyle \varphi (p)=p}
3989:{\displaystyle v,w\in T_{p}M}
3636:differentiable sphere theorem
241:, which is a multilinear map
5481:Raising and lowering indices
4977:
4656:
4610:
6420:Classification of manifolds
5137:is the induced metric, and
3623:For example, a key part of
1176:then one automatically has
1124:Additionally, note that if
6570:
5502:Pseudo-Riemannian manifold
4173:also acts transitively on
2543:pseudo-Riemannian manifold
2541:be a smooth Riemannian or
1146:{\displaystyle B=\kappa g}
6496:over commutative algebras
5631:Geometrization conjecture
419:, which is a real number
6212:Riemann curvature tensor
4591:{\displaystyle n\geq 3.}
3928:This means that for all
3857:the group of isometries
1169:{\displaystyle \kappa ,}
239:Riemann curvature tensor
5209:{\displaystyle \Sigma }
4840:{\displaystyle \Sigma }
4470:{\displaystyle p\in M.}
3921:{\displaystyle T_{p}M.}
3290:{\displaystyle p\in M,}
3177:{\displaystyle p\in M,}
3105:{\displaystyle \kappa }
3083:{\displaystyle p\in M,}
2982:{\displaystyle \kappa }
2879:{\displaystyle \kappa }
2677:{\displaystyle \kappa }
2664:If there is a function
2361:{\displaystyle T_{p}M,}
2169:{\displaystyle T_{p}M,}
2045:{\displaystyle \kappa }
2023:{\displaystyle T_{p}M,}
1870:{\displaystyle \kappa }
1597:{\displaystyle p\in M,}
1473:{\displaystyle p\in M,}
1401:{\displaystyle \kappa }
1281:{\displaystyle \kappa }
642:{\displaystyle \kappa }
486:{\displaystyle \kappa }
189:{\displaystyle T_{p}M,}
6004:Manifold with boundary
5719:Differential structure
5641:Uniformization theorem
5574:Nash embedding theorem
5507:Riemannian volume form
5466:Levi-Civita connection
5210:
5187:
5157:is the mean curvature
5151:
5131:
5111:
5091:
4940:
4902:
4870:
4841:
4818:
4619:
4592:
4566:
4515:cosmological principle
4503:
4471:
4442:
4415:
4367:
4332:
4276:
4217:
4167:
4129:
4081:
4046:
3990:
3948:
3947:{\displaystyle p\in M}
3922:
3889:
3851:
3850:{\displaystyle p\in M}
3825:
3790:
3770:
3750:
3687:
3609:
3601:
3566:
3565:{\displaystyle p\in M}
3540:
3439:
3438:{\displaystyle p\in M}
3413:
3311:
3291:
3262:
3178:
3149:
3106:
3084:
3055:
3003:
2983:
2959:
2939:
2903:
2880:
2860:
2859:{\displaystyle n>1}
2834:
2793:
2770:
2750:
2698:
2678:
2658:
2582:
2562:
2535:
2502:
2488:
2487:{\displaystyle p\in M}
2462:
2362:
2329:
2309:
2308:{\displaystyle p\in M}
2283:
2208:has constant curvature
2202:
2170:
2137:
2117:
2116:{\displaystyle p\in M}
2091:
2046:
2024:
1991:
1971:
1970:{\displaystyle p\in M}
1945:
1891:
1871:
1847:
1827:
1787:
1779:
1744:
1670:
1598:
1569:
1506:
1474:
1445:
1402:
1380:
1379:{\displaystyle p\in M}
1354:
1302:
1282:
1258:
1222:
1170:
1147:
1118:
957:
956:{\displaystyle (V,g).}
922:
902:
875:
828:
773:
750:
695:
683:
663:
643:
623:
585:
584:{\displaystyle p\in M}
559:
507:
487:
467:
440:
407:
334:
229:
190:
157:
129:
109:
89:
62:
5211:
5188:
5152:
5132:
5112:
5092:
4941:
4903:
4871:
4842:
4819:
4620:
4593:
4567:
4565:{\displaystyle (M,g)}
4504:
4502:{\displaystyle (M,g)}
4472:
4448:is constant for each
4443:
4416:
4368:
4333:
4282:there is an isometry
4277:
4218:
4168:
4130:
4082:
4047:
3996:there is an isometry
3991:
3949:
3923:
3895:acts transitively on
3890:
3852:
3826:
3824:{\displaystyle (M,g)}
3791:
3771:
3751:
3688:
3602:
3600:{\displaystyle (M,g)}
3567:
3541:
3440:
3414:
3312:
3292:
3263:
3179:
3150:
3107:
3085:
3056:
3004:
2984:
2960:
2940:
2938:{\displaystyle (M,g)}
2904:
2881:
2861:
2835:
2794:
2771:
2751:
2699:
2679:
2659:
2583:
2563:
2536:
2534:{\displaystyle (M,g)}
2489:
2463:
2363:
2330:
2310:
2284:
2203:
2201:{\displaystyle (M,g)}
2171:
2138:
2118:
2092:
2047:
2025:
1992:
1972:
1946:
1892:
1872:
1848:
1828:
1826:{\displaystyle (M,g)}
1780:
1778:{\displaystyle (M,g)}
1745:
1671:
1599:
1570:
1507:
1505:{\displaystyle (M,g)}
1475:
1446:
1403:
1381:
1355:
1303:
1283:
1259:
1257:{\displaystyle (M,g)}
1223:
1171:
1148:
1119:
958:
923:
903:
876:
829:
774:
751:
701:," which states that
684:
664:
644:
624:
586:
560:
508:
488:
468:
441:
408:
335:
230:
191:
158:
130:
110:
90:
63:
61:{\displaystyle (M,g)}
6544:Riemannian manifolds
6151:Covariant derivative
5702:Topological manifold
5564:Gauss–Bonnet theorem
5471:Covariant derivative
5330:J. Differential Geom
5292:J. Differential Geom
5254:J. Differential Geom
5200:
5161:
5141:
5121:
5101:
4950:
4912:
4892:
4851:
4831:
4629:
4602:
4576:
4544:
4481:
4452:
4425:
4377:
4342:
4286:
4227:
4177:
4139:
4091:
4056:
4000:
3958:
3932:
3899:
3861:
3835:
3803:
3780:
3760:
3697:
3649:
3579:
3550:
3450:
3423:
3323:
3301:
3272:
3189:
3159:
3116:
3096:
3065:
3013:
2993:
2973:
2969:there is a function
2949:
2917:
2890:
2870:
2844:
2803:
2780:
2760:
2708:
2688:
2668:
2596:
2572:
2549:
2513:
2472:
2373:
2339:
2319:
2293:
2214:
2180:
2147:
2127:
2101:
2056:
2036:
2001:
1981:
1955:
1901:
1881:
1861:
1857:There is a function
1837:
1805:
1757:
1681:
1610:
1579:
1519:
1484:
1455:
1412:
1392:
1364:
1312:
1292:
1272:
1268:There is a function
1236:
1180:
1157:
1128:
967:
932:
912:
892:
838:
783:
760:
705:
673:
653:
633:
595:
569:
517:
497:
477:
457:
423:
350:
245:
200:
167:
147:
119:
99:
76:
40:
6539:Riemannian geometry
6185:Exterior derivative
5787:Atiyah–Singer index
5736:Riemannian manifold
5636:Poincaré conjecture
5497:Riemannian manifold
5485:Musical isomorphism
5400:Riemannian geometry
5069:
4796:
4223:that is, for every
3643:mean curvature flow
3484:
3357:
2456:
2410:
1711:
1640:
1064:
994:
141:sectional curvature
70:Riemannian manifold
21:Riemannian geometry
6491:Secondary calculus
6445:Singularity theory
6400:Parallel transport
6168:De Rham cohomology
5807:Generalized Stokes
5626:General relativity
5569:Hopf–Rinow theorem
5516:Types of manifolds
5492:Parallel transport
5206:
5183:
5147:
5127:
5107:
5087:
5051:
4936:
4898:
4888:there is a number
4866:
4837:
4814:
4778:
4615:
4588:
4562:
4499:
4467:
4438:
4411:
4363:
4328:
4272:
4213:
4163:
4135:This implies that
4125:
4077:
4042:
3986:
3944:
3918:
3885:
3847:
3821:
3786:
3766:
3746:
3683:
3597:
3562:
3536:
3535:
3468:
3435:
3409:
3408:
3341:
3307:
3287:
3258:
3174:
3145:
3102:
3092:there is a number
3080:
3051:
2999:
2979:
2955:
2935:
2902:{\displaystyle M.}
2899:
2876:
2856:
2830:
2792:{\displaystyle M,}
2789:
2766:
2746:
2694:
2674:
2654:
2578:
2561:{\displaystyle n.}
2558:
2531:
2484:
2458:
2457:
2442:
2394:
2358:
2325:
2305:
2279:
2198:
2166:
2133:
2113:
2087:
2042:
2032:There is a number
2020:
1987:
1967:
1941:
1887:
1867:
1843:
1823:
1775:
1740:
1739:
1695:
1666:
1665:
1624:
1594:
1565:
1502:
1470:
1441:
1398:
1388:There is a number
1376:
1350:
1298:
1278:
1254:
1218:
1166:
1143:
1114:
998:
978:
953:
918:
898:
871:
870:
824:
772:{\displaystyle M.}
769:
746:
679:
659:
639:
619:
581:
555:
503:
483:
463:
436:
403:
330:
225:
186:
153:
125:
105:
88:{\displaystyle n.}
85:
58:
6526:
6525:
6408:
6407:
6173:Differential form
5827:Whitney embedding
5761:Differential form
5649:
5648:
5354:Barrett O'Neill.
5273:Ann. of Math. (2)
5150:{\displaystyle H}
5130:{\displaystyle g}
5110:{\displaystyle h}
5043:
4980:
4901:{\displaystyle C}
4878:Camillo De Lellis
4770:
4737:
4659:
4613:
4530:Camillo De Lellis
4513:which models the
4245:
4183:
3789:{\displaystyle H}
3769:{\displaystyle h}
3731:
3497:
3370:
3310:{\displaystyle h}
3213:
3002:{\displaystyle M}
2958:{\displaystyle h}
2769:{\displaystyle p}
2697:{\displaystyle M}
2581:{\displaystyle h}
2440:
2328:{\displaystyle V}
2267:
2136:{\displaystyle V}
1990:{\displaystyle V}
1890:{\displaystyle M}
1846:{\displaystyle n}
1724:
1653:
1543:
1301:{\displaystyle M}
1197:
1076:
1018:
921:{\displaystyle n}
901:{\displaystyle B}
859:
735:
691:Einstein manifold
682:{\displaystyle M}
662:{\displaystyle M}
506:{\displaystyle M}
466:{\displaystyle n}
156:{\displaystyle V}
128:{\displaystyle M}
108:{\displaystyle p}
6561:
6518:Stratified space
6476:Fréchet manifold
6190:Interior product
6083:
5780:
5676:
5669:
5662:
5653:
5393:
5386:
5379:
5370:
5338:
5337:
5325:
5319:
5318:
5317:(3–44): 347–354.
5306:
5300:
5299:
5287:
5281:
5280:
5268:
5262:
5261:
5249:
5243:
5242:
5230:
5215:
5213:
5212:
5207:
5192:
5190:
5189:
5184:
5173:
5172:
5156:
5154:
5153:
5148:
5136:
5134:
5133:
5128:
5116:
5114:
5113:
5108:
5096:
5094:
5093:
5088:
5083:
5082:
5068:
5063:
5058:
5057:
5044:
5036:
5028:
5027:
5021:
5020:
5005:
5004:
4991:
4990:
4981:
4973:
4962:
4961:
4945:
4943:
4942:
4937:
4932:
4931:
4926:
4907:
4905:
4904:
4899:
4875:
4873:
4872:
4867:
4865:
4864:
4859:
4846:
4844:
4843:
4838:
4823:
4821:
4820:
4815:
4810:
4809:
4795:
4790:
4785:
4784:
4771:
4763:
4755:
4754:
4748:
4747:
4738:
4736:
4735:
4734:
4712:
4689:
4684:
4683:
4670:
4669:
4660:
4652:
4641:
4640:
4624:
4622:
4621:
4616:
4614:
4606:
4597:
4595:
4594:
4589:
4571:
4569:
4568:
4563:
4508:
4506:
4505:
4500:
4476:
4474:
4473:
4468:
4447:
4445:
4444:
4439:
4437:
4436:
4420:
4418:
4417:
4412:
4392:
4391:
4372:
4370:
4369:
4364:
4337:
4335:
4334:
4329:
4281:
4279:
4278:
4273:
4265:
4264:
4246:
4243:
4222:
4220:
4219:
4214:
4203:
4202:
4184:
4181:
4172:
4170:
4169:
4164:
4134:
4132:
4131:
4126:
4106:
4105:
4086:
4084:
4083:
4078:
4051:
4049:
4048:
4043:
3995:
3993:
3992:
3987:
3982:
3981:
3953:
3951:
3950:
3945:
3927:
3925:
3924:
3919:
3911:
3910:
3894:
3892:
3891:
3886:
3856:
3854:
3853:
3848:
3830:
3828:
3827:
3822:
3795:
3793:
3792:
3787:
3775:
3773:
3772:
3767:
3755:
3753:
3752:
3747:
3742:
3741:
3732:
3724:
3719:
3718:
3713:
3704:
3692:
3690:
3689:
3684:
3682:
3681:
3670:
3661:
3660:
3625:Richard Hamilton
3606:
3604:
3603:
3598:
3571:
3569:
3568:
3563:
3545:
3543:
3542:
3537:
3534:
3533:
3524:
3523:
3511:
3510:
3498:
3490:
3483:
3478:
3473:
3467:
3466:
3457:
3444:
3442:
3441:
3436:
3418:
3416:
3415:
3410:
3407:
3406:
3397:
3396:
3384:
3383:
3371:
3363:
3356:
3351:
3346:
3340:
3339:
3330:
3316:
3314:
3313:
3308:
3296:
3294:
3293:
3288:
3267:
3265:
3264:
3259:
3257:
3256:
3247:
3243:
3242:
3241:
3229:
3228:
3214:
3206:
3201:
3200:
3183:
3181:
3180:
3175:
3154:
3152:
3151:
3146:
3144:
3143:
3128:
3127:
3111:
3109:
3108:
3103:
3089:
3087:
3086:
3081:
3060:
3058:
3057:
3052:
3050:
3049:
3025:
3024:
3008:
3006:
3005:
3000:
2988:
2986:
2985:
2980:
2964:
2962:
2961:
2956:
2944:
2942:
2941:
2936:
2908:
2906:
2905:
2900:
2885:
2883:
2882:
2877:
2865:
2863:
2862:
2857:
2839:
2837:
2836:
2831:
2798:
2796:
2795:
2790:
2775:
2773:
2772:
2767:
2755:
2753:
2752:
2747:
2745:
2744:
2720:
2719:
2703:
2701:
2700:
2695:
2683:
2681:
2680:
2675:
2663:
2661:
2660:
2655:
2650:
2649:
2637:
2636:
2627:
2626:
2608:
2607:
2590:Bianchi identity
2587:
2585:
2584:
2579:
2567:
2565:
2564:
2559:
2540:
2538:
2537:
2532:
2493:
2491:
2490:
2485:
2467:
2465:
2464:
2459:
2455:
2450:
2441:
2439:
2416:
2409:
2404:
2399:
2390:
2389:
2380:
2367:
2365:
2364:
2359:
2351:
2350:
2334:
2332:
2331:
2326:
2314:
2312:
2311:
2306:
2288:
2286:
2285:
2280:
2278:
2277:
2268:
2266:
2243:
2226:
2225:
2207:
2205:
2204:
2199:
2175:
2173:
2172:
2167:
2159:
2158:
2142:
2140:
2139:
2134:
2122:
2120:
2119:
2114:
2096:
2094:
2093:
2088:
2068:
2067:
2051:
2049:
2048:
2043:
2029:
2027:
2026:
2021:
2013:
2012:
1996:
1994:
1993:
1988:
1976:
1974:
1973:
1968:
1950:
1948:
1947:
1942:
1913:
1912:
1896:
1894:
1893:
1888:
1876:
1874:
1873:
1868:
1852:
1850:
1849:
1844:
1832:
1830:
1829:
1824:
1784:
1782:
1781:
1776:
1749:
1747:
1746:
1741:
1735:
1734:
1725:
1717:
1710:
1705:
1700:
1688:
1675:
1673:
1672:
1667:
1664:
1663:
1654:
1646:
1639:
1634:
1629:
1617:
1603:
1601:
1600:
1595:
1574:
1572:
1571:
1566:
1564:
1563:
1554:
1553:
1544:
1536:
1531:
1530:
1511:
1509:
1508:
1503:
1479:
1477:
1476:
1471:
1450:
1448:
1447:
1442:
1440:
1439:
1424:
1423:
1407:
1405:
1404:
1399:
1385:
1383:
1382:
1377:
1359:
1357:
1356:
1351:
1349:
1348:
1324:
1323:
1307:
1305:
1304:
1299:
1287:
1285:
1284:
1279:
1263:
1261:
1260:
1255:
1227:
1225:
1224:
1219:
1208:
1207:
1198:
1190:
1175:
1173:
1172:
1167:
1153:for some number
1152:
1150:
1149:
1144:
1123:
1121:
1120:
1115:
1110:
1109:
1104:
1100:
1093:
1092:
1077:
1069:
1063:
1058:
1053:
1049:
1045:
1041:
1034:
1033:
1019:
1011:
993:
988:
983:
974:
962:
960:
959:
954:
927:
925:
924:
919:
907:
905:
904:
899:
880:
878:
877:
872:
860:
852:
833:
831:
830:
825:
820:
819:
795:
794:
778:
776:
775:
770:
755:
753:
752:
747:
736:
728:
717:
716:
699:Bianchi identity
688:
686:
685:
680:
668:
666:
665:
660:
648:
646:
645:
640:
628:
626:
625:
620:
590:
588:
587:
582:
564:
562:
561:
556:
554:
553:
529:
528:
512:
510:
509:
504:
492:
490:
489:
484:
472:
470:
469:
464:
445:
443:
442:
437:
435:
434:
417:scalar curvature
412:
410:
409:
404:
402:
391:
390:
375:
374:
362:
361:
339:
337:
336:
331:
329:
318:
317:
302:
301:
286:
285:
270:
269:
257:
256:
234:
232:
231:
226:
212:
211:
195:
193:
192:
187:
179:
178:
162:
160:
159:
154:
134:
132:
131:
126:
114:
112:
111:
106:
94:
92:
91:
86:
67:
65:
64:
59:
6569:
6568:
6564:
6563:
6562:
6560:
6559:
6558:
6529:
6528:
6527:
6522:
6461:Banach manifold
6454:Generalizations
6449:
6404:
6341:
6238:
6200:Ricci curvature
6156:Cotangent space
6134:
6072:
5914:
5908:
5867:Exponential map
5831:
5776:
5770:
5690:
5680:
5650:
5645:
5614:
5593:Generalizations
5588:
5542:
5511:
5446:Exponential map
5407:
5397:
5342:
5341:
5327:
5326:
5322:
5308:
5307:
5303:
5289:
5288:
5284:
5279:(3): 1079–1097.
5270:
5269:
5265:
5251:
5250:
5246:
5232:
5231:
5227:
5222:
5198:
5197:
5164:
5159:
5158:
5139:
5138:
5119:
5118:
5099:
5098:
5074:
5012:
4996:
4982:
4953:
4948:
4947:
4921:
4910:
4909:
4890:
4889:
4854:
4849:
4848:
4829:
4828:
4801:
4739:
4726:
4713:
4690:
4675:
4661:
4632:
4627:
4626:
4600:
4599:
4574:
4573:
4542:
4541:
4523:
4479:
4478:
4450:
4449:
4428:
4423:
4422:
4383:
4375:
4374:
4340:
4339:
4284:
4283:
4256:
4225:
4224:
4194:
4175:
4174:
4137:
4136:
4097:
4089:
4088:
4054:
4053:
3998:
3997:
3973:
3956:
3955:
3930:
3929:
3902:
3897:
3896:
3859:
3858:
3833:
3832:
3801:
3800:
3778:
3777:
3758:
3757:
3733:
3708:
3695:
3694:
3665:
3652:
3647:
3646:
3632:Gerhard Huisken
3618:geometric flows
3614:
3577:
3576:
3548:
3547:
3525:
3515:
3502:
3458:
3448:
3447:
3421:
3420:
3398:
3388:
3375:
3331:
3321:
3320:
3299:
3298:
3270:
3269:
3248:
3233:
3220:
3219:
3215:
3192:
3187:
3186:
3157:
3156:
3135:
3119:
3114:
3113:
3094:
3093:
3063:
3062:
3041:
3016:
3011:
3010:
2991:
2990:
2971:
2970:
2947:
2946:
2915:
2914:
2888:
2887:
2868:
2867:
2842:
2841:
2801:
2800:
2778:
2777:
2758:
2757:
2736:
2711:
2706:
2705:
2686:
2685:
2666:
2665:
2628:
2599:
2594:
2593:
2570:
2569:
2547:
2546:
2511:
2510:
2507:
2470:
2469:
2420:
2381:
2371:
2370:
2342:
2337:
2336:
2317:
2316:
2291:
2290:
2269:
2247:
2217:
2212:
2211:
2178:
2177:
2150:
2145:
2144:
2125:
2124:
2099:
2098:
2059:
2054:
2053:
2034:
2033:
2004:
1999:
1998:
1979:
1978:
1953:
1952:
1904:
1899:
1898:
1879:
1878:
1859:
1858:
1835:
1834:
1803:
1802:
1795:
1755:
1754:
1726:
1679:
1678:
1655:
1608:
1607:
1577:
1576:
1555:
1545:
1522:
1517:
1516:
1482:
1481:
1453:
1452:
1431:
1415:
1410:
1409:
1390:
1389:
1362:
1361:
1340:
1315:
1310:
1309:
1290:
1289:
1270:
1269:
1234:
1233:
1199:
1178:
1177:
1155:
1154:
1126:
1125:
1084:
1083:
1079:
1078:
1025:
1024:
1020:
1003:
999:
965:
964:
930:
929:
910:
909:
890:
889:
886:
836:
835:
834:one finds that
811:
786:
781:
780:
758:
757:
708:
703:
702:
671:
670:
651:
650:
631:
630:
593:
592:
567:
566:
545:
520:
515:
514:
495:
494:
475:
474:
455:
454:
426:
421:
420:
382:
366:
353:
348:
347:
344:Ricci curvature
309:
293:
277:
261:
248:
243:
242:
203:
198:
197:
170:
165:
164:
145:
144:
117:
116:
97:
96:
74:
73:
72:with dimension
38:
37:
34:
17:
12:
11:
5:
6567:
6565:
6557:
6556:
6551:
6546:
6541:
6531:
6530:
6524:
6523:
6521:
6520:
6515:
6510:
6505:
6500:
6499:
6498:
6488:
6483:
6478:
6473:
6468:
6463:
6457:
6455:
6451:
6450:
6448:
6447:
6442:
6437:
6432:
6427:
6422:
6416:
6414:
6410:
6409:
6406:
6405:
6403:
6402:
6397:
6392:
6387:
6382:
6377:
6372:
6367:
6362:
6357:
6351:
6349:
6343:
6342:
6340:
6339:
6334:
6329:
6324:
6319:
6314:
6309:
6299:
6294:
6289:
6279:
6274:
6269:
6264:
6259:
6254:
6248:
6246:
6240:
6239:
6237:
6236:
6231:
6226:
6225:
6224:
6214:
6209:
6208:
6207:
6197:
6192:
6187:
6182:
6181:
6180:
6170:
6165:
6164:
6163:
6153:
6148:
6142:
6140:
6136:
6135:
6133:
6132:
6127:
6122:
6117:
6116:
6115:
6105:
6100:
6095:
6089:
6087:
6080:
6074:
6073:
6071:
6070:
6065:
6055:
6050:
6036:
6031:
6026:
6021:
6016:
6014:Parallelizable
6011:
6006:
6001:
6000:
5999:
5989:
5984:
5979:
5974:
5969:
5964:
5959:
5954:
5949:
5944:
5934:
5924:
5918:
5916:
5910:
5909:
5907:
5906:
5901:
5896:
5894:Lie derivative
5891:
5889:Integral curve
5886:
5881:
5876:
5875:
5874:
5864:
5859:
5858:
5857:
5850:Diffeomorphism
5847:
5841:
5839:
5833:
5832:
5830:
5829:
5824:
5819:
5814:
5809:
5804:
5799:
5794:
5789:
5783:
5781:
5772:
5771:
5769:
5768:
5763:
5758:
5753:
5748:
5743:
5738:
5733:
5728:
5727:
5726:
5721:
5711:
5710:
5709:
5698:
5696:
5695:Basic concepts
5692:
5691:
5681:
5679:
5678:
5671:
5664:
5656:
5647:
5646:
5644:
5643:
5638:
5633:
5628:
5622:
5620:
5616:
5615:
5613:
5612:
5610:Sub-Riemannian
5607:
5602:
5596:
5594:
5590:
5589:
5587:
5586:
5581:
5576:
5571:
5566:
5561:
5556:
5550:
5548:
5544:
5543:
5541:
5540:
5535:
5530:
5525:
5519:
5517:
5513:
5512:
5510:
5509:
5504:
5499:
5494:
5489:
5488:
5487:
5478:
5473:
5468:
5458:
5453:
5448:
5443:
5442:
5441:
5436:
5431:
5426:
5415:
5413:
5412:Basic concepts
5409:
5408:
5398:
5396:
5395:
5388:
5381:
5373:
5367:
5366:
5352:
5340:
5339:
5320:
5301:
5282:
5263:
5244:
5224:
5223:
5221:
5218:
5205:
5194:
5193:
5182:
5179:
5176:
5171:
5167:
5146:
5126:
5106:
5086:
5081:
5077:
5073:
5067:
5062:
5056:
5050:
5047:
5042:
5039:
5034:
5031:
5026:
5019:
5015:
5011:
5008:
5003:
4999:
4995:
4989:
4985:
4979:
4976:
4971:
4968:
4965:
4960:
4956:
4935:
4930:
4925:
4920:
4917:
4897:
4863:
4858:
4836:
4825:
4824:
4813:
4808:
4804:
4800:
4794:
4789:
4783:
4777:
4774:
4769:
4766:
4761:
4758:
4753:
4746:
4742:
4733:
4729:
4725:
4722:
4719:
4716:
4711:
4708:
4705:
4702:
4699:
4696:
4693:
4687:
4682:
4678:
4674:
4668:
4664:
4658:
4655:
4650:
4647:
4644:
4639:
4635:
4612:
4609:
4587:
4584:
4581:
4561:
4558:
4555:
4552:
4549:
4522:
4519:
4498:
4495:
4492:
4489:
4486:
4466:
4463:
4460:
4457:
4435:
4431:
4410:
4407:
4404:
4401:
4398:
4395:
4390:
4386:
4382:
4362:
4359:
4356:
4353:
4350:
4347:
4327:
4324:
4321:
4318:
4315:
4312:
4309:
4306:
4303:
4300:
4297:
4294:
4291:
4271:
4268:
4263:
4259:
4255:
4252:
4249:
4241:
4238:
4235:
4232:
4212:
4209:
4206:
4201:
4197:
4193:
4190:
4187:
4162:
4159:
4156:
4153:
4150:
4147:
4144:
4124:
4121:
4118:
4115:
4112:
4109:
4104:
4100:
4096:
4076:
4073:
4070:
4067:
4064:
4061:
4041:
4038:
4035:
4032:
4029:
4026:
4023:
4020:
4017:
4014:
4011:
4008:
4005:
3985:
3980:
3976:
3972:
3969:
3966:
3963:
3943:
3940:
3937:
3917:
3914:
3909:
3905:
3884:
3881:
3878:
3875:
3872:
3869:
3866:
3846:
3843:
3840:
3820:
3817:
3814:
3811:
3808:
3785:
3765:
3745:
3740:
3736:
3730:
3727:
3722:
3717:
3712:
3707:
3703:
3680:
3677:
3674:
3669:
3664:
3659:
3655:
3613:
3610:
3596:
3593:
3590:
3587:
3584:
3573:
3572:
3561:
3558:
3555:
3532:
3528:
3522:
3518:
3514:
3509:
3505:
3501:
3496:
3493:
3487:
3482:
3477:
3472:
3465:
3461:
3456:
3445:
3434:
3431:
3428:
3405:
3401:
3395:
3391:
3387:
3382:
3378:
3374:
3369:
3366:
3360:
3355:
3350:
3345:
3338:
3334:
3329:
3318:
3306:
3286:
3283:
3280:
3277:
3255:
3251:
3246:
3240:
3236:
3232:
3227:
3223:
3218:
3212:
3209:
3204:
3199:
3195:
3184:
3173:
3170:
3167:
3164:
3142:
3138:
3134:
3131:
3126:
3122:
3101:
3090:
3079:
3076:
3073:
3070:
3048:
3044:
3040:
3037:
3034:
3031:
3028:
3023:
3019:
2998:
2978:
2954:
2934:
2931:
2928:
2925:
2922:
2898:
2895:
2875:
2855:
2852:
2849:
2829:
2826:
2823:
2820:
2817:
2814:
2811:
2808:
2788:
2785:
2765:
2743:
2739:
2735:
2732:
2729:
2726:
2723:
2718:
2714:
2693:
2673:
2653:
2648:
2643:
2640:
2635:
2631:
2625:
2620:
2617:
2614:
2611:
2606:
2602:
2577:
2557:
2554:
2530:
2527:
2524:
2521:
2518:
2506:
2503:
2501:
2500:
2497:
2494:
2483:
2480:
2477:
2454:
2449:
2445:
2438:
2435:
2432:
2429:
2426:
2423:
2419:
2413:
2408:
2403:
2398:
2393:
2388:
2384:
2379:
2368:
2357:
2354:
2349:
2345:
2324:
2304:
2301:
2298:
2276:
2272:
2265:
2262:
2259:
2256:
2253:
2250:
2246:
2241:
2238:
2235:
2232:
2229:
2224:
2220:
2209:
2197:
2194:
2191:
2188:
2185:
2165:
2162:
2157:
2153:
2132:
2112:
2109:
2106:
2086:
2083:
2080:
2077:
2074:
2071:
2066:
2062:
2041:
2030:
2019:
2016:
2011:
2007:
1986:
1966:
1963:
1960:
1940:
1937:
1934:
1931:
1928:
1925:
1922:
1919:
1916:
1911:
1907:
1886:
1866:
1842:
1822:
1819:
1816:
1813:
1810:
1794:
1791:
1774:
1771:
1768:
1765:
1762:
1751:
1750:
1738:
1733:
1729:
1723:
1720:
1714:
1709:
1704:
1699:
1694:
1691:
1687:
1676:
1662:
1658:
1652:
1649:
1643:
1638:
1633:
1628:
1623:
1620:
1616:
1605:
1593:
1590:
1587:
1584:
1562:
1558:
1552:
1548:
1542:
1539:
1534:
1529:
1525:
1513:
1501:
1498:
1495:
1492:
1489:
1469:
1466:
1463:
1460:
1438:
1434:
1430:
1427:
1422:
1418:
1397:
1386:
1375:
1372:
1369:
1347:
1343:
1339:
1336:
1333:
1330:
1327:
1322:
1318:
1297:
1277:
1253:
1250:
1247:
1244:
1241:
1217:
1214:
1211:
1206:
1202:
1196:
1193:
1188:
1185:
1165:
1162:
1142:
1139:
1136:
1133:
1113:
1108:
1103:
1099:
1096:
1091:
1087:
1082:
1075:
1072:
1067:
1062:
1057:
1052:
1048:
1044:
1040:
1037:
1032:
1028:
1023:
1017:
1014:
1009:
1006:
1002:
997:
992:
987:
982:
977:
973:
952:
949:
946:
943:
940:
937:
917:
897:
885:
882:
869:
866:
863:
858:
855:
850:
847:
844:
823:
818:
814:
810:
807:
804:
801:
798:
793:
789:
768:
765:
745:
742:
739:
734:
731:
726:
723:
720:
715:
711:
678:
658:
638:
629:Equivalently,
618:
615:
612:
609:
606:
603:
600:
580:
577:
574:
552:
548:
544:
541:
538:
535:
532:
527:
523:
502:
482:
462:
447:
446:
433:
429:
413:
401:
397:
394:
389:
385:
381:
378:
373:
369:
365:
360:
356:
340:
328:
324:
321:
316:
312:
308:
305:
300:
296:
292:
289:
284:
280:
276:
273:
268:
264:
260:
255:
251:
235:
224:
221:
218:
215:
210:
206:
196:a real number
185:
182:
177:
173:
152:
124:
104:
84:
81:
57:
54:
51:
48:
45:
33:
30:
15:
13:
10:
9:
6:
4:
3:
2:
6566:
6555:
6552:
6550:
6547:
6545:
6542:
6540:
6537:
6536:
6534:
6519:
6516:
6514:
6513:Supermanifold
6511:
6509:
6506:
6504:
6501:
6497:
6494:
6493:
6492:
6489:
6487:
6484:
6482:
6479:
6477:
6474:
6472:
6469:
6467:
6464:
6462:
6459:
6458:
6456:
6452:
6446:
6443:
6441:
6438:
6436:
6433:
6431:
6428:
6426:
6423:
6421:
6418:
6417:
6415:
6411:
6401:
6398:
6396:
6393:
6391:
6388:
6386:
6383:
6381:
6378:
6376:
6373:
6371:
6368:
6366:
6363:
6361:
6358:
6356:
6353:
6352:
6350:
6348:
6344:
6338:
6335:
6333:
6330:
6328:
6325:
6323:
6320:
6318:
6315:
6313:
6310:
6308:
6304:
6300:
6298:
6295:
6293:
6290:
6288:
6284:
6280:
6278:
6275:
6273:
6270:
6268:
6265:
6263:
6260:
6258:
6255:
6253:
6250:
6249:
6247:
6245:
6241:
6235:
6234:Wedge product
6232:
6230:
6227:
6223:
6220:
6219:
6218:
6215:
6213:
6210:
6206:
6203:
6202:
6201:
6198:
6196:
6193:
6191:
6188:
6186:
6183:
6179:
6178:Vector-valued
6176:
6175:
6174:
6171:
6169:
6166:
6162:
6159:
6158:
6157:
6154:
6152:
6149:
6147:
6144:
6143:
6141:
6137:
6131:
6128:
6126:
6123:
6121:
6118:
6114:
6111:
6110:
6109:
6108:Tangent space
6106:
6104:
6101:
6099:
6096:
6094:
6091:
6090:
6088:
6084:
6081:
6079:
6075:
6069:
6066:
6064:
6060:
6056:
6054:
6051:
6049:
6045:
6041:
6037:
6035:
6032:
6030:
6027:
6025:
6022:
6020:
6017:
6015:
6012:
6010:
6007:
6005:
6002:
5998:
5995:
5994:
5993:
5990:
5988:
5985:
5983:
5980:
5978:
5975:
5973:
5970:
5968:
5965:
5963:
5960:
5958:
5955:
5953:
5950:
5948:
5945:
5943:
5939:
5935:
5933:
5929:
5925:
5923:
5920:
5919:
5917:
5911:
5905:
5902:
5900:
5897:
5895:
5892:
5890:
5887:
5885:
5882:
5880:
5877:
5873:
5872:in Lie theory
5870:
5869:
5868:
5865:
5863:
5860:
5856:
5853:
5852:
5851:
5848:
5846:
5843:
5842:
5840:
5838:
5834:
5828:
5825:
5823:
5820:
5818:
5815:
5813:
5810:
5808:
5805:
5803:
5800:
5798:
5795:
5793:
5790:
5788:
5785:
5784:
5782:
5779:
5775:Main results
5773:
5767:
5764:
5762:
5759:
5757:
5756:Tangent space
5754:
5752:
5749:
5747:
5744:
5742:
5739:
5737:
5734:
5732:
5729:
5725:
5722:
5720:
5717:
5716:
5715:
5712:
5708:
5705:
5704:
5703:
5700:
5699:
5697:
5693:
5688:
5684:
5677:
5672:
5670:
5665:
5663:
5658:
5657:
5654:
5642:
5639:
5637:
5634:
5632:
5629:
5627:
5624:
5623:
5621:
5617:
5611:
5608:
5606:
5603:
5601:
5598:
5597:
5595:
5591:
5585:
5584:Schur's lemma
5582:
5580:
5577:
5575:
5572:
5570:
5567:
5565:
5562:
5560:
5559:Gauss's lemma
5557:
5555:
5552:
5551:
5549:
5545:
5539:
5536:
5534:
5531:
5529:
5526:
5524:
5521:
5520:
5518:
5514:
5508:
5505:
5503:
5500:
5498:
5495:
5493:
5490:
5486:
5482:
5479:
5477:
5474:
5472:
5469:
5467:
5464:
5463:
5462:
5461:Metric tensor
5459:
5457:
5456:Inner product
5454:
5452:
5449:
5447:
5444:
5440:
5437:
5435:
5432:
5430:
5427:
5425:
5422:
5421:
5420:
5417:
5416:
5414:
5410:
5405:
5401:
5394:
5389:
5387:
5382:
5380:
5375:
5374:
5371:
5365:
5364:0-12-526740-1
5361:
5357:
5353:
5350:
5349:
5344:
5343:
5335:
5331:
5324:
5321:
5316:
5312:
5305:
5302:
5298:(1): 237–266.
5297:
5293:
5286:
5283:
5278:
5274:
5267:
5264:
5259:
5255:
5248:
5245:
5241:(2): 255–306.
5240:
5236:
5229:
5226:
5219:
5217:
5180:
5177:
5174:
5169:
5165:
5144:
5124:
5104:
5084:
5079:
5075:
5071:
5065:
5060:
5048:
5045:
5040:
5037:
5032:
5029:
5013:
5009:
5006:
5001:
4997:
4993:
4987:
4974:
4969:
4966:
4954:
4933:
4928:
4918:
4895:
4887:
4886:
4885:
4883:
4882:Stefan Müller
4879:
4861:
4811:
4806:
4802:
4798:
4792:
4787:
4775:
4772:
4767:
4764:
4759:
4756:
4744:
4740:
4731:
4723:
4720:
4717:
4706:
4703:
4700:
4694:
4691:
4685:
4680:
4676:
4672:
4666:
4653:
4648:
4645:
4637:
4633:
4607:
4585:
4582:
4579:
4556:
4553:
4550:
4539:
4538:
4537:
4535:
4534:Peter Topping
4531:
4526:
4520:
4518:
4516:
4512:
4511:any spacetime
4493:
4490:
4487:
4464:
4461:
4458:
4455:
4433:
4429:
4408:
4405:
4402:
4396:
4388:
4384:
4380:
4360:
4357:
4351:
4345:
4322:
4319:
4316:
4304:
4301:
4298:
4292:
4289:
4266:
4261:
4257:
4253:
4250:
4239:
4236:
4233:
4230:
4210:
4204:
4199:
4195:
4191:
4188:
4157:
4154:
4151:
4145:
4142:
4122:
4119:
4116:
4110:
4102:
4098:
4094:
4074:
4071:
4065:
4059:
4036:
4033:
4030:
4018:
4015:
4012:
4006:
4003:
3983:
3978:
3974:
3970:
3967:
3964:
3961:
3941:
3938:
3935:
3915:
3912:
3907:
3903:
3879:
3876:
3873:
3867:
3864:
3844:
3841:
3838:
3815:
3812:
3809:
3797:
3783:
3763:
3743:
3738:
3734:
3728:
3725:
3720:
3715:
3705:
3678:
3675:
3672:
3657:
3653:
3644:
3639:
3637:
3633:
3628:
3626:
3621:
3619:
3611:
3608:
3591:
3588:
3585:
3559:
3556:
3553:
3530:
3520:
3516:
3512:
3507:
3503:
3494:
3491:
3485:
3480:
3475:
3463:
3459:
3446:
3432:
3429:
3426:
3403:
3393:
3389:
3385:
3380:
3376:
3367:
3364:
3358:
3353:
3348:
3336:
3332:
3319:
3304:
3284:
3281:
3278:
3275:
3253:
3249:
3244:
3238:
3234:
3230:
3225:
3221:
3216:
3210:
3207:
3202:
3197:
3193:
3185:
3171:
3168:
3165:
3162:
3140:
3136:
3132:
3129:
3124:
3120:
3099:
3091:
3077:
3074:
3071:
3068:
3046:
3042:
3035:
3029:
3026:
3021:
3017:
2996:
2976:
2968:
2967:
2966:
2952:
2929:
2926:
2923:
2910:
2896:
2893:
2873:
2866:implies that
2853:
2850:
2847:
2827:
2824:
2821:
2818:
2815:
2812:
2809:
2806:
2786:
2783:
2763:
2741:
2737:
2730:
2724:
2721:
2716:
2712:
2691:
2671:
2651:
2641:
2638:
2633:
2629:
2618:
2615:
2612:
2609:
2604:
2600:
2591:
2575:
2555:
2552:
2545:of dimension
2544:
2525:
2522:
2519:
2504:
2498:
2495:
2481:
2478:
2475:
2452:
2447:
2443:
2433:
2430:
2427:
2421:
2417:
2411:
2406:
2401:
2391:
2386:
2382:
2369:
2355:
2352:
2347:
2343:
2322:
2302:
2299:
2296:
2274:
2270:
2260:
2257:
2254:
2248:
2244:
2239:
2233:
2227:
2222:
2218:
2210:
2192:
2189:
2186:
2163:
2160:
2155:
2151:
2130:
2110:
2107:
2104:
2084:
2081:
2075:
2069:
2064:
2060:
2039:
2031:
2017:
2014:
2009:
2005:
1984:
1964:
1961:
1958:
1935:
1929:
1926:
1920:
1914:
1909:
1905:
1884:
1864:
1856:
1855:
1854:
1840:
1817:
1814:
1811:
1798:
1792:
1790:
1786:
1769:
1766:
1763:
1736:
1731:
1727:
1721:
1718:
1712:
1707:
1702:
1692:
1689:
1677:
1660:
1656:
1650:
1647:
1641:
1636:
1631:
1621:
1618:
1606:
1591:
1588:
1585:
1582:
1560:
1556:
1550:
1546:
1540:
1537:
1532:
1527:
1523:
1514:
1496:
1493:
1490:
1467:
1464:
1461:
1458:
1436:
1432:
1428:
1425:
1420:
1416:
1395:
1387:
1373:
1370:
1367:
1345:
1341:
1334:
1328:
1325:
1320:
1316:
1295:
1275:
1267:
1266:
1265:
1248:
1245:
1242:
1229:
1215:
1212:
1209:
1204:
1200:
1194:
1191:
1186:
1183:
1163:
1160:
1140:
1137:
1134:
1131:
1111:
1106:
1101:
1097:
1094:
1089:
1085:
1080:
1073:
1070:
1065:
1060:
1055:
1050:
1046:
1042:
1038:
1035:
1030:
1026:
1021:
1015:
1012:
1007:
1004:
1000:
995:
990:
985:
975:
950:
944:
941:
938:
915:
895:
883:
881:
867:
864:
861:
856:
853:
848:
845:
842:
821:
816:
812:
805:
799:
796:
791:
787:
766:
763:
743:
740:
737:
732:
729:
724:
721:
718:
713:
709:
700:
694:
692:
676:
656:
636:
616:
613:
607:
601:
598:
578:
575:
572:
550:
546:
539:
533:
530:
525:
521:
500:
480:
460:
453:Suppose that
450:
431:
418:
414:
392:
387:
383:
379:
376:
371:
367:
363:
358:
354:
345:
341:
319:
314:
310:
306:
303:
298:
294:
290:
287:
282:
278:
274:
271:
266:
262:
258:
253:
249:
240:
236:
219:
213:
208:
204:
183:
180:
175:
171:
150:
142:
138:
137:
136:
122:
102:
82:
79:
71:
52:
49:
46:
31:
29:
26:
25:Schur's lemma
22:
6440:Moving frame
6435:Morse theory
6425:Gauge theory
6217:Tensor field
6146:Closed/Exact
6125:Vector field
6093:Distribution
6034:Hypercomplex
6029:Quaternionic
5766:Vector field
5724:Smooth atlas
5619:Applications
5583:
5547:Main results
5355:
5346:
5336:(1): 75–110.
5333:
5329:
5323:
5314:
5310:
5304:
5295:
5291:
5285:
5276:
5272:
5266:
5257:
5253:
5247:
5238:
5234:
5228:
5195:
4826:
4598:Then, where
4527:
4524:
3798:
3640:
3629:
3622:
3615:
3612:Applications
3574:
2912:
2508:
1800:
1796:
1788:
1752:
1231:
887:
696:
452:
448:
68:is a smooth
35:
24:
18:
6385:Levi-Civita
6375:Generalized
6347:Connections
6297:Lie algebra
6229:Volume form
6130:Vector flow
6103:Pushforward
6098:Lie bracket
5997:Lie algebra
5962:G-structure
5751:Pushforward
5731:Submanifold
5260:(1): 47–62.
1512:is Einstein
6533:Categories
6508:Stratifold
6466:Diffeology
6262:Associated
6063:Symplectic
6048:Riemannian
5977:Hyperbolic
5904:Submersion
5812:Hopf–Rinow
5746:Submersion
5741:Smooth map
5579:Ricci flow
5528:Hyperbolic
5220:References
4338:such that
4052:such that
3112:such that
3009:such that
2704:such that
2052:such that
1897:such that
1408:such that
1308:such that
513:such that
6390:Principal
6365:Ehresmann
6322:Subbundle
6312:Principal
6287:Fibration
6267:Cotangent
6139:Covectors
5992:Lie group
5972:Hermitian
5915:manifolds
5884:Immersion
5879:Foliation
5817:Noether's
5802:Frobenius
5797:De Rham's
5792:Darboux's
5683:Manifolds
5523:Hermitian
5476:Signature
5439:Sectional
5419:Curvature
5204:Σ
5175:
5076:μ
5033:−
5018:Σ
5014:∫
5007:≤
4998:μ
4978:¯
4970:−
4959:Σ
4955:∫
4919:⊆
4916:Σ
4835:Σ
4803:μ
4760:−
4741:∫
4721:−
4704:−
4686:≤
4677:μ
4657:¯
4649:−
4634:∫
4611:¯
4583:≥
4521:Stability
4459:∈
4385:φ
4346:φ
4311:→
4290:φ
4240:∈
4146:
4099:φ
4060:φ
4025:→
4004:φ
3971:∈
3939:∈
3868:
3842:∈
3663:→
3557:∈
3513:
3486:≤
3430:∈
3386:
3279:∈
3231:
3166:∈
3133:κ
3100:κ
3072:∈
3030:κ
2977:κ
2874:κ
2825:κ
2819:⋅
2810:κ
2725:κ
2672:κ
2639:
2610:
2479:∈
2431:−
2392:
2300:∈
2258:−
2228:
2176:that is,
2108:∈
2085:κ
2070:
2040:κ
1962:∈
1930:κ
1915:
1865:κ
1713:≤
1693:
1622:
1586:∈
1480:that is,
1462:∈
1429:κ
1396:κ
1371:∈
1329:κ
1276:κ
1210:
1184:κ
1161:κ
1138:κ
1095:
1036:
1008:−
865:κ
846:κ
800:κ
719:
637:κ
602:κ
576:∈
534:κ
481:κ
396:→
380:×
323:→
307:×
291:×
275:×
214:
6486:Orbifold
6481:K-theory
6471:Diffiety
6195:Pullback
6009:Oriented
5987:Kenmotsu
5967:Hadamard
5913:Types of
5862:Geodesic
5687:Glossary
5538:Kenmotsu
5451:Geodesic
5404:Glossary
4946:one has
4540:Suppose
3954:and all
3546:for all
3419:for all
3268:for all
3155:for all
3061:for all
2756:for all
2468:for all
2289:for all
2097:for all
1951:for all
1575:for all
1515:One has
1451:for all
1360:for all
565:for all
36:Suppose
6430:History
6413:Related
6327:Tangent
6305:)
6285:)
6252:Adjoint
6244:Bundles
6222:density
6120:Torsion
6086:Vectors
6078:Tensors
6061:)
6046:)
6042:,
6040:Pseudo−
6019:Poisson
5952:Finsler
5947:Fibered
5942:Contact
5940:)
5932:Complex
5930:)
5899:Section
5605:Hilbert
5600:Finsler
3317:is zero
6554:Lemmas
6395:Vector
6380:Koszul
6360:Cartan
6355:Affine
6337:Vector
6332:Tensor
6317:Spinor
6307:Normal
6303:Stable
6257:Affine
6161:bundle
6113:bundle
6059:Almost
5982:Kähler
5938:Almost
5928:Almost
5922:Closed
5822:Sard's
5778:(list)
5533:Kähler
5429:Scalar
5424:tensor
5362:
5097:where
3756:where
2840:Hence
689:is an
6503:Sheaf
6277:Fiber
6053:Rizza
6024:Prime
5855:Local
5845:Curve
5707:Atlas
5434:Ricci
3693:with
963:Then
591:then
6370:Form
6272:Dual
6205:flow
6068:Tame
6044:Sub−
5957:Flat
5837:Maps
5360:ISBN
4880:and
4532:and
4373:and
4143:Isom
4087:and
3865:Isom
2913:Let
2851:>
2568:Let
2509:Let
1801:Let
1232:Let
888:Let
415:the
342:the
237:the
139:the
6292:Jet
5315:443
5277:167
4757:Ric
4430:sec
3575:If
2989:on
2776:in
2684:on
2601:div
2335:of
2219:sec
2143:of
2061:sec
1997:of
1906:sec
1877:on
1753:If
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