24:
2338:
4130:
953:
only by identifications with objects of the tensor algebra - but likewise there are identifications with concepts in the
Clifford-algebra. Let us note that these three axioms of a curvature structure give rise to a well-developed structure theory, formulated in terms of projectors (a Weyl projector,
3937:
Although individually, the Weyl tensor and Ricci tensor do not in general determine the full curvature tensor, the
Riemann curvature tensor can be decomposed into a Weyl part and a Ricci part. This decomposition is known as the Ricci decomposition, and plays an important role in the
969:
The three identities form a complete list of symmetries of the curvature tensor, i.e. given any tensor which satisfies the identities above, one could find a
Riemannian manifold with such a curvature tensor at some point. Simple calculations show that such a tensor has
3553:
2121:
1617:
3978:
301:
2682:
3278:
800:
1120:
1247:
558:
3748:
2856:
4173:
3067:
2770:
3402:
1685:
2429:
901:
2924:
427:
383:
2726:
2579:
3283:
It is possible to do this precisely because of the symmetries of the curvature tensor (namely antisymmetry in the first and last pairs of indices, and block-symmetry of those pairs).
2966:
958:
and an
Einstein projector, needed for the setup of the Einsteinian gravitational equations). This structure theory is compatible with the action of the pseudo-orthogonal groups plus
2465:
2800:
2504:
702:
3002:
1025:
612:
3829:
3183:
3884:
3619:
3578:
3351:
3103:
1290:
3653:
3391:
1482:
2333:{\displaystyle \langle R(u,v)w,z\rangle ={\frac {1}{6}}\left.{\frac {\partial ^{2}}{\partial s\partial t}}\left(K(u+sz,v+tw)-K(u+sw,v+tz)\right)\right|_{(s,t)=(0,0)}}
2111:
1898:
1474:
1417:
1389:
1365:
1337:
1313:
142:
3970:
339:
2538:
3917:
3329:
1454:
3753:
The result does not depend on the choice of orthonormal basis. With four or more dimensions, Ricci curvature does not describe the curvature tensor completely.
3584:. The result does not depend on the choice of orthonormal basis. Starting with dimension 3, scalar curvature does not describe the curvature tensor completely.
465:
4323:
3849:
3143:
177:
4125:{\displaystyle e^{2f}\left(R+\left({\text{Hess}}(f)-df\otimes df+{\frac {1}{2}}\|{\text{grad}}(f)\|^{2}g\right){~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~}g\right)}
185:
4486:
4336:
2587:
3786:
has the same symmetries as the
Riemann curvature tensor, but with one extra constraint: its trace (as used to define the Ricci curvature) must vanish.
3191:
4199:
710:
4516:
4491:
1033:
4378:
1420:
1139:
473:
116:
The curvature of a
Riemannian manifold can be described in various ways; the most standard one is the curvature tensor, given in terms of a
4241:
1263:
Sectional curvature is a further, equivalent but more geometrical, description of the curvature of
Riemannian manifolds. It is a function
4606:
4653:
3291:
In general the following tensors and functions do not describe the curvature tensor completely, however they play an important role.
3665:
4678:
4316:
4250:
86:
4815:
4628:
3889:
In dimensions 2 and 3 the Weyl tensor vanishes, but in 4 or more dimensions the Weyl tensor can be non-zero. For a manifold of
2805:
4820:
4138:
3011:
2734:
4830:
3548:{\displaystyle S=\sum _{i,j}\langle R(e_{i},e_{j})e_{j},e_{i}\rangle =\sum _{i}\langle {\text{Ric}}(e_{i}),e_{i}\rangle ,}
3112:
4825:
4413:
4309:
3942:
of
Riemannian manifolds. In particular, it can be used to show that if the metric is rescaled by a conformal factor of
4708:
4176:
1628:
4648:
2858:
and hence does not necessarily vanish. The following describes relation between curvature form and curvature tensor:
2540:
be a local section of orthonormal bases. Then one can define the connection form, an antisymmetric matrix of 1-forms
2387:
808:
4434:
94:
2864:
4563:
4265:
121:
4584:
4496:
388:
344:
4771:
4746:
4668:
4356:
2690:
2543:
907:
111:
81:
48:
2939:
4718:
4599:
4542:
4192:
2438:
959:
4501:
2779:
2477:
4723:
4713:
4573:
4568:
4506:
4439:
4398:
3761:
3354:
3123:
2366:
642:
117:
2974:
4620:
60:
973:
573:
4403:
4203:
3796:
3152:
2374:
3854:
1622:
The following formula indicates that sectional curvature describes the curvature tensor completely:
914:, just because it looks similar to the Bianchi identity below. The first two should be addressed as
4761:
4733:
4688:
4460:
4429:
4417:
4388:
4371:
4332:
3932:
3765:
1258:
341:
is a linear transformation of the tangent space of the manifold; it is linear in each argument. If
68:
36:
3602:
3561:
79:
introduced an abstract and rigorous way to define curvature for these manifolds, now known as the
4693:
4643:
4592:
4558:
4455:
4424:
4282:
3939:
3890:
3334:
1612:{\displaystyle K(\sigma )=K(u,v)/|u\wedge v|^{2}{\text{ where }}K(u,v)=\langle R(u,v)v,u\rangle }
28:
17:
4465:
3079:
1266:
3625:
3363:
4470:
4246:
4210:
3358:
1904:
1691:
1459:
1402:
1374:
1350:
1322:
1298:
127:
4756:
4658:
4537:
4532:
4408:
4361:
4274:
3945:
3920:
3300:
2472:
2362:
309:
2516:
966:
and algebras, Lie triples and Jordan algebras. See the references given in the discussion.
4751:
4663:
4366:
4221:
3593:
2507:
2354:
3896:
3308:
1433:
432:
4784:
4779:
4698:
4214:
3834:
3128:
2348:
145:
75:
greater than 2 is too complicated to be described by a single number at a given point.
945:
are elements of the pseudo-orthogonal Lie algebra. All three together should be named
150:
4809:
4703:
4393:
3790:
3598:
Ricci curvature is a linear operator on tangent space at a point, usually denoted by
3146:
2358:
64:
296:{\displaystyle R(u,v)w=\nabla _{u}\nabla _{v}w-\nabla _{v}\nabla _{u}w-\nabla _{}w.}
3757:
3581:
2677:{\displaystyle \omega _{\ j}^{k}(e_{i})=\langle \nabla _{e_{i}}e_{j},e_{k}\rangle }
23:
4301:
3273:{\displaystyle \langle Q(u\wedge v),w\wedge z\rangle =\langle R(u,v)z,w\rangle .}
4789:
4260:
3777:
3305:
Scalar curvature is a function on any
Riemannian manifold, denoted variously by
2468:
628:
There are a few books where the curvature tensor is defined with opposite sign.
56:
32:
3972:, then the Riemann curvature tensor changes to (seen as a (0, 4)-tensor):
4511:
1027:
independent components. Yet another useful identity follows from these three:
4638:
4616:
2357:
gives an alternative way to describe curvature. It is used more for general
1316:
963:
90:
72:
795:{\displaystyle \langle R(u,v)w,z\rangle =-\langle R(u,v)z,w\rangle _{}^{}}
4794:
4383:
4225:
1115:{\displaystyle \langle R(u,v)w,z\rangle =\langle R(w,z)u,v\rangle _{}^{}}
44:
47:
may have different curvatures in different directions, described by the
4286:
1242:{\displaystyle \nabla _{u}R(v,w)+\nabla _{v}R(w,u)+\nabla _{w}R(u,v)=0}
553:{\displaystyle R(u,v)w=\nabla _{u}\nabla _{v}w-\nabla _{v}\nabla _{u}w}
76:
2929:
This approach builds in all symmetries of curvature tensor except the
2432:
40:
4278:
22:
97:
can be expressed in the same way with only slight modifications.
4588:
4305:
3743:{\displaystyle {\text{Ric}}(u)=\sum _{i}R(u,e_{i})e_{i}._{}^{}}
2365:, but it works just as well for the tangent bundle with the
2173:
3122:
It is sometimes convenient to think about curvature as an
1371:
is a locally defined piece of surface which has the plane
2851:{\displaystyle \omega _{\ j}^{i}\wedge \omega _{\ k}^{j}}
3185:), which is uniquely defined by the following identity:
85:. Similar notions have found applications everywhere in
4224:
can help if one knows something about the behavior of
4168:{\displaystyle {~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~}}
3919:
if and only if the metric is locally conformal to the
3062:{\displaystyle \theta ^{i}(v)=\langle e_{i},v\rangle }
2765:{\displaystyle \Omega =d\omega +\omega \wedge \omega }
101:
Ways to express the curvature of a
Riemannian manifold
4141:
3981:
3948:
3899:
3857:
3837:
3799:
3668:
3628:
3605:
3564:
3405:
3366:
3337:
3311:
3194:
3155:
3131:
3082:
3014:
2977:
2942:
2867:
2808:
2782:
2737:
2693:
2590:
2546:
2519:
2480:
2441:
2390:
2124:
1907:
1694:
1631:
1485:
1462:
1436:
1405:
1377:
1353:
1325:
1301:
1269:
1142:
1036:
976:
811:
713:
645:
576:
476:
435:
391:
347:
312:
188:
153:
130:
4770:
4732:
4677:
4627:
4551:
4525:
4479:
4448:
4344:
636:The curvature tensor has the following symmetries:
4167:
4124:
3964:
3911:
3878:
3843:
3823:
3742:
3647:
3613:
3572:
3547:
3385:
3345:
3323:
3272:
3177:
3137:
3097:
3061:
2996:
2960:
2918:
2850:
2794:
2764:
2720:
2676:
2573:
2532:
2498:
2459:
2423:
2332:
2105:
1892:
1679:
1611:
1468:
1448:
1411:
1383:
1359:
1331:
1315:(i.e. a 2-plane in the tangent spaces). It is the
1307:
1284:
1241:
1114:
1019:
895:
794:
696:
606:
552:
459:
421:
377:
333:
295:
171:
136:
4156:
4155:
4154:
4153:
4152:
4151:
4150:
4149:
4105:
4104:
4103:
4102:
4101:
4100:
4099:
4098:
2510:of the tangent bundle of a Riemannian manifold).
1680:{\displaystyle 6\langle R(u,v)w,z\rangle =_{}^{}}
39:), and a surface of positive Gaussian curvature (
3793:change of metric: if two metrics are related as
2373:-dimensional Riemannian manifold is given by an
3789:The Weyl tensor is invariant with respect to a
2424:{\displaystyle \Omega _{}^{}=\Omega _{\ j}^{i}}
896:{\displaystyle R(u,v)w+R(v,w)u+R(w,u)v=0_{}^{}}
4239:Kobayashi, Shoshichi; Nomizu, Katsumi (1996).
922:respectively, since the second means that the
4600:
4317:
8:
4242:Foundations of Differential Geometry, Vol. 1
4074:
4056:
3642:
3629:
3539:
3502:
3486:
3428:
3380:
3367:
3264:
3234:
3228:
3195:
3056:
3037:
2919:{\displaystyle R(u,v)w=\Omega (u\wedge v)w.}
2671:
2628:
2155:
2125:
1665:
1635:
1606:
1576:
1104:
1073:
1067:
1037:
784:
753:
744:
714:
565:noncommutativity of the covariant derivative
4607:
4593:
4585:
4487:Fundamental theorem of Riemannian geometry
4324:
4310:
4302:
4157:
4106:
2581:which satisfy from the following identity
422:{\displaystyle v=\partial /\partial x_{j}}
378:{\displaystyle u=\partial /\partial x_{i}}
27:From left to right: a surface of negative
4142:
4140:
4091:
4077:
4059:
4046:
4011:
3986:
3980:
3953:
3947:
3898:
3859:
3858:
3856:
3836:
3801:
3800:
3798:
3737:
3735:
3725:
3712:
3690:
3669:
3667:
3636:
3627:
3606:
3604:
3565:
3563:
3533:
3517:
3505:
3496:
3480:
3467:
3454:
3441:
3416:
3404:
3374:
3365:
3338:
3336:
3310:
3193:
3160:
3154:
3130:
3081:
3044:
3019:
3013:
2988:
2976:
2941:
2866:
2842:
2834:
2821:
2813:
2807:
2781:
2736:
2721:{\displaystyle \Omega =\Omega _{\ j}^{i}}
2712:
2704:
2692:
2665:
2652:
2640:
2635:
2616:
2603:
2595:
2589:
2574:{\displaystyle \omega =\omega _{\ j}^{i}}
2565:
2557:
2545:
2524:
2518:
2479:
2440:
2435:(or equivalently a 2-form with values in
2415:
2407:
2397:
2395:
2389:
2294:
2182:
2176:
2161:
2123:
2100:
2098:
1906:
1887:
1885:
1693:
1674:
1672:
1630:
1550:
1544:
1539:
1524:
1519:
1484:
1461:
1435:
1404:
1395:, obtained from geodesics which start at
1376:
1352:
1324:
1300:
1268:
1209:
1178:
1147:
1141:
1109:
1107:
1035:
1009:
994:
981:
975:
890:
888:
810:
789:
787:
712:
691:
689:
644:
575:
541:
531:
515:
505:
475:
434:
413:
401:
390:
369:
357:
346:
311:
269:
253:
243:
227:
217:
187:
152:
129:
35:), a surface of zero Gaussian curvature (
1456:are two linearly independent vectors in
962:. It has strong ties with the theory of
467:and therefore the formula simplifies to
4263:(1901). "Space of constant curvature".
4200:list of formulas in Riemannian geometry
2961:{\displaystyle \Omega \wedge \theta =0}
4191:of hypersurfaces and submanifolds see
2460:{\displaystyle \operatorname {so} (n)}
1133:) involves the covariant derivatives:
16:For a more elementary discussion, see
3893:, the Weyl tensor is zero. Moreover,
2795:{\displaystyle \omega \wedge \omega }
2499:{\displaystyle \operatorname {O} (n)}
947:pseudo-orthogonal curvature structure
7:
4245:(New ed.). Wiley-Interscience.
906:The last identity was discovered by
697:{\displaystyle R(u,v)=-R(v,u)_{}^{}}
2997:{\displaystyle \theta =\theta ^{i}}
563:i.e. the curvature tensor measures
4654:Radius of curvature (applications)
3831:for some positive scalar function
3357:of the curvature tensor; given an
3157:
3086:
2943:
2892:
2738:
2701:
2694:
2632:
2481:
2404:
2392:
2195:
2189:
2179:
1399:in the directions of the image of
1206:
1175:
1144:
538:
528:
512:
502:
429:are coordinate vector fields then
406:
398:
362:
354:
266:
250:
240:
224:
214:
131:
14:
4742:Curvature of Riemannian manifolds
3393:in the tangent space at a point
1020:{\displaystyle n^{2}(n^{2}-1)/12}
87:differential geometry of surfaces
607:{\displaystyle w\mapsto R(u,v)w}
3824:{\displaystyle {\tilde {g}}=fg}
3622:. Given an orthonormal basis
3178:{\displaystyle \Lambda ^{2}(T)}
4070:
4064:
4022:
4016:
3879:{\displaystyle {\tilde {W}}=W}
3864:
3806:
3718:
3699:
3680:
3674:
3523:
3510:
3460:
3434:
3252:
3240:
3213:
3201:
3172:
3166:
3031:
3025:
3008:-vector of 1-forms defined by
2907:
2895:
2883:
2871:
2622:
2609:
2493:
2487:
2454:
2448:
2325:
2313:
2307:
2295:
2281:
2251:
2242:
2212:
2143:
2131:
2091:
2088:
2076:
2067:
2055:
2046:
2028:
2019:
2001:
1992:
1974:
1965:
1947:
1938:
1914:
1908:
1878:
1875:
1863:
1854:
1842:
1833:
1815:
1806:
1788:
1779:
1761:
1752:
1734:
1725:
1701:
1695:
1653:
1641:
1594:
1582:
1570:
1558:
1540:
1525:
1516:
1504:
1495:
1489:
1279:
1273:
1230:
1218:
1199:
1187:
1168:
1156:
1091:
1079:
1055:
1043:
1006:
987:
875:
863:
851:
839:
827:
815:
771:
759:
732:
720:
686:
673:
661:
649:
598:
586:
580:
492:
480:
448:
436:
328:
316:
282:
270:
204:
192:
166:
154:
1:
4187:For calculation of curvature
3756:Explicit expressions for the
3113:exterior covariant derivative
4414:Raising and lowering indices
3614:{\displaystyle {\text{Ric}}}
3573:{\displaystyle {\text{Ric}}}
106:The Riemann curvature tensor
3764:is given in the article on
3346:{\displaystyle {\text{Sc}}}
43:). In higher dimensions, a
4847:
4435:Pseudo-Riemannian manifold
3930:
3775:
3591:
3298:
3098:{\displaystyle D\Omega =0}
2776:Note that the expression "
2346:
1285:{\displaystyle K(\sigma )}
1256:
910:, but is often called the
570:The linear transformation
179:by the following formula:
109:
95:pseudo-Riemannian manifold
15:
4564:Geometrization conjecture
4266:The Annals of Mathematics
4179:and Hess is the Hessian.
3648:{\displaystyle \{e_{i}\}}
3386:{\displaystyle \{e_{i}\}}
3287:Further curvature tensors
2116:Or in a simpler formula:
632:Symmetries and identities
122:covariant differentiation
4772:Curvature of connections
4747:Riemann curvature tensor
4669:Total absolute curvature
4183:Calculation of curvature
3655:in the tangent space at
2687:Then the curvature form
616:curvature transformation
112:Riemann curvature tensor
82:Riemann curvature tensor
49:Riemann curvature tensor
4816:Curvature (mathematics)
4719:Second fundamental form
4709:Gauss–Codazzi equations
4198:in coordinates see the
4193:second fundamental form
4177:Kulkarni–Nomizu product
3071:second Bianchi identity
2106:{\displaystyle ._{}^{}}
1893:{\displaystyle -_{}^{}}
1469:{\displaystyle \sigma }
1412:{\displaystyle \sigma }
1384:{\displaystyle \sigma }
1360:{\displaystyle \sigma }
1332:{\displaystyle \sigma }
1308:{\displaystyle \sigma }
1131:second Bianchi identity
137:{\displaystyle \nabla }
89:and other objects. The
4724:Third fundamental form
4714:First fundamental form
4679:Differential geometry
4649:Frenet–Serret formulas
4629:Differential geometry
4574:Uniformization theorem
4507:Nash embedding theorem
4440:Riemannian volume form
4399:Levi-Civita connection
4169:
4126:
3966:
3965:{\displaystyle e^{2f}}
3913:
3880:
3845:
3825:
3762:Levi-Civita connection
3744:
3649:
3615:
3574:
3549:
3387:
3347:
3325:
3274:
3179:
3139:
3118:The curvature operator
3099:
3063:
2998:
2962:
2931:first Bianchi identity
2920:
2852:
2796:
2766:
2722:
2678:
2575:
2534:
2500:
2461:
2425:
2369:. The curvature of an
2367:Levi-Civita connection
2334:
2107:
1894:
1681:
1613:
1470:
1450:
1413:
1391:as a tangent plane at
1385:
1361:
1333:
1309:
1286:
1243:
1116:
1021:
949:. They give rise to a
912:first Bianchi identity
897:
796:
698:
608:
554:
461:
423:
379:
335:
334:{\displaystyle R(u,v)}
297:
173:
138:
118:Levi-Civita connection
52:
4821:Differential geometry
4621:differential geometry
4209:by moving frames see
4170:
4127:
3967:
3914:
3881:
3846:
3826:
3784:Weyl curvature tensor
3772:Weyl curvature tensor
3745:
3650:
3616:
3575:
3550:
3388:
3348:
3326:
3275:
3180:
3140:
3100:
3064:
2999:
2963:
2921:
2853:
2797:
2767:
2723:
2679:
2576:
2535:
2533:{\displaystyle e_{i}}
2501:
2462:
2426:
2335:
2108:
1895:
1682:
1614:
1471:
1451:
1414:
1386:
1362:
1334:
1310:
1287:
1244:
1117:
1022:
898:
797:
699:
609:
555:
462:
424:
380:
336:
298:
174:
139:
61:differential geometry
26:
4831:Riemannian manifolds
4689:Principal curvatures
4497:Gauss–Bonnet theorem
4404:Covariant derivative
4204:covariant derivative
4139:
3979:
3946:
3897:
3855:
3835:
3797:
3666:
3626:
3603:
3562:
3403:
3364:
3335:
3309:
3192:
3153:
3129:
3080:
3012:
2975:
2940:
2865:
2806:
2780:
2735:
2691:
2588:
2544:
2517:
2478:
2439:
2388:
2122:
1905:
1692:
1629:
1483:
1460:
1434:
1403:
1375:
1351:
1323:
1299:
1267:
1140:
1034:
974:
920:Lie algebra property
809:
711:
643:
574:
474:
433:
389:
345:
310:
186:
151:
128:
69:Riemannian manifolds
4826:Riemannian geometry
4762:Sectional curvature
4734:Riemannian geometry
4615:Various notions of
4569:Poincaré conjecture
4430:Riemannian manifold
4418:Musical isomorphism
4333:Riemannian geometry
3933:Ricci decomposition
3927:Ricci decomposition
3912:{\displaystyle W=0}
3766:Christoffel symbols
3739:
3324:{\displaystyle S,R}
2933:, which takes form
2847:
2826:
2802:" is shorthand for
2717:
2608:
2570:
2420:
2399:
2102:
1889:
1676:
1449:{\displaystyle v,u}
1292:which depends on a
1259:Sectional curvature
1253:Sectional curvature
1111:
892:
791:
693:
614:is also called the
4694:Gaussian curvature
4644:Torsion of a curve
4559:General relativity
4502:Hopf–Rinow theorem
4449:Types of manifolds
4425:Parallel transport
4165:
4122:
3962:
3940:conformal geometry
3909:
3891:constant curvature
3876:
3841:
3821:
3740:
3731:
3695:
3645:
3611:
3570:
3545:
3501:
3427:
3383:
3353:. It is the full
3343:
3321:
3270:
3175:
3135:
3095:
3059:
2994:
2958:
2916:
2848:
2830:
2809:
2792:
2762:
2718:
2700:
2674:
2591:
2571:
2553:
2530:
2496:
2457:
2421:
2403:
2391:
2330:
2103:
2094:
1890:
1881:
1677:
1668:
1609:
1466:
1446:
1409:
1381:
1357:
1329:
1305:
1282:
1239:
1112:
1103:
1017:
893:
884:
792:
783:
694:
685:
604:
550:
460:{\displaystyle =0}
457:
419:
375:
331:
293:
169:
134:
53:
29:Gaussian curvature
18:Curvature of space
4803:
4802:
4582:
4581:
4211:Cartan connection
4163:
4145:
4112:
4094:
4062:
4054:
4014:
3867:
3844:{\displaystyle f}
3809:
3686:
3672:
3609:
3568:
3508:
3492:
3412:
3359:orthonormal basis
3341:
3138:{\displaystyle Q}
2837:
2816:
2707:
2598:
2560:
2410:
2363:principal bundles
2202:
2169:
1553:
1552: where
4838:
4757:Scalar curvature
4659:Affine curvature
4609:
4602:
4595:
4586:
4326:
4319:
4312:
4303:
4290:
4256:
4174:
4172:
4171:
4166:
4164:
4161:
4143:
4131:
4129:
4128:
4123:
4121:
4117:
4113:
4110:
4092:
4090:
4086:
4082:
4081:
4063:
4060:
4055:
4047:
4015:
4012:
3994:
3993:
3971:
3969:
3968:
3963:
3961:
3960:
3921:Euclidean metric
3918:
3916:
3915:
3910:
3885:
3883:
3882:
3877:
3869:
3868:
3860:
3850:
3848:
3847:
3842:
3830:
3828:
3827:
3822:
3811:
3810:
3802:
3760:in terms of the
3749:
3747:
3746:
3741:
3738:
3736:
3730:
3729:
3717:
3716:
3694:
3673:
3670:
3654:
3652:
3651:
3646:
3641:
3640:
3620:
3618:
3617:
3612:
3610:
3607:
3579:
3577:
3576:
3571:
3569:
3566:
3554:
3552:
3551:
3546:
3538:
3537:
3522:
3521:
3509:
3506:
3500:
3485:
3484:
3472:
3471:
3459:
3458:
3446:
3445:
3426:
3392:
3390:
3389:
3384:
3379:
3378:
3352:
3350:
3349:
3344:
3342:
3339:
3330:
3328:
3327:
3322:
3301:Scalar curvature
3295:Scalar curvature
3279:
3277:
3276:
3271:
3184:
3182:
3181:
3176:
3165:
3164:
3144:
3142:
3141:
3136:
3104:
3102:
3101:
3096:
3068:
3066:
3065:
3060:
3049:
3048:
3024:
3023:
3003:
3001:
3000:
2995:
2993:
2992:
2967:
2965:
2964:
2959:
2925:
2923:
2922:
2917:
2857:
2855:
2854:
2849:
2846:
2841:
2835:
2825:
2820:
2814:
2801:
2799:
2798:
2793:
2771:
2769:
2768:
2763:
2727:
2725:
2724:
2719:
2716:
2711:
2705:
2683:
2681:
2680:
2675:
2670:
2669:
2657:
2656:
2647:
2646:
2645:
2644:
2621:
2620:
2607:
2602:
2596:
2580:
2578:
2577:
2572:
2569:
2564:
2558:
2539:
2537:
2536:
2531:
2529:
2528:
2505:
2503:
2502:
2497:
2473:orthogonal group
2466:
2464:
2463:
2458:
2430:
2428:
2427:
2422:
2419:
2414:
2408:
2398:
2396:
2339:
2337:
2336:
2331:
2329:
2328:
2293:
2289:
2288:
2284:
2203:
2201:
2187:
2186:
2177:
2170:
2162:
2112:
2110:
2109:
2104:
2101:
2099:
1899:
1897:
1896:
1891:
1888:
1886:
1686:
1684:
1683:
1678:
1675:
1673:
1618:
1616:
1615:
1610:
1554:
1551:
1549:
1548:
1543:
1528:
1523:
1475:
1473:
1472:
1467:
1455:
1453:
1452:
1447:
1418:
1416:
1415:
1410:
1390:
1388:
1387:
1382:
1366:
1364:
1363:
1358:
1338:
1336:
1335:
1330:
1314:
1312:
1311:
1306:
1291:
1289:
1288:
1283:
1248:
1246:
1245:
1240:
1214:
1213:
1183:
1182:
1152:
1151:
1127:Bianchi identity
1121:
1119:
1118:
1113:
1110:
1108:
1026:
1024:
1023:
1018:
1013:
999:
998:
986:
985:
936:
902:
900:
899:
894:
891:
889:
801:
799:
798:
793:
790:
788:
703:
701:
700:
695:
692:
690:
613:
611:
610:
605:
559:
557:
556:
551:
546:
545:
536:
535:
520:
519:
510:
509:
466:
464:
463:
458:
428:
426:
425:
420:
418:
417:
405:
384:
382:
381:
376:
374:
373:
361:
340:
338:
337:
332:
302:
300:
299:
294:
286:
285:
258:
257:
248:
247:
232:
231:
222:
221:
178:
176:
175:
172:{\displaystyle }
170:
143:
141:
140:
135:
4846:
4845:
4841:
4840:
4839:
4837:
4836:
4835:
4806:
4805:
4804:
4799:
4766:
4752:Ricci curvature
4728:
4680:
4673:
4664:Total curvature
4630:
4623:
4613:
4583:
4578:
4547:
4526:Generalizations
4521:
4475:
4444:
4379:Exponential map
4340:
4330:
4299:
4297:
4279:10.2307/1967636
4273:(1/4): 71–112.
4259:
4253:
4238:
4235:
4222:Jacobi equation
4185:
4137:
4136:
4073:
4010:
4006:
3999:
3995:
3982:
3977:
3976:
3949:
3944:
3943:
3935:
3929:
3895:
3894:
3853:
3852:
3833:
3832:
3795:
3794:
3780:
3774:
3721:
3708:
3664:
3663:
3632:
3624:
3623:
3601:
3600:
3596:
3594:Ricci curvature
3590:
3588:Ricci curvature
3560:
3559:
3529:
3513:
3476:
3463:
3450:
3437:
3401:
3400:
3370:
3362:
3361:
3333:
3332:
3307:
3306:
3303:
3297:
3289:
3190:
3189:
3156:
3151:
3150:
3127:
3126:
3120:
3078:
3077:
3040:
3015:
3010:
3009:
2984:
2973:
2972:
2938:
2937:
2863:
2862:
2804:
2803:
2778:
2777:
2733:
2732:
2689:
2688:
2661:
2648:
2636:
2631:
2612:
2586:
2585:
2542:
2541:
2520:
2515:
2514:
2508:structure group
2506:, which is the
2476:
2475:
2437:
2436:
2386:
2385:
2355:connection form
2351:
2345:
2208:
2204:
2188:
2178:
2175:
2172:
2171:
2120:
2119:
1903:
1902:
1690:
1689:
1627:
1626:
1538:
1481:
1480:
1458:
1457:
1432:
1431:
1421:exponential map
1401:
1400:
1373:
1372:
1349:
1348:
1321:
1320:
1317:Gauss curvature
1297:
1296:
1265:
1264:
1261:
1255:
1205:
1174:
1143:
1138:
1137:
1032:
1031:
990:
977:
972:
971:
954:giving rise to
923:
807:
806:
709:
708:
641:
640:
634:
572:
571:
537:
527:
511:
501:
472:
471:
431:
430:
409:
387:
386:
365:
343:
342:
308:
307:
265:
249:
239:
223:
213:
184:
183:
149:
148:
126:
125:
114:
108:
103:
59:, specifically
21:
12:
11:
5:
4844:
4842:
4834:
4833:
4828:
4823:
4818:
4808:
4807:
4801:
4800:
4798:
4797:
4792:
4787:
4785:Torsion tensor
4782:
4780:Curvature form
4776:
4774:
4768:
4767:
4765:
4764:
4759:
4754:
4749:
4744:
4738:
4736:
4730:
4729:
4727:
4726:
4721:
4716:
4711:
4706:
4701:
4699:Mean curvature
4696:
4691:
4685:
4683:
4675:
4674:
4672:
4671:
4666:
4661:
4656:
4651:
4646:
4641:
4635:
4633:
4625:
4624:
4614:
4612:
4611:
4604:
4597:
4589:
4580:
4579:
4577:
4576:
4571:
4566:
4561:
4555:
4553:
4549:
4548:
4546:
4545:
4543:Sub-Riemannian
4540:
4535:
4529:
4527:
4523:
4522:
4520:
4519:
4514:
4509:
4504:
4499:
4494:
4489:
4483:
4481:
4477:
4476:
4474:
4473:
4468:
4463:
4458:
4452:
4450:
4446:
4445:
4443:
4442:
4437:
4432:
4427:
4422:
4421:
4420:
4411:
4406:
4401:
4391:
4386:
4381:
4376:
4375:
4374:
4369:
4364:
4359:
4348:
4346:
4345:Basic concepts
4342:
4341:
4331:
4329:
4328:
4321:
4314:
4306:
4296:
4293:
4292:
4291:
4257:
4251:
4234:
4231:
4230:
4229:
4218:
4215:curvature form
4207:
4196:
4184:
4181:
4160:
4148:
4133:
4132:
4120:
4116:
4109:
4097:
4089:
4085:
4080:
4076:
4072:
4069:
4066:
4058:
4053:
4050:
4045:
4042:
4039:
4036:
4033:
4030:
4027:
4024:
4021:
4018:
4009:
4005:
4002:
3998:
3992:
3989:
3985:
3959:
3956:
3952:
3931:Main article:
3928:
3925:
3908:
3905:
3902:
3875:
3872:
3866:
3863:
3840:
3820:
3817:
3814:
3808:
3805:
3776:Main article:
3773:
3770:
3751:
3750:
3734:
3728:
3724:
3720:
3715:
3711:
3707:
3704:
3701:
3698:
3693:
3689:
3685:
3682:
3679:
3676:
3644:
3639:
3635:
3631:
3592:Main article:
3589:
3586:
3556:
3555:
3544:
3541:
3536:
3532:
3528:
3525:
3520:
3516:
3512:
3504:
3499:
3495:
3491:
3488:
3483:
3479:
3475:
3470:
3466:
3462:
3457:
3453:
3449:
3444:
3440:
3436:
3433:
3430:
3425:
3422:
3419:
3415:
3411:
3408:
3382:
3377:
3373:
3369:
3320:
3317:
3314:
3299:Main article:
3296:
3293:
3288:
3285:
3281:
3280:
3269:
3266:
3263:
3260:
3257:
3254:
3251:
3248:
3245:
3242:
3239:
3236:
3233:
3230:
3227:
3224:
3221:
3218:
3215:
3212:
3209:
3206:
3203:
3200:
3197:
3174:
3171:
3168:
3163:
3159:
3134:
3119:
3116:
3106:
3105:
3094:
3091:
3088:
3085:
3058:
3055:
3052:
3047:
3043:
3039:
3036:
3033:
3030:
3027:
3022:
3018:
2991:
2987:
2983:
2980:
2969:
2968:
2957:
2954:
2951:
2948:
2945:
2927:
2926:
2915:
2912:
2909:
2906:
2903:
2900:
2897:
2894:
2891:
2888:
2885:
2882:
2879:
2876:
2873:
2870:
2845:
2840:
2833:
2829:
2824:
2819:
2812:
2791:
2788:
2785:
2774:
2773:
2761:
2758:
2755:
2752:
2749:
2746:
2743:
2740:
2728:is defined by
2715:
2710:
2703:
2699:
2696:
2685:
2684:
2673:
2668:
2664:
2660:
2655:
2651:
2643:
2639:
2634:
2630:
2627:
2624:
2619:
2615:
2611:
2606:
2601:
2594:
2568:
2563:
2556:
2552:
2549:
2527:
2523:
2495:
2492:
2489:
2486:
2483:
2456:
2453:
2450:
2447:
2444:
2418:
2413:
2406:
2402:
2394:
2359:vector bundles
2349:Curvature form
2347:Main article:
2344:
2343:Curvature form
2341:
2327:
2324:
2321:
2318:
2315:
2312:
2309:
2306:
2303:
2300:
2297:
2292:
2287:
2283:
2280:
2277:
2274:
2271:
2268:
2265:
2262:
2259:
2256:
2253:
2250:
2247:
2244:
2241:
2238:
2235:
2232:
2229:
2226:
2223:
2220:
2217:
2214:
2211:
2207:
2200:
2197:
2194:
2191:
2185:
2181:
2174:
2168:
2165:
2160:
2157:
2154:
2151:
2148:
2145:
2142:
2139:
2136:
2133:
2130:
2127:
2114:
2113:
2097:
2093:
2090:
2087:
2084:
2081:
2078:
2075:
2072:
2069:
2066:
2063:
2060:
2057:
2054:
2051:
2048:
2045:
2042:
2039:
2036:
2033:
2030:
2027:
2024:
2021:
2018:
2015:
2012:
2009:
2006:
2003:
2000:
1997:
1994:
1991:
1988:
1985:
1982:
1979:
1976:
1973:
1970:
1967:
1964:
1961:
1958:
1955:
1952:
1949:
1946:
1943:
1940:
1937:
1934:
1931:
1928:
1925:
1922:
1919:
1916:
1913:
1910:
1900:
1884:
1880:
1877:
1874:
1871:
1868:
1865:
1862:
1859:
1856:
1853:
1850:
1847:
1844:
1841:
1838:
1835:
1832:
1829:
1826:
1823:
1820:
1817:
1814:
1811:
1808:
1805:
1802:
1799:
1796:
1793:
1790:
1787:
1784:
1781:
1778:
1775:
1772:
1769:
1766:
1763:
1760:
1757:
1754:
1751:
1748:
1745:
1742:
1739:
1736:
1733:
1730:
1727:
1724:
1721:
1718:
1715:
1712:
1709:
1706:
1703:
1700:
1697:
1687:
1671:
1667:
1664:
1661:
1658:
1655:
1652:
1649:
1646:
1643:
1640:
1637:
1634:
1620:
1619:
1608:
1605:
1602:
1599:
1596:
1593:
1590:
1587:
1584:
1581:
1578:
1575:
1572:
1569:
1566:
1563:
1560:
1557:
1547:
1542:
1537:
1534:
1531:
1527:
1522:
1518:
1515:
1512:
1509:
1506:
1503:
1500:
1497:
1494:
1491:
1488:
1465:
1445:
1442:
1439:
1408:
1380:
1356:
1328:
1304:
1281:
1278:
1275:
1272:
1257:Main article:
1254:
1251:
1250:
1249:
1238:
1235:
1232:
1229:
1226:
1223:
1220:
1217:
1212:
1208:
1204:
1201:
1198:
1195:
1192:
1189:
1186:
1181:
1177:
1173:
1170:
1167:
1164:
1161:
1158:
1155:
1150:
1146:
1123:
1122:
1106:
1102:
1099:
1096:
1093:
1090:
1087:
1084:
1081:
1078:
1075:
1072:
1069:
1066:
1063:
1060:
1057:
1054:
1051:
1048:
1045:
1042:
1039:
1016:
1012:
1008:
1005:
1002:
997:
993:
989:
984:
980:
956:Weyl curvature
904:
903:
887:
883:
880:
877:
874:
871:
868:
865:
862:
859:
856:
853:
850:
847:
844:
841:
838:
835:
832:
829:
826:
823:
820:
817:
814:
803:
802:
786:
782:
779:
776:
773:
770:
767:
764:
761:
758:
755:
752:
749:
746:
743:
740:
737:
734:
731:
728:
725:
722:
719:
716:
705:
704:
688:
684:
681:
678:
675:
672:
669:
666:
663:
660:
657:
654:
651:
648:
633:
630:
603:
600:
597:
594:
591:
588:
585:
582:
579:
561:
560:
549:
544:
540:
534:
530:
526:
523:
518:
514:
508:
504:
500:
497:
494:
491:
488:
485:
482:
479:
456:
453:
450:
447:
444:
441:
438:
416:
412:
408:
404:
400:
397:
394:
372:
368:
364:
360:
356:
353:
350:
330:
327:
324:
321:
318:
315:
304:
303:
292:
289:
284:
281:
278:
275:
272:
268:
264:
261:
256:
252:
246:
242:
238:
235:
230:
226:
220:
216:
212:
209:
206:
203:
200:
197:
194:
191:
168:
165:
162:
159:
156:
133:
110:Main article:
107:
104:
102:
99:
13:
10:
9:
6:
4:
3:
2:
4843:
4832:
4829:
4827:
4824:
4822:
4819:
4817:
4814:
4813:
4811:
4796:
4793:
4791:
4788:
4786:
4783:
4781:
4778:
4777:
4775:
4773:
4769:
4763:
4760:
4758:
4755:
4753:
4750:
4748:
4745:
4743:
4740:
4739:
4737:
4735:
4731:
4725:
4722:
4720:
4717:
4715:
4712:
4710:
4707:
4705:
4704:Darboux frame
4702:
4700:
4697:
4695:
4692:
4690:
4687:
4686:
4684:
4682:
4676:
4670:
4667:
4665:
4662:
4660:
4657:
4655:
4652:
4650:
4647:
4645:
4642:
4640:
4637:
4636:
4634:
4632:
4626:
4622:
4618:
4610:
4605:
4603:
4598:
4596:
4591:
4590:
4587:
4575:
4572:
4570:
4567:
4565:
4562:
4560:
4557:
4556:
4554:
4550:
4544:
4541:
4539:
4536:
4534:
4531:
4530:
4528:
4524:
4518:
4517:Schur's lemma
4515:
4513:
4510:
4508:
4505:
4503:
4500:
4498:
4495:
4493:
4492:Gauss's lemma
4490:
4488:
4485:
4484:
4482:
4478:
4472:
4469:
4467:
4464:
4462:
4459:
4457:
4454:
4453:
4451:
4447:
4441:
4438:
4436:
4433:
4431:
4428:
4426:
4423:
4419:
4415:
4412:
4410:
4407:
4405:
4402:
4400:
4397:
4396:
4395:
4394:Metric tensor
4392:
4390:
4389:Inner product
4387:
4385:
4382:
4380:
4377:
4373:
4370:
4368:
4365:
4363:
4360:
4358:
4355:
4354:
4353:
4350:
4349:
4347:
4343:
4338:
4334:
4327:
4322:
4320:
4315:
4313:
4308:
4307:
4304:
4300:
4294:
4288:
4284:
4280:
4276:
4272:
4268:
4267:
4262:
4258:
4254:
4252:0-471-15733-3
4248:
4244:
4243:
4237:
4236:
4232:
4227:
4223:
4219:
4216:
4212:
4208:
4205:
4201:
4197:
4194:
4190:
4189:
4188:
4182:
4180:
4178:
4158:
4146:
4118:
4114:
4107:
4095:
4087:
4083:
4078:
4067:
4051:
4048:
4043:
4040:
4037:
4034:
4031:
4028:
4025:
4019:
4007:
4003:
4000:
3996:
3990:
3987:
3983:
3975:
3974:
3973:
3957:
3954:
3950:
3941:
3934:
3926:
3924:
3922:
3906:
3903:
3900:
3892:
3887:
3873:
3870:
3861:
3838:
3818:
3815:
3812:
3803:
3792:
3787:
3785:
3779:
3771:
3769:
3767:
3763:
3759:
3754:
3732:
3726:
3722:
3713:
3709:
3705:
3702:
3696:
3691:
3687:
3683:
3677:
3662:
3661:
3660:
3658:
3637:
3633:
3621:
3595:
3587:
3585:
3583:
3542:
3534:
3530:
3526:
3518:
3514:
3497:
3493:
3489:
3481:
3477:
3473:
3468:
3464:
3455:
3451:
3447:
3442:
3438:
3431:
3423:
3420:
3417:
3413:
3409:
3406:
3399:
3398:
3397:
3394:
3375:
3371:
3360:
3356:
3318:
3315:
3312:
3302:
3294:
3292:
3286:
3284:
3267:
3261:
3258:
3255:
3249:
3246:
3243:
3237:
3231:
3225:
3222:
3219:
3216:
3210:
3207:
3204:
3198:
3188:
3187:
3186:
3169:
3161:
3149:(elements of
3148:
3132:
3125:
3117:
3115:
3114:
3110:
3092:
3089:
3083:
3076:
3075:
3074:
3072:
3053:
3050:
3045:
3041:
3034:
3028:
3020:
3016:
3007:
2989:
2985:
2981:
2978:
2955:
2952:
2949:
2946:
2936:
2935:
2934:
2932:
2913:
2910:
2904:
2901:
2898:
2889:
2886:
2880:
2877:
2874:
2868:
2861:
2860:
2859:
2843:
2838:
2831:
2827:
2822:
2817:
2810:
2789:
2786:
2783:
2759:
2756:
2753:
2750:
2747:
2744:
2741:
2731:
2730:
2729:
2713:
2708:
2697:
2666:
2662:
2658:
2653:
2649:
2641:
2637:
2625:
2617:
2613:
2604:
2599:
2592:
2584:
2583:
2582:
2566:
2561:
2554:
2550:
2547:
2525:
2521:
2511:
2509:
2490:
2484:
2474:
2470:
2451:
2445:
2442:
2434:
2416:
2411:
2400:
2383:
2379:
2376:
2375:antisymmetric
2372:
2368:
2364:
2360:
2356:
2350:
2342:
2340:
2322:
2319:
2316:
2310:
2304:
2301:
2298:
2290:
2285:
2278:
2275:
2272:
2269:
2266:
2263:
2260:
2257:
2254:
2248:
2245:
2239:
2236:
2233:
2230:
2227:
2224:
2221:
2218:
2215:
2209:
2205:
2198:
2192:
2183:
2166:
2163:
2158:
2152:
2149:
2146:
2140:
2137:
2134:
2128:
2117:
2095:
2085:
2082:
2079:
2073:
2070:
2064:
2061:
2058:
2052:
2049:
2043:
2040:
2037:
2034:
2031:
2025:
2022:
2016:
2013:
2010:
2007:
2004:
1998:
1995:
1989:
1986:
1983:
1980:
1977:
1971:
1968:
1962:
1959:
1956:
1953:
1950:
1944:
1941:
1935:
1932:
1929:
1926:
1923:
1920:
1917:
1911:
1901:
1882:
1872:
1869:
1866:
1860:
1857:
1851:
1848:
1845:
1839:
1836:
1830:
1827:
1824:
1821:
1818:
1812:
1809:
1803:
1800:
1797:
1794:
1791:
1785:
1782:
1776:
1773:
1770:
1767:
1764:
1758:
1755:
1749:
1746:
1743:
1740:
1737:
1731:
1728:
1722:
1719:
1716:
1713:
1710:
1707:
1704:
1698:
1688:
1669:
1662:
1659:
1656:
1650:
1647:
1644:
1638:
1632:
1625:
1624:
1623:
1603:
1600:
1597:
1591:
1588:
1585:
1579:
1573:
1567:
1564:
1561:
1555:
1545:
1535:
1532:
1529:
1520:
1513:
1510:
1507:
1501:
1498:
1492:
1486:
1479:
1478:
1477:
1463:
1443:
1440:
1437:
1428:
1426:
1422:
1406:
1398:
1394:
1378:
1370:
1354:
1346:
1342:
1326:
1318:
1302:
1295:
1276:
1270:
1260:
1252:
1236:
1233:
1227:
1224:
1221:
1215:
1210:
1202:
1196:
1193:
1190:
1184:
1179:
1171:
1165:
1162:
1159:
1153:
1148:
1136:
1135:
1134:
1132:
1128:
1100:
1097:
1094:
1088:
1085:
1082:
1076:
1070:
1064:
1061:
1058:
1052:
1049:
1046:
1040:
1030:
1029:
1028:
1014:
1010:
1003:
1000:
995:
991:
982:
978:
967:
965:
961:
957:
952:
948:
944:
940:
934:
930:
926:
921:
917:
913:
909:
885:
881:
878:
872:
869:
866:
860:
857:
854:
848:
845:
842:
836:
833:
830:
824:
821:
818:
812:
805:
804:
780:
777:
774:
768:
765:
762:
756:
750:
747:
741:
738:
735:
729:
726:
723:
717:
707:
706:
682:
679:
676:
670:
667:
664:
658:
655:
652:
646:
639:
638:
637:
631:
629:
627:
623:
621:
617:
601:
595:
592:
589:
583:
577:
568:
566:
547:
542:
532:
524:
521:
516:
506:
498:
495:
489:
486:
483:
477:
470:
469:
468:
454:
451:
445:
442:
439:
414:
410:
402:
395:
392:
370:
366:
358:
351:
348:
325:
322:
319:
313:
290:
287:
279:
276:
273:
262:
259:
254:
244:
236:
233:
228:
218:
210:
207:
201:
198:
195:
189:
182:
181:
180:
163:
160:
157:
147:
123:
119:
113:
105:
100:
98:
96:
92:
88:
84:
83:
78:
74:
70:
66:
65:infinitesimal
62:
58:
50:
46:
42:
38:
34:
30:
25:
19:
4741:
4552:Applications
4480:Main results
4351:
4298:
4270:
4264:
4261:Woods, F. S.
4240:
4186:
4175:denotes the
4134:
3936:
3888:
3788:
3783:
3781:
3758:Ricci tensor
3755:
3752:
3656:
3599:
3597:
3582:Ricci tensor
3580:denotes the
3557:
3395:
3304:
3290:
3282:
3121:
3111:denotes the
3108:
3107:
3070:
3005:
2970:
2930:
2928:
2775:
2686:
2512:
2381:
2377:
2370:
2352:
2118:
2115:
1621:
1429:
1424:
1396:
1392:
1368:
1344:
1340:
1293:
1262:
1130:
1126:
1124:
968:
955:
950:
946:
942:
938:
932:
928:
924:
919:
916:antisymmetry
915:
911:
905:
635:
625:
624:
620:endomorphism
619:
615:
569:
564:
562:
305:
115:
80:
67:geometry of
54:
4790:Cocurvature
4681:of surfaces
4619:defined in
3778:Weyl tensor
3145:on tangent
3073:takes form
2469:Lie algebra
1129:(often the
146:Lie bracket
57:mathematics
33:hyperboloid
4810:Categories
4512:Ricci flow
4461:Hyperbolic
4233:References
2361:, and for
1419:under the
964:Lie groups
4639:Curvature
4631:of curves
4617:curvature
4456:Hermitian
4409:Signature
4372:Sectional
4352:Curvature
4226:geodesics
4159:◯
4147:∧
4108:◯
4096:∧
4075:‖
4057:‖
4035:⊗
4026:−
3865:~
3807:~
3791:conformal
3688:∑
3540:⟩
3503:⟨
3494:∑
3487:⟩
3429:⟨
3414:∑
3265:⟩
3235:⟨
3229:⟩
3223:∧
3208:∧
3196:⟨
3158:Λ
3147:bivectors
3087:Ω
3057:⟩
3038:⟨
3017:θ
2986:θ
2979:θ
2950:θ
2947:∧
2944:Ω
2902:∧
2893:Ω
2832:ω
2828:∧
2811:ω
2790:ω
2787:∧
2784:ω
2760:ω
2757:∧
2754:ω
2748:ω
2739:Ω
2702:Ω
2695:Ω
2672:⟩
2633:∇
2629:⟨
2593:ω
2555:ω
2548:ω
2485:
2446:
2405:Ω
2393:Ω
2246:−
2196:∂
2190:∂
2180:∂
2156:⟩
2126:⟨
2023:−
1996:−
1969:−
1942:−
1883:−
1810:−
1783:−
1756:−
1729:−
1666:⟩
1636:⟨
1607:⟩
1577:⟨
1533:∧
1493:σ
1464:σ
1407:σ
1379:σ
1355:σ
1327:σ
1303:σ
1277:σ
1207:∇
1176:∇
1145:∇
1105:⟩
1074:⟨
1068:⟩
1038:⟨
1001:−
960:dilations
785:⟩
754:⟨
751:−
745:⟩
715:⟨
668:−
581:↦
539:∇
529:∇
525:−
513:∇
503:∇
407:∂
399:∂
363:∂
355:∂
267:∇
263:−
251:∇
241:∇
237:−
225:∇
215:∇
164:⋅
158:⋅
132:∇
91:curvature
73:dimension
4795:Holonomy
4471:Kenmotsu
4384:Geodesic
4337:Glossary
3659:we have
3396:we have
3124:operator
937:for all
45:manifold
37:cylinder
4538:Hilbert
4533:Finsler
4287:1967636
3851:, then
2471:of the
2433:2-forms
2384:matrix
1369:section
1347:; here
1341:section
1319:of the
1294:section
77:Riemann
4466:Kähler
4362:Scalar
4357:tensor
4285:
4249:
4162:
4144:
4135:where
4111:
4093:
3558:where
3069:. The
3004:is an
2971:where
2836:
2815:
2706:
2597:
2559:
2467:, the
2409:
2380:×
951:tensor
63:, the
41:sphere
4367:Ricci
4295:Notes
4283:JSTOR
3355:trace
1476:then
908:Ricci
306:Here
93:of a
71:with
4247:ISBN
4220:the
4213:and
4061:grad
4013:Hess
3782:The
2513:Let
2353:The
1125:The
918:and
626:N.B.
385:and
144:and
120:(or
4275:doi
4202:or
3671:Ric
3608:Ric
3567:Ric
3507:Ric
3331:or
2431:of
1430:If
1423:at
1343:at
618:or
55:In
4812::
4281:.
4269:.
3923:.
3886:.
3768:.
3340:Sc
2443:so
1427:.
1015:12
941:,
931:,
622:.
567:.
124:)
4608:e
4601:t
4594:v
4416:/
4339:)
4335:(
4325:e
4318:t
4311:v
4289:.
4277::
4271:3
4255:.
4228:.
4217:.
4206:,
4195:,
4119:)
4115:g
4088:)
4084:g
4079:2
4071:)
4068:f
4065:(
4052:2
4049:1
4044:+
4041:f
4038:d
4032:f
4029:d
4023:)
4020:f
4017:(
4008:(
4004:+
4001:R
3997:(
3991:f
3988:2
3984:e
3958:f
3955:2
3951:e
3907:0
3904:=
3901:W
3874:W
3871:=
3862:W
3839:f
3819:g
3816:f
3813:=
3804:g
3733:.
3727:i
3723:e
3719:)
3714:i
3710:e
3706:,
3703:u
3700:(
3697:R
3692:i
3684:=
3681:)
3678:u
3675:(
3657:p
3643:}
3638:i
3634:e
3630:{
3543:,
3535:i
3531:e
3527:,
3524:)
3519:i
3515:e
3511:(
3498:i
3490:=
3482:i
3478:e
3474:,
3469:j
3465:e
3461:)
3456:j
3452:e
3448:,
3443:i
3439:e
3435:(
3432:R
3424:j
3421:,
3418:i
3410:=
3407:S
3381:}
3376:i
3372:e
3368:{
3319:R
3316:,
3313:S
3268:.
3262:w
3259:,
3256:z
3253:)
3250:v
3247:,
3244:u
3241:(
3238:R
3232:=
3226:z
3220:w
3217:,
3214:)
3211:v
3205:u
3202:(
3199:Q
3173:)
3170:T
3167:(
3162:2
3133:Q
3109:D
3093:0
3090:=
3084:D
3054:v
3051:,
3046:i
3042:e
3035:=
3032:)
3029:v
3026:(
3021:i
3006:n
2990:i
2982:=
2956:0
2953:=
2914:.
2911:w
2908:)
2905:v
2899:u
2896:(
2890:=
2887:w
2884:)
2881:v
2878:,
2875:u
2872:(
2869:R
2844:j
2839:k
2823:i
2818:j
2772:.
2751:+
2745:d
2742:=
2714:i
2709:j
2698:=
2667:k
2663:e
2659:,
2654:j
2650:e
2642:i
2638:e
2626:=
2623:)
2618:i
2614:e
2610:(
2605:k
2600:j
2567:i
2562:j
2551:=
2526:i
2522:e
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