2361:
1727:
1355:
2356:{\displaystyle {\begin{aligned}X\left(g(Y,Z)\right)&+Y\left(g(X,Z)\right)-Z\left(g(X,Y)\right)\\&={\Big (}g(\nabla _{X}Y,Z)+g(Y,\nabla _{X}Z){\Big )}+{\Big (}g(\nabla _{Y}X,Z)+g(X,\nabla _{Y}Z){\Big )}-{\Big (}g(\nabla _{Z}X,Y)+g(X,\nabla _{Z}Y){\Big )}\\&=g(\nabla _{X}Y+\nabla _{Y}X,Z)+g(\nabla _{X}Z-\nabla _{Z}X,Y)+g(\nabla _{Y}Z-\nabla _{Z}Y,X)\\&=g(2\nabla _{X}Y+,Z)+g(,Y)+g(,X).\end{aligned}}}
978:
504:. Regardless of the presentation, the idea is to use the metric-compatibility and torsion-freeness conditions to obtain a direct formula for any connection that is both metric-compatible and torsion-free. This establishes the uniqueness claim in the fundamental theorem. To establish the existence claim, it must be directly checked that the formula obtained does define a connection as desired.
1350:{\displaystyle {\begin{aligned}\partial _{i}g_{jl}+\partial _{j}g_{il}-\partial _{l}g_{ij}&=\left(\Gamma _{ij}^{p}g_{pl}+\Gamma _{il}^{p}g_{jp}\right)+\left(\Gamma _{ji}^{p}g_{pl}+\Gamma _{jl}^{p}g_{ip}\right)-\left(\Gamma _{li}^{p}g_{pj}+\Gamma _{lj}^{p}g_{ip}\right)\\&=2\Gamma _{ij}^{p}g_{pl}\end{aligned}}}
1523:
This proves the uniqueness of a torsion-free and metric-compatible condition; that is, any such connection must be given by the above formula. To prove the existence, it must be checked that the above formula defines a connection that is torsion-free and metric-compatible. This can be done directly.
974:
In this way, it is seen that the conditions of torsion-freeness and metric-compatibility can be viewed as a linear system of equations for the connection, in which the coefficients and 'right-hand side' of the system are given by the metric and its first derivative. The fundamental theorem of
1521:
2634:
2717:
699:
972:
228:
1710:
1375:
1357:
in which the metric-compatibility condition is used three times for the first equality and the torsion-free condition is used three times for the second equality. The resulting formula is sometimes known as the
828:
2387:
1732:
983:
582:
1597:
344:
587:
465:
as obtained in the proofs below. This explicit definition expresses the Levi-Civita connection in terms of the metric and its first derivatives. As such, if the metric is
866:
437:, with any given vector-valued 2-form as its torsion. The difference between an arbitrary connection (with torsion) and the corresponding Levi-Civita connection is the
116:
1602:
742:
2974:
4574:
3765:
3469:
58:
of the given metric. Because it is canonically defined by such properties, often this connection is automatically used when given a metric.
4569:
2682:
of the metric. In the local formulation above, this key property of the metric was implicitly used, in the same way, via the existence of
383:âparallel vector fields along the curve is constant. It may also be equivalently phrased as saying that the metric tensor is preserved by
3081:
3856:
3482:
975:
Riemannian geometry can be viewed as saying that this linear system has a unique solution. This is seen via the following computation:
541:
3880:
4632:
4075:
433:
An extension of the fundamental theorem states that given a pseudo-Riemannian manifold there is a unique connection preserving the
1535:
282:
3662:
3637:
1516:{\displaystyle \Gamma _{ij}^{k}={\tfrac {1}{2}}g^{kl}\left(\partial _{i}g_{jl}+\partial _{j}g_{il}-\partial _{l}g_{ij}\right).}
3945:
3524:
3378:
3209:
496:
The proof of the theorem can be presented in various ways. Here the proof is first given in the language of coordinates and
4171:
3243:
457:. However, the existence result is extremely direct, as the connection in question may be explicitly defined by either the
3131:
4224:
3752:
3039:. Mathematics: Theory & Applications. Translated from the second Portuguese edition by Francis Flaherty. Boston, MA:
4508:
4617:
3497:
3462:
3422:
3327:
3159:
3105:
3047:
2629:{\displaystyle 2g(\nabla _{X}Y,Z)=X\left(g(Y,Z)\right)+Y\left(g(X,Z)\right)-Z\left(g(X,Y)\right)-g(,Z)-g(,Y)-g(,X).}
4273:
529:
4256:
3865:
3139:
419:
42:
2902:
1532:
The above proof can also be expressed in terms of vector fields. Torsion-freeness refers to the condition that
4627:
4468:
3875:
3358:
3143:
513:
400:
4622:
4453:
4176:
3950:
3559:
3455:
444:
The fundamental theorem asserts both existence and uniqueness of a certain connection, which is called the
347:
4498:
3414:
485:
2708:, and it is routine to check that this defines a connection that is torsion-free and metric-compatible.
4503:
4473:
4181:
4137:
4118:
3885:
3829:
3580:
3280:
3249:
3087:
87:
34:
4040:
3905:
3709:
3408:
517:
1724:
are arbitrary vector fields. The computation previously done in local coordinates can be written as
4425:
4290:
3982:
3824:
3642:
3502:
2829:
4122:
4092:
4016:
4006:
3962:
3792:
3745:
3688:
4463:
4082:
3977:
3890:
3797:
3719:
3652:
3647:
3585:
3544:
446:
47:
3714:
2688:. Furthermore, by the same reasoning, the Koszul formula can be used to define a vector field
4112:
4107:
2375:
are coordinate vector fields. The equations displayed above can be rearranged to produce the
4443:
4381:
4229:
3933:
3923:
3895:
3870:
3780:
3549:
3432:
3388:
3337:
3288:
3256:
3219:
3169:
3115:
3057:
501:
3440:
3396:
3345:
3296:
3264:
3227:
3177:
3123:
3065:
8:
4581:
4263:
4141:
4126:
4055:
3814:
3606:
3575:
3563:
3534:
3517:
3478:
3235:
480:
The Levi-Civita connection can also be characterized in other ways, for instance via the
83:
30:
22:
4554:
387:, which is to say that the metric is parallel when considering the natural extension of
4523:
4478:
4375:
4246:
4050:
3738:
3704:
3601:
3570:
3304:
3077:
2636:
This proves the uniqueness of a torsion-free and metric-compatible condition, since if
497:
481:
384:
4060:
3611:
3319:
4458:
4438:
4433:
4340:
4251:
4065:
4045:
3900:
3839:
3616:
3418:
3374:
3323:
3205:
3155:
3101:
3043:
3032:
694:{\displaystyle (\nabla _{X}Y)^{i}=X^{j}\partial _{j}Y^{i}+X^{j}Y^{k}\Gamma _{jk}^{i}}
438:
38:
4596:
4390:
4345:
4268:
4239:
4097:
4030:
4025:
4020:
4010:
3802:
3785:
3683:
3678:
3554:
3507:
3436:
3392:
3366:
3341:
3315:
3292:
3260:
3223:
3197:
3173:
3147:
3119:
3091:
3061:
3142:. Vol. 34 (Corrected reprint of the 1978 original ed.). Providence, RI:
967:{\displaystyle \partial _{k}g_{ij}=\Gamma _{ki}^{l}g_{lj}+\Gamma _{kj}^{l}g_{il}.}
863:. Similarly, the condition of metric-compatibility is equivalent to the condition
4539:
4448:
4278:
4234:
4000:
3512:
3447:
3428:
3384:
3333:
3284:
3252:
3215:
3165:
3111:
3073:
3053:
2785:
4405:
4330:
4300:
4198:
4191:
4131:
4102:
3972:
3967:
3928:
3362:
3312:
3239:
3193:
2813:
2377:
3370:
3201:
3185:
3086:. Cambridge Monographs on Mathematical Physics. Vol. 1. LondonâNew York:
4611:
4591:
4415:
4410:
4395:
4385:
4335:
4312:
4186:
4146:
4087:
4035:
3834:
3539:
3096:
2769:
434:
376:
3040:
2801:
2363:
This reduces immediately to the first
Christoffel identity in the case that
223:{\displaystyle X{\big (}g(Y,Z){\big )}=g(\nabla _{X}Y,Z)+g(Y,\nabla _{X}Z),}
4518:
4513:
4355:
4322:
4295:
4203:
3844:
404:
99:
399:. It is further equivalent to require that the connection is induced by a
4361:
4350:
4307:
4208:
3809:
3404:
3272:
1705:{\displaystyle X\left(g(Y,Z)\right)=g(\nabla _{X}Y,Z)+g(Y,\nabla _{X}Z),}
3006:
4586:
4544:
4370:
4283:
3915:
3819:
3730:
3657:
469:-times continuously differentiable, then the Levi-Civita connection is
3151:
371:. It may be equivalently expressed by saying that, given any curve in
4400:
4365:
4070:
3957:
4564:
4559:
4549:
3940:
3761:
3529:
2958:
709:. Torsion-freeness of the connection refers to the condition that
3279:. Annals of Mathematics Studies. Vol. 51. Princeton, N.J.:
823:{\displaystyle 0=X^{j}Y^{k}(\Gamma _{jk}^{i}-\Gamma _{kj}^{i}),}
4156:
2757:
739:. Written in terms of local coordinates, this is equivalent to
516:
will be used, which is to say that an index repeated as both
3192:. Universitext (Seventh edition of 1995 original ed.).
3361:. Vol. 171 (Third edition of 1998 original ed.).
2861:
2849:
3311:. Pure and Applied Mathematics. Vol. 103. New York:
3309:
Semi-Riemannian geometry. With applications to relativity
2890:
3136:
Differential geometry, Lie groups, and symmetric spaces
1362:. It can be contracted with the inverse of the metric,
2946:
2922:
1599:
and metric-compatibility refers to the condition that
1401:
2390:
1730:
1605:
1538:
1378:
981:
869:
745:
590:
544:
285:
119:
2934:
3477:
2887:for presentations differing from those given here.
2628:
2355:
1704:
1591:
1515:
1349:
966:
822:
693:
576:
338:
222:
2068:
2002:
1992:
1926:
1916:
1850:
4609:
577:{\displaystyle \left\{\Gamma _{ij}^{l}\right\},}
3234:
2996:
2912:
2884:
2807:
2735:
500:, and then in the coordinate-free language of
18:Unique existence of the Levi-Civita connection
3746:
3463:
150:
125:
3072:
2984:
2867:
2835:
528:. Recall that, relative to a local chart, a
3303:
3245:Foundations of differential geometry. Vol I
3016:
3000:
2743:
1592:{\displaystyle \nabla _{X}Y-\nabla _{Y}X=,}
430:. There are alternative characterizations.
339:{\displaystyle \nabla _{X}Y-\nabla _{Y}X=,}
68:Fundamental theorem of Riemannian Geometry.
61:
3753:
3739:
3633:Fundamental theorem of Riemannian geometry
3470:
3456:
3190:Riemannian geometry and geometric analysis
94:which satisfies the following conditions:
27:fundamental theorem of Riemannian geometry
3095:
410:The second condition is sometimes called
3760:
3352:
3130:
3031:
2988:
2980:
2896:
2880:
2839:
2819:
2791:
2775:
2747:
2727:
2723:
1527:
3083:The large scale structure of space-time
491:
249:denotes the derivative of the function
4610:
3271:
2964:
2952:
2928:
2916:
2823:
2779:
2739:
418:. It expresses the condition that the
3734:
3451:
520:is being summed over all values. Let
90:). Then there is a unique connection
3403:
3184:
3012:
2992:
2968:
2940:
2908:
2855:
2843:
2795:
2763:
2751:
2731:
507:
477:-times continuously differentiable.
426:is zero, and as such is also called
13:
2401:
2241:
2200:
2184:
2153:
2137:
2106:
2090:
2051:
2014:
1975:
1938:
1899:
1862:
1684:
1647:
1556:
1540:
1483:
1457:
1431:
1380:
1313:
1264:
1230:
1186:
1152:
1108:
1074:
1039:
1013:
987:
931:
897:
871:
797:
776:
674:
631:
595:
550:
303:
287:
202:
165:
45:and metric-compatible, called the
14:
4644:
2879:See for instance pages 54-55 of
4633:Theorems in Riemannian geometry
3140:Graduate Studies in Mathematics
2873:
2678:. This is a consequence of the
838:is equivalent to the condition
391:to act on (0,2)-tensor fields:
3793:Differentiable/Smooth manifold
2620:
2611:
2599:
2596:
2587:
2578:
2566:
2563:
2554:
2545:
2533:
2530:
2516:
2504:
2482:
2470:
2448:
2436:
2419:
2397:
2343:
2334:
2322:
2319:
2310:
2301:
2289:
2286:
2277:
2268:
2256:
2234:
2218:
2180:
2171:
2133:
2124:
2086:
2063:
2041:
2032:
2010:
1987:
1965:
1956:
1934:
1911:
1889:
1880:
1858:
1830:
1818:
1796:
1784:
1758:
1746:
1696:
1674:
1665:
1643:
1629:
1617:
1583:
1571:
814:
772:
608:
591:
363:The first condition is called
330:
318:
214:
192:
183:
161:
145:
133:
1:
3359:Graduate Texts in Mathematics
3320:10.1016/s0079-8169(08)x6002-7
3144:American Mathematical Society
3025:
2885:Kobayashi & Nomizu (1963)
514:Einstein summation convention
21:In the mathematical field of
3560:Raising and lowering indices
7:
4499:Classification of manifolds
3415:University of Chicago Press
3250:John Wiley & Sons, Inc.
3033:do Carmo, Manfredo PerdigĂŁo
2997:Kobayashi & Nomizu 1963
2913:Kobayashi & Nomizu 1963
2808:Kobayashi & Nomizu 1963
2736:Kobayashi & Nomizu 1963
1370:second Christoffel identity
459:second Christoffel identity
401:principal bundle connection
10:
4649:
3581:Pseudo-Riemannian manifold
3281:Princeton University Press
3088:Cambridge University Press
1360:first Christoffel identity
830:which by arbitrariness of
88:pseudo-Riemannian manifold
35:pseudo-Riemannian manifold
4575:over commutative algebras
4532:
4491:
4424:
4321:
4217:
4164:
4155:
3991:
3914:
3853:
3773:
3710:Geometrization conjecture
3697:
3671:
3625:
3594:
3490:
3371:10.1007/978-3-319-26654-1
3202:10.1007/978-3-319-61860-9
518:subscript and superscript
4618:Connection (mathematics)
4291:Riemann curvature tensor
3353:Petersen, Peter (2016).
3097:10.1017/CBO9780511524646
2985:Hawking & Ellis 1973
2868:Hawking & Ellis 1973
2836:Hawking & Ellis 1973
2711:
524:denote the dimension of
405:orthonormal frame bundle
62:Statement of the theorem
3041:BirkhÀuser Boston, Inc.
2915:, Proposition III.7.6;
486:EinsteinâHilbert action
4083:Manifold with boundary
3798:Differential structure
3720:Uniformization theorem
3653:Nash embedding theorem
3586:Riemannian volume form
3545:Levi-Civita connection
2630:
2357:
1706:
1593:
1517:
1351:
968:
824:
701:for any vector fields
695:
578:
447:Levi-Civita connection
361:
340:
271:for any vector fields
224:
48:Levi-Civita connection
2810:, Proposition IV.2.1.
2794:, Proposition 2.2.5;
2631:
2358:
1707:
1594:
1528:Invariant formulation
1518:
1352:
969:
825:
696:
579:
502:covariant derivatives
455:Riemannian connection
341:
225:
65:
56:Riemannian connection
4628:Riemannian manifolds
4230:Covariant derivative
3781:Topological manifold
3643:GaussâBonnet theorem
3550:Covariant derivative
3313:Academic Press, Inc.
3236:Kobayashi, Shoshichi
2883:or pages 158-159 of
2388:
1728:
1603:
1536:
1376:
979:
867:
743:
588:
542:
492:Proof of the theorem
365:metric-compatibility
283:
117:
37:) there is a unique
4623:Riemannian geometry
4264:Exterior derivative
3866:AtiyahâSinger index
3815:Riemannian manifold
3715:Poincaré conjecture
3576:Riemannian manifold
3564:Musical isomorphism
3479:Riemannian geometry
3355:Riemannian geometry
3248:. New YorkâLondon:
3037:Riemannian geometry
2846:, Definition 4.1.7.
2766:, Definition 4.2.1.
1396:
1329:
1280:
1246:
1202:
1168:
1124:
1090:
947:
913:
813:
792:
690:
566:
498:Christoffel symbols
346:where denotes the
264:along vector field
84:Riemannian manifold
31:Riemannian manifold
29:states that on any
23:Riemannian geometry
4570:Secondary calculus
4524:Singularity theory
4479:Parallel transport
4247:De Rham cohomology
3886:Generalized Stokes
3705:General relativity
3648:HopfâRinow theorem
3595:Types of manifolds
3571:Parallel transport
3410:General relativity
3132:Helgason, Sigurdur
2738:, Theorem IV.2.2;
2626:
2353:
2351:
1702:
1589:
1513:
1410:
1379:
1347:
1345:
1312:
1263:
1229:
1185:
1151:
1107:
1073:
964:
930:
896:
820:
796:
775:
691:
673:
574:
549:
482:Palatini variation
385:parallel transport
336:
220:
4605:
4604:
4487:
4486:
4252:Differential form
3906:Whitney embedding
3840:Differential form
3728:
3727:
3380:978-3-319-26652-7
3211:978-3-319-61859-3
2826:, Definition 8.5.
2750:, Theorem 2.2.2;
2734:, Theorem 4.3.1;
2730:, Theorem I.9.1;
2726:, Theorem 2.3.6;
1409:
538:smooth functions
508:Local coordinates
439:contorsion tensor
39:affine connection
4640:
4597:Stratified space
4555:Fréchet manifold
4269:Interior product
4162:
4161:
3859:
3755:
3748:
3741:
3732:
3731:
3472:
3465:
3458:
3449:
3448:
3444:
3400:
3349:
3305:O'Neill, Barrett
3300:
3268:
3231:
3181:
3127:
3099:
3069:
3020:
3010:
3004:
2978:
2972:
2962:
2956:
2950:
2944:
2938:
2932:
2926:
2920:
2906:
2900:
2894:
2888:
2877:
2871:
2865:
2859:
2853:
2847:
2833:
2827:
2817:
2811:
2805:
2799:
2789:
2783:
2773:
2767:
2761:
2755:
2754:, Theorem 3.1.1.
2746:, Theorem 3.11;
2721:
2707:
2703:
2699:
2687:
2677:
2673:
2669:
2665:
2650:
2635:
2633:
2632:
2627:
2523:
2519:
2489:
2485:
2455:
2451:
2409:
2408:
2374:
2370:
2366:
2362:
2360:
2359:
2354:
2352:
2249:
2248:
2224:
2208:
2207:
2192:
2191:
2161:
2160:
2145:
2144:
2114:
2113:
2098:
2097:
2076:
2072:
2071:
2059:
2058:
2022:
2021:
2006:
2005:
1996:
1995:
1983:
1982:
1946:
1945:
1930:
1929:
1920:
1919:
1907:
1906:
1870:
1869:
1854:
1853:
1841:
1837:
1833:
1803:
1799:
1765:
1761:
1723:
1719:
1715:
1711:
1709:
1708:
1703:
1692:
1691:
1655:
1654:
1636:
1632:
1598:
1596:
1595:
1590:
1564:
1563:
1548:
1547:
1522:
1520:
1519:
1514:
1509:
1505:
1504:
1503:
1491:
1490:
1478:
1477:
1465:
1464:
1452:
1451:
1439:
1438:
1424:
1423:
1411:
1402:
1395:
1390:
1367:
1356:
1354:
1353:
1348:
1346:
1342:
1341:
1328:
1323:
1302:
1298:
1294:
1293:
1292:
1279:
1274:
1259:
1258:
1245:
1240:
1220:
1216:
1215:
1214:
1201:
1196:
1181:
1180:
1167:
1162:
1142:
1138:
1137:
1136:
1123:
1118:
1103:
1102:
1089:
1084:
1060:
1059:
1047:
1046:
1034:
1033:
1021:
1020:
1008:
1007:
995:
994:
973:
971:
970:
965:
960:
959:
946:
941:
926:
925:
912:
907:
892:
891:
879:
878:
862:
861:
860:
850:
849:
837:
833:
829:
827:
826:
821:
812:
807:
791:
786:
771:
770:
761:
760:
738:
734:
730:
708:
704:
700:
698:
697:
692:
689:
684:
672:
671:
662:
661:
649:
648:
639:
638:
629:
628:
616:
615:
603:
602:
583:
581:
580:
575:
570:
565:
560:
537:
527:
523:
476:
468:
454:
428:torsion-freeness
425:
417:
398:
390:
382:
374:
370:
357:
353:
345:
343:
342:
337:
311:
310:
295:
294:
278:
274:
267:
263:
248:
229:
227:
226:
221:
210:
209:
173:
172:
154:
153:
129:
128:
112:
108:
104:
93:
81:
55:
4648:
4647:
4643:
4642:
4641:
4639:
4638:
4637:
4608:
4607:
4606:
4601:
4540:Banach manifold
4533:Generalizations
4528:
4483:
4420:
4317:
4279:Ricci curvature
4235:Cotangent space
4213:
4151:
3993:
3987:
3946:Exponential map
3910:
3855:
3849:
3769:
3759:
3729:
3724:
3693:
3672:Generalizations
3667:
3621:
3590:
3525:Exponential map
3486:
3476:
3425:
3413:. Chicago, IL:
3405:Wald, Robert M.
3381:
3330:
3240:Nomizu, Katsumi
3212:
3162:
3152:10.1090/gsm/034
3108:
3078:Ellis, G. F. R.
3050:
3028:
3023:
3011:
3007:
2979:
2975:
2963:
2959:
2951:
2947:
2939:
2935:
2927:
2923:
2911:, Lemma 4.1.1;
2907:
2903:
2895:
2891:
2881:Petersen (2016)
2878:
2874:
2866:
2862:
2854:
2850:
2834:
2830:
2818:
2814:
2806:
2802:
2790:
2786:
2774:
2770:
2762:
2758:
2722:
2718:
2714:
2705:
2701:
2695:
2689:
2683:
2675:
2671:
2667:
2652:
2637:
2500:
2496:
2466:
2462:
2432:
2428:
2404:
2400:
2389:
2386:
2385:
2372:
2368:
2364:
2350:
2349:
2244:
2240:
2222:
2221:
2203:
2199:
2187:
2183:
2156:
2152:
2140:
2136:
2109:
2105:
2093:
2089:
2074:
2073:
2067:
2066:
2054:
2050:
2017:
2013:
2001:
2000:
1991:
1990:
1978:
1974:
1941:
1937:
1925:
1924:
1915:
1914:
1902:
1898:
1865:
1861:
1849:
1848:
1839:
1838:
1814:
1810:
1780:
1776:
1766:
1742:
1738:
1731:
1729:
1726:
1725:
1721:
1717:
1713:
1687:
1683:
1650:
1646:
1613:
1609:
1604:
1601:
1600:
1559:
1555:
1543:
1539:
1537:
1534:
1533:
1530:
1496:
1492:
1486:
1482:
1470:
1466:
1460:
1456:
1444:
1440:
1434:
1430:
1429:
1425:
1416:
1412:
1400:
1391:
1383:
1377:
1374:
1373:
1363:
1344:
1343:
1334:
1330:
1324:
1316:
1300:
1299:
1285:
1281:
1275:
1267:
1251:
1247:
1241:
1233:
1228:
1224:
1207:
1203:
1197:
1189:
1173:
1169:
1163:
1155:
1150:
1146:
1129:
1125:
1119:
1111:
1095:
1091:
1085:
1077:
1072:
1068:
1061:
1052:
1048:
1042:
1038:
1026:
1022:
1016:
1012:
1000:
996:
990:
986:
982:
980:
977:
976:
952:
948:
942:
934:
918:
914:
908:
900:
884:
880:
874:
870:
868:
865:
864:
859:
854:
853:
852:
848:
843:
842:
841:
839:
835:
831:
808:
800:
787:
779:
766:
762:
756:
752:
744:
741:
740:
736:
732:
725:
716:
710:
706:
702:
685:
677:
667:
663:
657:
653:
644:
640:
634:
630:
624:
620:
611:
607:
598:
594:
589:
586:
585:
561:
553:
545:
543:
540:
539:
533:
525:
521:
510:
494:
470:
466:
452:
423:
415:
392:
388:
380:
372:
368:
355:
351:
306:
302:
290:
286:
284:
281:
280:
276:
272:
265:
250:
231:
205:
201:
168:
164:
149:
148:
124:
123:
118:
115:
114:
110:
106:
102:
91:
71:
64:
53:
19:
12:
11:
5:
4646:
4636:
4635:
4630:
4625:
4620:
4603:
4602:
4600:
4599:
4594:
4589:
4584:
4579:
4578:
4577:
4567:
4562:
4557:
4552:
4547:
4542:
4536:
4534:
4530:
4529:
4527:
4526:
4521:
4516:
4511:
4506:
4501:
4495:
4493:
4489:
4488:
4485:
4484:
4482:
4481:
4476:
4471:
4466:
4461:
4456:
4451:
4446:
4441:
4436:
4430:
4428:
4422:
4421:
4419:
4418:
4413:
4408:
4403:
4398:
4393:
4388:
4378:
4373:
4368:
4358:
4353:
4348:
4343:
4338:
4333:
4327:
4325:
4319:
4318:
4316:
4315:
4310:
4305:
4304:
4303:
4293:
4288:
4287:
4286:
4276:
4271:
4266:
4261:
4260:
4259:
4249:
4244:
4243:
4242:
4232:
4227:
4221:
4219:
4215:
4214:
4212:
4211:
4206:
4201:
4196:
4195:
4194:
4184:
4179:
4174:
4168:
4166:
4159:
4153:
4152:
4150:
4149:
4144:
4134:
4129:
4115:
4110:
4105:
4100:
4095:
4093:Parallelizable
4090:
4085:
4080:
4079:
4078:
4068:
4063:
4058:
4053:
4048:
4043:
4038:
4033:
4028:
4023:
4013:
4003:
3997:
3995:
3989:
3988:
3986:
3985:
3980:
3975:
3973:Lie derivative
3970:
3968:Integral curve
3965:
3960:
3955:
3954:
3953:
3943:
3938:
3937:
3936:
3929:Diffeomorphism
3926:
3920:
3918:
3912:
3911:
3909:
3908:
3903:
3898:
3893:
3888:
3883:
3878:
3873:
3868:
3862:
3860:
3851:
3850:
3848:
3847:
3842:
3837:
3832:
3827:
3822:
3817:
3812:
3807:
3806:
3805:
3800:
3790:
3789:
3788:
3777:
3775:
3774:Basic concepts
3771:
3770:
3758:
3757:
3750:
3743:
3735:
3726:
3725:
3723:
3722:
3717:
3712:
3707:
3701:
3699:
3695:
3694:
3692:
3691:
3689:Sub-Riemannian
3686:
3681:
3675:
3673:
3669:
3668:
3666:
3665:
3660:
3655:
3650:
3645:
3640:
3635:
3629:
3627:
3623:
3622:
3620:
3619:
3614:
3609:
3604:
3598:
3596:
3592:
3591:
3589:
3588:
3583:
3578:
3573:
3568:
3567:
3566:
3557:
3552:
3547:
3537:
3532:
3527:
3522:
3521:
3520:
3515:
3510:
3505:
3494:
3492:
3491:Basic concepts
3488:
3487:
3475:
3474:
3467:
3460:
3452:
3446:
3445:
3423:
3401:
3379:
3363:Springer, Cham
3350:
3328:
3301:
3269:
3232:
3210:
3194:Springer, Cham
3182:
3160:
3128:
3106:
3074:Hawking, S. W.
3070:
3048:
3027:
3024:
3022:
3021:
3005:
2973:
2957:
2945:
2933:
2921:
2901:
2889:
2872:
2860:
2858:, section 3.1.
2848:
2828:
2812:
2800:
2784:
2768:
2756:
2715:
2713:
2710:
2691:
2680:non-degeneracy
2666:for arbitrary
2625:
2622:
2619:
2616:
2613:
2610:
2607:
2604:
2601:
2598:
2595:
2592:
2589:
2586:
2583:
2580:
2577:
2574:
2571:
2568:
2565:
2562:
2559:
2556:
2553:
2550:
2547:
2544:
2541:
2538:
2535:
2532:
2529:
2526:
2522:
2518:
2515:
2512:
2509:
2506:
2503:
2499:
2495:
2492:
2488:
2484:
2481:
2478:
2475:
2472:
2469:
2465:
2461:
2458:
2454:
2450:
2447:
2444:
2441:
2438:
2435:
2431:
2427:
2424:
2421:
2418:
2415:
2412:
2407:
2403:
2399:
2396:
2393:
2378:Koszul formula
2348:
2345:
2342:
2339:
2336:
2333:
2330:
2327:
2324:
2321:
2318:
2315:
2312:
2309:
2306:
2303:
2300:
2297:
2294:
2291:
2288:
2285:
2282:
2279:
2276:
2273:
2270:
2267:
2264:
2261:
2258:
2255:
2252:
2247:
2243:
2239:
2236:
2233:
2230:
2227:
2225:
2223:
2220:
2217:
2214:
2211:
2206:
2202:
2198:
2195:
2190:
2186:
2182:
2179:
2176:
2173:
2170:
2167:
2164:
2159:
2155:
2151:
2148:
2143:
2139:
2135:
2132:
2129:
2126:
2123:
2120:
2117:
2112:
2108:
2104:
2101:
2096:
2092:
2088:
2085:
2082:
2079:
2077:
2075:
2070:
2065:
2062:
2057:
2053:
2049:
2046:
2043:
2040:
2037:
2034:
2031:
2028:
2025:
2020:
2016:
2012:
2009:
2004:
1999:
1994:
1989:
1986:
1981:
1977:
1973:
1970:
1967:
1964:
1961:
1958:
1955:
1952:
1949:
1944:
1940:
1936:
1933:
1928:
1923:
1918:
1913:
1910:
1905:
1901:
1897:
1894:
1891:
1888:
1885:
1882:
1879:
1876:
1873:
1868:
1864:
1860:
1857:
1852:
1847:
1844:
1842:
1840:
1836:
1832:
1829:
1826:
1823:
1820:
1817:
1813:
1809:
1806:
1802:
1798:
1795:
1792:
1789:
1786:
1783:
1779:
1775:
1772:
1769:
1767:
1764:
1760:
1757:
1754:
1751:
1748:
1745:
1741:
1737:
1734:
1733:
1701:
1698:
1695:
1690:
1686:
1682:
1679:
1676:
1673:
1670:
1667:
1664:
1661:
1658:
1653:
1649:
1645:
1642:
1639:
1635:
1631:
1628:
1625:
1622:
1619:
1616:
1612:
1608:
1588:
1585:
1582:
1579:
1576:
1573:
1570:
1567:
1562:
1558:
1554:
1551:
1546:
1542:
1529:
1526:
1512:
1508:
1502:
1499:
1495:
1489:
1485:
1481:
1476:
1473:
1469:
1463:
1459:
1455:
1450:
1447:
1443:
1437:
1433:
1428:
1422:
1419:
1415:
1408:
1405:
1399:
1394:
1389:
1386:
1382:
1368:, to find the
1340:
1337:
1333:
1327:
1322:
1319:
1315:
1311:
1308:
1305:
1303:
1301:
1297:
1291:
1288:
1284:
1278:
1273:
1270:
1266:
1262:
1257:
1254:
1250:
1244:
1239:
1236:
1232:
1227:
1223:
1219:
1213:
1210:
1206:
1200:
1195:
1192:
1188:
1184:
1179:
1176:
1172:
1166:
1161:
1158:
1154:
1149:
1145:
1141:
1135:
1132:
1128:
1122:
1117:
1114:
1110:
1106:
1101:
1098:
1094:
1088:
1083:
1080:
1076:
1071:
1067:
1064:
1062:
1058:
1055:
1051:
1045:
1041:
1037:
1032:
1029:
1025:
1019:
1015:
1011:
1006:
1003:
999:
993:
989:
985:
984:
963:
958:
955:
951:
945:
940:
937:
933:
929:
924:
921:
917:
911:
906:
903:
899:
895:
890:
887:
883:
877:
873:
855:
844:
819:
816:
811:
806:
803:
799:
795:
790:
785:
782:
778:
774:
769:
765:
759:
755:
751:
748:
731:for arbitrary
721:
712:
688:
683:
680:
676:
670:
666:
660:
656:
652:
647:
643:
637:
633:
627:
623:
619:
614:
610:
606:
601:
597:
593:
573:
569:
564:
559:
556:
552:
548:
509:
506:
493:
490:
463:Koszul formula
360:
359:
335:
332:
329:
326:
323:
320:
317:
314:
309:
305:
301:
298:
293:
289:
269:
219:
216:
213:
208:
204:
200:
197:
194:
191:
188:
185:
182:
179:
176:
171:
167:
163:
160:
157:
152:
147:
144:
141:
138:
135:
132:
127:
122:
63:
60:
17:
9:
6:
4:
3:
2:
4645:
4634:
4631:
4629:
4626:
4624:
4621:
4619:
4616:
4615:
4613:
4598:
4595:
4593:
4592:Supermanifold
4590:
4588:
4585:
4583:
4580:
4576:
4573:
4572:
4571:
4568:
4566:
4563:
4561:
4558:
4556:
4553:
4551:
4548:
4546:
4543:
4541:
4538:
4537:
4535:
4531:
4525:
4522:
4520:
4517:
4515:
4512:
4510:
4507:
4505:
4502:
4500:
4497:
4496:
4494:
4490:
4480:
4477:
4475:
4472:
4470:
4467:
4465:
4462:
4460:
4457:
4455:
4452:
4450:
4447:
4445:
4442:
4440:
4437:
4435:
4432:
4431:
4429:
4427:
4423:
4417:
4414:
4412:
4409:
4407:
4404:
4402:
4399:
4397:
4394:
4392:
4389:
4387:
4383:
4379:
4377:
4374:
4372:
4369:
4367:
4363:
4359:
4357:
4354:
4352:
4349:
4347:
4344:
4342:
4339:
4337:
4334:
4332:
4329:
4328:
4326:
4324:
4320:
4314:
4313:Wedge product
4311:
4309:
4306:
4302:
4299:
4298:
4297:
4294:
4292:
4289:
4285:
4282:
4281:
4280:
4277:
4275:
4272:
4270:
4267:
4265:
4262:
4258:
4257:Vector-valued
4255:
4254:
4253:
4250:
4248:
4245:
4241:
4238:
4237:
4236:
4233:
4231:
4228:
4226:
4223:
4222:
4220:
4216:
4210:
4207:
4205:
4202:
4200:
4197:
4193:
4190:
4189:
4188:
4187:Tangent space
4185:
4183:
4180:
4178:
4175:
4173:
4170:
4169:
4167:
4163:
4160:
4158:
4154:
4148:
4145:
4143:
4139:
4135:
4133:
4130:
4128:
4124:
4120:
4116:
4114:
4111:
4109:
4106:
4104:
4101:
4099:
4096:
4094:
4091:
4089:
4086:
4084:
4081:
4077:
4074:
4073:
4072:
4069:
4067:
4064:
4062:
4059:
4057:
4054:
4052:
4049:
4047:
4044:
4042:
4039:
4037:
4034:
4032:
4029:
4027:
4024:
4022:
4018:
4014:
4012:
4008:
4004:
4002:
3999:
3998:
3996:
3990:
3984:
3981:
3979:
3976:
3974:
3971:
3969:
3966:
3964:
3961:
3959:
3956:
3952:
3951:in Lie theory
3949:
3948:
3947:
3944:
3942:
3939:
3935:
3932:
3931:
3930:
3927:
3925:
3922:
3921:
3919:
3917:
3913:
3907:
3904:
3902:
3899:
3897:
3894:
3892:
3889:
3887:
3884:
3882:
3879:
3877:
3874:
3872:
3869:
3867:
3864:
3863:
3861:
3858:
3854:Main results
3852:
3846:
3843:
3841:
3838:
3836:
3835:Tangent space
3833:
3831:
3828:
3826:
3823:
3821:
3818:
3816:
3813:
3811:
3808:
3804:
3801:
3799:
3796:
3795:
3794:
3791:
3787:
3784:
3783:
3782:
3779:
3778:
3776:
3772:
3767:
3763:
3756:
3751:
3749:
3744:
3742:
3737:
3736:
3733:
3721:
3718:
3716:
3713:
3711:
3708:
3706:
3703:
3702:
3700:
3696:
3690:
3687:
3685:
3682:
3680:
3677:
3676:
3674:
3670:
3664:
3663:Schur's lemma
3661:
3659:
3656:
3654:
3651:
3649:
3646:
3644:
3641:
3639:
3638:Gauss's lemma
3636:
3634:
3631:
3630:
3628:
3624:
3618:
3615:
3613:
3610:
3608:
3605:
3603:
3600:
3599:
3597:
3593:
3587:
3584:
3582:
3579:
3577:
3574:
3572:
3569:
3565:
3561:
3558:
3556:
3553:
3551:
3548:
3546:
3543:
3542:
3541:
3540:Metric tensor
3538:
3536:
3535:Inner product
3533:
3531:
3528:
3526:
3523:
3519:
3516:
3514:
3511:
3509:
3506:
3504:
3501:
3500:
3499:
3496:
3495:
3493:
3489:
3484:
3480:
3473:
3468:
3466:
3461:
3459:
3454:
3453:
3450:
3442:
3438:
3434:
3430:
3426:
3424:0-226-87032-4
3420:
3416:
3412:
3411:
3406:
3402:
3398:
3394:
3390:
3386:
3382:
3376:
3372:
3368:
3364:
3360:
3356:
3351:
3347:
3343:
3339:
3335:
3331:
3329:0-12-526740-1
3325:
3321:
3317:
3314:
3310:
3306:
3302:
3298:
3294:
3290:
3286:
3282:
3278:
3274:
3270:
3266:
3262:
3258:
3254:
3251:
3247:
3246:
3241:
3237:
3233:
3229:
3225:
3221:
3217:
3213:
3207:
3203:
3199:
3195:
3191:
3187:
3183:
3179:
3175:
3171:
3167:
3163:
3161:0-8218-2848-7
3157:
3153:
3149:
3145:
3141:
3137:
3133:
3129:
3125:
3121:
3117:
3113:
3109:
3107:9780521099066
3103:
3098:
3093:
3089:
3085:
3084:
3079:
3075:
3071:
3067:
3063:
3059:
3055:
3051:
3049:0-8176-3490-8
3045:
3042:
3038:
3034:
3030:
3029:
3018:
3014:
3009:
3002:
2998:
2994:
2990:
2989:Helgason 2001
2986:
2982:
2981:do Carmo 1992
2977:
2970:
2966:
2961:
2954:
2949:
2942:
2937:
2930:
2925:
2918:
2914:
2910:
2905:
2898:
2897:Petersen 2016
2893:
2886:
2882:
2876:
2869:
2864:
2857:
2852:
2845:
2841:
2840:Helgason 2001
2837:
2832:
2825:
2821:
2820:do Carmo 1992
2816:
2809:
2804:
2797:
2793:
2792:Petersen 2016
2788:
2781:
2777:
2776:do Carmo 1992
2772:
2765:
2760:
2753:
2749:
2748:Petersen 2016
2745:
2742:, Lemma 8.6;
2741:
2737:
2733:
2729:
2728:Helgason 2001
2725:
2724:do Carmo 1992
2720:
2716:
2709:
2698:
2694:
2686:
2681:
2663:
2659:
2655:
2648:
2644:
2640:
2623:
2617:
2614:
2608:
2605:
2602:
2593:
2590:
2584:
2581:
2575:
2572:
2569:
2560:
2557:
2551:
2548:
2542:
2539:
2536:
2527:
2524:
2520:
2513:
2510:
2507:
2501:
2497:
2493:
2490:
2486:
2479:
2476:
2473:
2467:
2463:
2459:
2456:
2452:
2445:
2442:
2439:
2433:
2429:
2425:
2422:
2416:
2413:
2410:
2405:
2394:
2391:
2384:
2380:
2379:
2346:
2340:
2337:
2331:
2328:
2325:
2316:
2313:
2307:
2304:
2298:
2295:
2292:
2283:
2280:
2274:
2271:
2265:
2262:
2259:
2253:
2250:
2245:
2237:
2231:
2228:
2226:
2215:
2212:
2209:
2204:
2196:
2193:
2188:
2177:
2174:
2168:
2165:
2162:
2157:
2149:
2146:
2141:
2130:
2127:
2121:
2118:
2115:
2110:
2102:
2099:
2094:
2083:
2080:
2078:
2060:
2055:
2047:
2044:
2038:
2035:
2029:
2026:
2023:
2018:
2007:
1997:
1984:
1979:
1971:
1968:
1962:
1959:
1953:
1950:
1947:
1942:
1931:
1921:
1908:
1903:
1895:
1892:
1886:
1883:
1877:
1874:
1871:
1866:
1855:
1845:
1843:
1834:
1827:
1824:
1821:
1815:
1811:
1807:
1804:
1800:
1793:
1790:
1787:
1781:
1777:
1773:
1770:
1768:
1762:
1755:
1752:
1749:
1743:
1739:
1735:
1699:
1693:
1688:
1680:
1677:
1671:
1668:
1662:
1659:
1656:
1651:
1640:
1637:
1633:
1626:
1623:
1620:
1614:
1610:
1606:
1586:
1580:
1577:
1574:
1568:
1565:
1560:
1552:
1549:
1544:
1525:
1510:
1506:
1500:
1497:
1493:
1487:
1479:
1474:
1471:
1467:
1461:
1453:
1448:
1445:
1441:
1435:
1426:
1420:
1417:
1413:
1406:
1403:
1397:
1392:
1387:
1384:
1371:
1366:
1361:
1338:
1335:
1331:
1325:
1320:
1317:
1309:
1306:
1304:
1295:
1289:
1286:
1282:
1276:
1271:
1268:
1260:
1255:
1252:
1248:
1242:
1237:
1234:
1225:
1221:
1217:
1211:
1208:
1204:
1198:
1193:
1190:
1182:
1177:
1174:
1170:
1164:
1159:
1156:
1147:
1143:
1139:
1133:
1130:
1126:
1120:
1115:
1112:
1104:
1099:
1096:
1092:
1086:
1081:
1078:
1069:
1065:
1063:
1056:
1053:
1049:
1043:
1035:
1030:
1027:
1023:
1017:
1009:
1004:
1001:
997:
991:
961:
956:
953:
949:
943:
938:
935:
927:
922:
919:
915:
909:
904:
901:
893:
888:
885:
881:
875:
858:
847:
817:
809:
804:
801:
793:
788:
783:
780:
767:
763:
757:
753:
749:
746:
728:
724:
719:
715:
686:
681:
678:
668:
664:
658:
654:
650:
645:
641:
635:
625:
621:
617:
612:
604:
599:
571:
567:
562:
557:
554:
546:
536:
531:
519:
515:
505:
503:
499:
489:
487:
483:
478:
474:
464:
460:
456:
449:
448:
442:
440:
436:
435:metric tensor
431:
429:
421:
413:
408:
406:
402:
396:
386:
378:
377:inner product
366:
349:
333:
327:
324:
321:
315:
312:
307:
299:
296:
291:
270:
261:
257:
253:
246:
242:
238:
234:
217:
211:
206:
198:
195:
189:
186:
180:
177:
174:
169:
158:
155:
142:
139:
136:
130:
120:
101:
100:vector fields
97:
96:
95:
89:
85:
79:
75:
69:
59:
57:
50:
49:
44:
40:
36:
32:
28:
24:
16:
4519:Moving frame
4514:Morse theory
4504:Gauge theory
4296:Tensor field
4225:Closed/Exact
4204:Vector field
4172:Distribution
4113:Hypercomplex
4108:Quaternionic
3845:Vector field
3803:Smooth atlas
3698:Applications
3632:
3626:Main results
3409:
3354:
3308:
3277:Morse theory
3276:
3244:
3189:
3186:Jost, JĂŒrgen
3135:
3082:
3036:
3017:O'Neill 1983
3008:
3001:O'Neill 1983
2976:
2960:
2948:
2936:
2924:
2904:
2892:
2875:
2863:
2851:
2831:
2815:
2803:
2787:
2778:, pp.53-54;
2771:
2759:
2744:O'Neill 1983
2719:
2696:
2692:
2684:
2679:
2661:
2657:
2653:
2651:is equal to
2646:
2642:
2638:
2382:
2376:
1531:
1369:
1364:
1359:
856:
845:
726:
722:
717:
713:
534:
532:is given by
511:
495:
479:
472:
462:
458:
451:
445:
443:
432:
427:
411:
409:
394:
364:
362:
259:
255:
251:
244:
240:
236:
232:
77:
73:
67:
66:
52:
46:
43:torsion-free
26:
20:
15:
4464:Levi-Civita
4454:Generalized
4426:Connections
4376:Lie algebra
4308:Volume form
4209:Vector flow
4182:Pushforward
4177:Lie bracket
4076:Lie algebra
4041:G-structure
3830:Pushforward
3810:Submanifold
2965:Milnor 1963
2953:Milnor 1963
2929:Milnor 1963
2917:Milnor 1963
2824:Milnor 1963
2782:, pp.47-48.
2780:Milnor 1963
2740:Milnor 1963
2700:when given
2674:must equal
379:of any two
348:Lie bracket
4612:Categories
4587:Stratifold
4545:Diffeology
4341:Associated
4142:Symplectic
4127:Riemannian
4056:Hyperbolic
3983:Submersion
3891:HopfâRinow
3825:Submersion
3820:Smooth map
3658:Ricci flow
3607:Hyperbolic
3441:0549.53001
3397:1417.53001
3346:0531.53051
3297:0108.10401
3273:Milnor, J.
3265:0119.37502
3228:1380.53001
3178:0993.53002
3124:0265.53054
3066:0752.53001
3026:References
530:connection
4469:Principal
4444:Ehresmann
4401:Subbundle
4391:Principal
4366:Fibration
4346:Cotangent
4218:Covectors
4071:Lie group
4051:Hermitian
3994:manifolds
3963:Immersion
3958:Foliation
3896:Noether's
3881:Frobenius
3876:De Rham's
3871:Darboux's
3762:Manifolds
3602:Hermitian
3555:Signature
3518:Sectional
3498:Curvature
3015:, p.194;
3013:Jost 2017
2999:, p.160;
2995:, p.194;
2993:Jost 2017
2969:Wald 1984
2941:Wald 1984
2909:Jost 2017
2856:Wald 1984
2844:Jost 2017
2796:Wald 1984
2764:Jost 2017
2752:Wald 1984
2732:Jost 2017
2591:−
2558:−
2525:−
2491:−
2402:∇
2242:∇
2201:∇
2197:−
2185:∇
2154:∇
2150:−
2138:∇
2107:∇
2091:∇
2052:∇
2015:∇
1998:−
1976:∇
1939:∇
1900:∇
1863:∇
1805:−
1685:∇
1648:∇
1557:∇
1553:−
1541:∇
1484:∂
1480:−
1458:∂
1432:∂
1381:Γ
1314:Γ
1265:Γ
1231:Γ
1222:−
1187:Γ
1153:Γ
1109:Γ
1075:Γ
1040:∂
1036:−
1014:∂
988:∂
932:Γ
898:Γ
872:∂
798:Γ
794:−
777:Γ
675:Γ
632:∂
596:∇
551:Γ
512:Here the
453:(pseudo-)
304:∇
300:−
288:∇
203:∇
166:∇
54:(pseudo-)
4565:Orbifold
4560:K-theory
4550:Diffiety
4274:Pullback
4088:Oriented
4066:Kenmotsu
4046:Hadamard
3992:Types of
3941:Geodesic
3766:Glossary
3617:Kenmotsu
3530:Geodesic
3483:Glossary
3407:(1984).
3307:(1983).
3275:(1963).
3242:(1963).
3188:(2017).
3134:(2001).
3080:(1973).
3035:(1992).
2991:, p.48;
2987:, p.40;
2983:, p.55;
2967:, p.49;
2842:, p.43;
2838:, p.34;
2822:, p.54;
2383:identity
412:symmetry
113:we have
98:for any
41:that is
4509:History
4492:Related
4406:Tangent
4384:)
4364:)
4331:Adjoint
4323:Bundles
4301:density
4199:Torsion
4165:Vectors
4157:Tensors
4140:)
4125:)
4121:,
4119:Pseudoâ
4098:Poisson
4031:Finsler
4026:Fibered
4021:Contact
4019:)
4011:Complex
4009:)
3978:Section
3684:Hilbert
3679:Finsler
3433:0757180
3389:3469435
3338:0719023
3289:0163331
3257:0152974
3220:3726907
3170:1834454
3116:0424186
3058:1138207
3019:, p.61.
3003:, p.61.
2971:, p.36.
2955:, p.49.
2943:, p.35.
2931:, p.48.
2919:, p.48.
2899:, p.66.
2870:, p.41.
2798:, p.35.
2670:, then
484:of the
420:torsion
403:on the
4474:Vector
4459:Koszul
4439:Cartan
4434:Affine
4416:Vector
4411:Tensor
4396:Spinor
4386:Normal
4382:Stable
4336:Affine
4240:bundle
4192:bundle
4138:Almost
4061:KĂ€hler
4017:Almost
4007:Almost
4001:Closed
3901:Sard's
3857:(list)
3612:KĂ€hler
3508:Scalar
3503:tensor
3439:
3431:
3421:
3395:
3387:
3377:
3344:
3336:
3326:
3295:
3287:
3263:
3255:
3226:
3218:
3208:
3176:
3168:
3158:
3122:
3114:
3104:
3064:
3056:
3046:
2371:, and
1720:, and
1712:where
375:, the
230:where
109:, and
25:, the
4582:Sheaf
4356:Fiber
4132:Rizza
4103:Prime
3934:Local
3924:Curve
3786:Atlas
3513:Ricci
2712:Notes
584:with
82:be a
4449:Form
4351:Dual
4284:flow
4147:Tame
4123:Subâ
4036:Flat
3916:Maps
3419:ISBN
3375:ISBN
3324:ISBN
3206:ISBN
3156:ISBN
3102:ISBN
3044:ISBN
2704:and
834:and
735:and
720:â â
705:and
475:â 1)
354:and
86:(or
70:Let
33:(or
4371:Jet
3437:Zbl
3393:Zbl
3367:doi
3342:Zbl
3316:doi
3293:Zbl
3261:Zbl
3224:Zbl
3198:doi
3174:Zbl
3148:doi
3120:Zbl
3092:doi
3062:Zbl
2381:or
851:= Î
461:or
450:or
422:of
414:of
397:= 0
367:of
350:of
51:or
4614::
4362:Co
3435:.
3429:MR
3427:.
3417:.
3391:.
3385:MR
3383:.
3373:.
3365:.
3357:.
3340:.
3334:MR
3332:.
3322:.
3291:.
3285:MR
3283:.
3259:.
3253:MR
3238:;
3222:.
3216:MR
3214:.
3204:.
3196:.
3172:.
3166:MR
3164:.
3154:.
3146:.
3138:.
3118:.
3112:MR
3110:.
3100:.
3090:.
3076:;
3060:.
3054:MR
3052:.
2660:,
2645:,
2367:,
1716:,
1372::
857:kj
846:jk
729:=
488:.
441:.
407:.
358:.
279:,
275:,
258:,
247:))
243:,
105:,
76:,
4380:(
4360:(
4136:(
4117:(
4015:(
4005:(
3768:)
3764:(
3754:e
3747:t
3740:v
3562:/
3485:)
3481:(
3471:e
3464:t
3457:v
3443:.
3399:.
3369::
3348:.
3318::
3299:.
3267:.
3230:.
3200::
3180:.
3150::
3126:.
3094::
3068:.
2706:Y
2702:X
2697:Y
2693:X
2690:â
2685:g
2676:U
2672:W
2668:Z
2664:)
2662:Z
2658:U
2656:(
2654:g
2649:)
2647:Z
2643:W
2641:(
2639:g
2624:.
2621:)
2618:X
2615:,
2612:]
2609:Z
2606:,
2603:Y
2600:[
2597:(
2594:g
2588:)
2585:Y
2582:,
2579:]
2576:Z
2573:,
2570:X
2567:[
2564:(
2561:g
2555:)
2552:Z
2549:,
2546:]
2543:X
2540:,
2537:Y
2534:[
2531:(
2528:g
2521:)
2517:)
2514:Y
2511:,
2508:X
2505:(
2502:g
2498:(
2494:Z
2487:)
2483:)
2480:Z
2477:,
2474:X
2471:(
2468:g
2464:(
2460:Y
2457:+
2453:)
2449:)
2446:Z
2443:,
2440:Y
2437:(
2434:g
2430:(
2426:X
2423:=
2420:)
2417:Z
2414:,
2411:Y
2406:X
2398:(
2395:g
2392:2
2373:Z
2369:Y
2365:X
2347:.
2344:)
2341:X
2338:,
2335:]
2332:Z
2329:,
2326:Y
2323:[
2320:(
2317:g
2314:+
2311:)
2308:Y
2305:,
2302:]
2299:Z
2296:,
2293:X
2290:[
2287:(
2284:g
2281:+
2278:)
2275:Z
2272:,
2269:]
2266:X
2263:,
2260:Y
2257:[
2254:+
2251:Y
2246:X
2238:2
2235:(
2232:g
2229:=
2219:)
2216:X
2213:,
2210:Y
2205:Z
2194:Z
2189:Y
2181:(
2178:g
2175:+
2172:)
2169:Y
2166:,
2163:X
2158:Z
2147:Z
2142:X
2134:(
2131:g
2128:+
2125:)
2122:Z
2119:,
2116:X
2111:Y
2103:+
2100:Y
2095:X
2087:(
2084:g
2081:=
2069:)
2064:)
2061:Y
2056:Z
2048:,
2045:X
2042:(
2039:g
2036:+
2033:)
2030:Y
2027:,
2024:X
2019:Z
2011:(
2008:g
2003:(
1993:)
1988:)
1985:Z
1980:Y
1972:,
1969:X
1966:(
1963:g
1960:+
1957:)
1954:Z
1951:,
1948:X
1943:Y
1935:(
1932:g
1927:(
1922:+
1917:)
1912:)
1909:Z
1904:X
1896:,
1893:Y
1890:(
1887:g
1884:+
1881:)
1878:Z
1875:,
1872:Y
1867:X
1859:(
1856:g
1851:(
1846:=
1835:)
1831:)
1828:Y
1825:,
1822:X
1819:(
1816:g
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