2782:
43:
2414:
2777:{\displaystyle g_{ik}{\Big (}\gamma (t),{\dot {\gamma }}(t){\Big )}{\ddot {\gamma }}^{i}(t)+\left({\frac {\partial g_{ik}}{\partial x^{j}}}{\Big (}\gamma (t),{\dot {\gamma }}(t){\Big )}-{\frac {1}{2}}{\frac {\partial g_{ij}}{\partial x^{k}}}{\Big (}\gamma (t),{\dot {\gamma }}(t){\Big )}\right){\dot {\gamma }}^{i}(t){\dot {\gamma }}^{j}(t)=0,}
1993:
3488:
3252:
2949:
3715:
810:
3860:
1748:
3267:
3062:
314:
2325:
1542:
2808:
2141:
1682:
3576:
1295:
1091:
652:
1988:{\displaystyle d_{L}(x,y):=\inf \left\{\ \left.\int _{0}^{1}F\left(\gamma (t),{\dot {\gamma }}(t)\right)\,dt\ \right|\ \gamma \in C^{1}(,M)\ ,\ \gamma (0)=x\ ,\ \gamma (1)=y\ \right\},}
3730:
3483:{\displaystyle G^{i}(x,v):={\frac {1}{4}}g^{ij}(x,v)\left(2{\frac {\partial g_{jk}}{\partial x^{\ell }}}(x,v)-{\frac {\partial g_{k\ell }}{\partial x^{j}}}(x,v)\right)v^{k}v^{\ell }.}
1132:
3247:{\displaystyle \left.H\right|_{(x,v)}:=\left.v^{i}{\frac {\partial }{\partial x^{i}}}\right|_{(x,v)}\!\!-\left.2G^{i}(x,v){\frac {\partial }{\partial v^{i}}}\right|_{(x,v)},}
211:
2210:
1172:
1338:
990:
1424:
2944:{\displaystyle g_{ij}(x,v):=g_{v}\left(\left.{\frac {\partial }{\partial x^{i}}}\right|_{x},\left.{\frac {\partial }{\partial x^{j}}}\right|_{x}\right).}
2043:
3905:). Length minimizing curves can always be positively reparametrized to be geodesics, and any geodesic must satisfy the Euler–Lagrange equation for
1582:
3710:{\displaystyle v:T(\mathrm {T} M\setminus \{0\})\to T(\mathrm {T} M\setminus \{0\});\quad v:={\frac {1}{2}}{\big (}I+{\mathcal {L}}_{H}J{\big )}.}
5190:
3882:
4175:
4381:
5245:
5185:
4207:
3526:
3522:
1184:
5408:
5258:
4472:
4496:
4691:
1014:
5438:
5413:
107:
5300:
4561:
4264:
4114:
805:{\displaystyle \mathbf {g} _{v}(X,Y):={\frac {1}{2}}\left.{\frac {\partial ^{2}}{\partial s\partial t}}\left\right|_{s=t=0},}
79:
4787:
4840:
4368:
86:
5124:
4074: – topological space modeled on a Fréchet space in much the same way as a manifold is modeled on a Euclidean space
5273:
5238:
4299:
4152:
126:
60:
4889:
4251:. Die Grundlehren der Mathematischen Wissenschaften. Vol. 101. Berlin–Göttingen–Heidelberg: Springer-Verlag.
3855:{\displaystyle D_{\dot {\gamma }}D_{\dot {\gamma }}X(t)+R_{\dot {\gamma }}\left({\dot {\gamma }}(t),X(t)\right)=0}
93:
4872:
4481:
4324:
5516:
64:
5531:
5084:
4491:
4334:
75:
17:
5526:
5335:
5231:
5069:
4792:
4566:
2955:
5114:
4329:
2381:
1099:
1711:
obtained in this way restricts to an asymmetric (typically non-Minkowski) norm on each tangent space of
5356:
5119:
5089:
4797:
4753:
4734:
4501:
4445:
2162:
948:
941:
of finite dimension are
Finsler manifolds if the norm of the vector space is smooth outside the origin.
521:
5536:
5521:
5485:
4656:
4521:
337:
when the distance between two points is defined as the infimum length of the curves that join them.
309:{\displaystyle L(\gamma )=\int _{a}^{b}F\left(\gamma (t),{\dot {\gamma }}(t)\right)\,\mathrm {d} t.}
5418:
5278:
5041:
4906:
4598:
4440:
621:
2320:{\displaystyle E:={\frac {1}{2}}\int _{a}^{b}F^{2}\left(\gamma (t),{\dot {\gamma }}(t)\right)\,dt}
334:
5464:
4738:
4708:
4632:
4622:
4578:
4408:
4361:
1369:
368:
152:
53:
5495:
5428:
5423:
5361:
5320:
5079:
4698:
4593:
4506:
4413:
3894:
1377:
5490:
1141:
100:
4728:
4723:
2331:
144:
5325:
5059:
4997:
4845:
4549:
4539:
4511:
4486:
4396:
4309:
4274:
4220:
4162:
4124:
3878:
3568:
3550:
2147:
1311:
963:
497:
166:
4236:
4188:
1537:{\displaystyle {\frac {1}{C}}\|\phi (y)-\phi (x)\|\leq d(x,y)\leq C\|\phi (y)-\phi (x)\|.}
8:
5382:
5351:
5339:
5310:
5293:
5254:
5197:
4879:
4757:
4742:
4671:
4430:
4001:
3897:
there always exist length minimizing curves (at least in small enough neighborhoods) on (
3721:
3538:
993:
944:
938:
320:
5170:
4071:
5480:
5377:
5346:
5139:
5094:
4991:
4862:
4666:
4354:
4100:
925:
5387:
4676:
5392:
5074:
5054:
5049:
4956:
4867:
4681:
4661:
4516:
4455:
4295:
4260:
4195:
4148:
4132:
4110:
1175:
1001:
5459:
5330:
5283:
5212:
5006:
4961:
4884:
4855:
4713:
4641:
4636:
4626:
4418:
4401:
4287:
4252:
4232:
4184:
4140:
4086:
4042:
4034:
2007:
1716:
641:
331:
30:"Finsler" redirects here. For the mathematician this manifold is named after, see
5288:
5223:
5155:
5064:
4894:
4850:
4616:
4305:
4270:
4216:
4158:
4120:
4104:
4080:
4065:
4026:
3965:
3866:
1561:
575:
540:
161:
4200:"Finsler geometry is just Riemannian geometry without the quadratic restriction"
4083:– which uses Hilbert manifolds and other kinds of infinite-dimensional manifolds
2136:{\displaystyle L:=\int _{a}^{b}F\left(\gamma (t),{\dot {\gamma }}(t)\right)\,dt}
5021:
4946:
4916:
4814:
4807:
4747:
4718:
4588:
4583:
4544:
4199:
4021:
3938:
3042:
1135:
605:
392:
4256:
4170:
4144:
4047:
340:
5510:
5315:
5207:
5031:
5026:
5011:
5001:
4951:
4928:
4802:
4762:
4703:
4651:
4450:
4024:(1941). "On an Asymmetrical Metric in the Four-Space of General Relativity".
1677:{\displaystyle F(x,v):=\lim _{t\to 0+}{\frac {d(\gamma (0),\gamma (t))}{t}},}
644:
464:
324:
5134:
5129:
4971:
4938:
4911:
4819:
4460:
4139:. Graduate Texts in Mathematics. Vol. 200. New York: Springer-Verlag.
4038:
3554:
3046:
348:
193:
31:
4977:
4966:
4923:
4824:
4425:
2010:
1739:
1361:
140:
5433:
5202:
5160:
4986:
4899:
4531:
4435:
4346:
4244:
2802:
is the coordinate representation of the fundamental tensor, defined as
5016:
4981:
4686:
4573:
3888:
42:
5305:
5180:
5175:
5165:
4556:
4377:
2170:
1290:{\displaystyle F(x,v):={\sqrt {a_{ij}(x)v^{i}v^{j}}}+b_{i}(x)v^{i}}
2375:
4291:
3956:
are length-minimizing among nearby curves until the first point
4772:
4341:
589:
The subadditivity axiom may then be replaced by the following
2204:
is a geodesic if it is stationary for the energy functional
3165:
3100:
3068:
2899:
2860:
1792:
1086:{\displaystyle \|b\|_{a}:={\sqrt {a^{ij}b_{i}b_{j}}}<1,}
698:
1344:, a special case of a non-reversible Finsler manifold.
837:
implies the subadditivity with a strict inequality if
3733:
3579:
3270:
3065:
2811:
2417:
2213:
2046:
1751:
1585:
1427:
1314:
1187:
1144:
1102:
1017:
966:
655:
214:
4076:
Pages displaying wikidata descriptions as a fallback
2173:of a Finsler manifold if its short enough segments
67:. Unsourced material may be challenged and removed.
5253:
3854:
3709:
3482:
3246:
2943:
2776:
2319:
2135:
1987:
1676:
1536:
1332:
1289:
1166:
1126:
1085:
984:
937:Smooth submanifolds (including open subsets) of a
804:
308:
3889:Uniqueness and minimizing properties of geodesics
3159:
3158:
2699:
2656:
2599:
2556:
2476:
2433:
351:, who studied this geometry in his dissertation (
5508:
1780:
1608:
2376:Canonical spray structure on a Finsler manifold
1564:in some punctured neighborhood of the diagonal.
323:since the tangent norms need not be induced by
192:, that enables one to define the length of any
3000:) is invertible and its inverse is denoted by
5239:
4362:
4229:Über Kurven und Flächen in allgemeinen Räumen
3699:
3669:
380:, which is a continuous nonnegative function
4208:Notices of the American Mathematical Society
4130:
3641:
3635:
3609:
3603:
1528:
1498:
1468:
1438:
1347:
1025:
1018:
928:(in the usual sense) on each tangent space.
4249:The differential geometry of Finsler spaces
4137:An introduction to Riemann–Finsler geometry
4089: – Manifold modelled on Hilbert spaces
5409:Fundamental theorem of Riemannian geometry
5246:
5232:
4369:
4355:
4099:
4068: – Manifold modeled on Banach spaces
4046:
2310:
2126:
1859:
951:) are special cases of Finsler manifolds.
578:on the complement of the zero section of
294:
127:Learn how and when to remove this message
4376:
1707:(0) = v. The Finsler function
319:Finsler manifolds are more general than
4226:
4106:Handbook of Finsler geometry. Vol. 1, 2
4020:
3984:there always exist shorter curves from
3913:there exists a unique maximal geodesic
2161:is invariant under positively oriented
1568:Then one can define a Finsler function
352:
14:
5509:
4173:(1933), "Sur les espaces de Finsler",
4169:
4109:, Boston: Kluwer Academic Publishers,
924:A reversible Finsler metric defines a
344:
27:Generalization of Riemannian manifolds
5227:
4350:
4194:
2334:vanishes among differentiable curves
4281:
4243:
2006: → [0, ∞) defines an
955:
65:adding citations to reliable sources
36:
3909:. Assuming the strong convexity of
3257:where the local spray coefficients
3030:) if and only if its tangent curve
1408: ≥ 1 such that for every
1127:{\displaystyle \left(a^{ij}\right)}
24:
3683:
3625:
3593:
3533:∖{0}. Hence, by definition,
3421:
3403:
3366:
3348:
3202:
3198:
3119:
3115:
2907:
2903:
2868:
2864:
2638:
2620:
2538:
2520:
720:
714:
704:
330:Every Finsler manifold becomes an
296:
180:is provided on each tangent space
25:
5548:
4317:
3632:
3600:
1998:and in fact any Finsler function
883:is strongly convex, then it is a
4240:(Reprinted by Birkhäuser (1951))
3937:∖{0} by the uniqueness of
2388:reads in the local coordinates (
658:
347:) named Finsler manifolds after
41:
4286:. Singapore: World Scientific.
3869:for a general spray structure (
3650:
52:needs additional citations for
4409:Differentiable/Smooth manifold
4014:
3948:is strongly convex, geodesics
3883:nonlinear covariant derivative
3838:
3832:
3823:
3817:
3777:
3771:
3644:
3621:
3615:
3612:
3589:
3449:
3437:
3394:
3382:
3334:
3322:
3293:
3281:
3236:
3224:
3193:
3181:
3153:
3141:
3090:
3078:
3056:∖{0} locally defined by
2837:
2825:
2762:
2756:
2734:
2728:
2694:
2688:
2670:
2664:
2594:
2588:
2570:
2564:
2506:
2500:
2471:
2465:
2447:
2441:
2302:
2296:
2278:
2272:
2223:
2217:
2118:
2112:
2094:
2088:
2056:
2050:
1965:
1959:
1938:
1932:
1917:
1908:
1896:
1893:
1851:
1845:
1827:
1821:
1774:
1762:
1662:
1659:
1653:
1644:
1638:
1632:
1615:
1601:
1589:
1525:
1519:
1510:
1504:
1489:
1477:
1465:
1459:
1450:
1444:
1327:
1315:
1274:
1268:
1230:
1224:
1203:
1191:
1161:
1145:
979:
967:
762:
737:
680:
668:
492:(but not necessarily for
286:
280:
262:
256:
224:
218:
13:
1:
4093:
1396:there exists a smooth chart (
358:
5336:Raising and lowering indices
4342:The (New) Finsler Newsletter
4325:"Finsler space, generalized"
4284:Lectures on Finsler geometry
2028:
7:
5115:Classification of manifolds
4330:Encyclopedia of Mathematics
4231:, Dissertation, Göttingen,
4059:
3925:(0) = v for any (
949:pseudo-Riemannian manifolds
931:
10:
5553:
5357:Pseudo-Riemannian manifold
2384:for the energy functional
2033:Due to the homogeneity of
591:strong convexity condition
29:
5486:Geometrization conjecture
5473:
5447:
5401:
5370:
5266:
5191:over commutative algebras
5148:
5107:
5040:
4937:
4833:
4780:
4771:
4607:
4530:
4469:
4389:
4257:10.1007/978-3-642-51610-8
4145:10.1007/978-1-4612-1268-3
4135:; Shen, Zhongmin (2000).
3724:case, there is a version
2180:are length-minimizing in
2165:. A constant speed curve
1348:Smooth quasimetric spaces
4907:Riemann curvature tensor
4007:
1717:induced intrinsic metric
1384:in the following sense:
1167:{\displaystyle (a_{ij})}
916:for all tangent vectors
597:For each tangent vector
4282:Shen, Zhongmin (2001).
2382:Euler–Lagrange equation
1376:is compatible with the
1370:differentiable manifold
887:on each tangent space.
561:is also required to be
441:for every two vectors
395:so that for each point
369:differentiable manifold
153:differentiable manifold
5496:Uniformization theorem
5429:Nash embedding theorem
5362:Riemannian volume form
5321:Levi-Civita connection
4699:Manifold with boundary
4414:Differential structure
4227:Finsler, Paul (1918),
4176:C. R. Acad. Sci. Paris
4039:10.1103/PhysRev.59.195
3921:(0) = x and
3856:
3711:
3527:canonical vector field
3523:canonical endomorphism
3484:
3248:
2945:
2778:
2330:in the sense that its
2321:
2137:
1989:
1742:can be recovered from
1678:
1538:
1378:differential structure
1334:
1291:
1168:
1128:
1087:
986:
831:. Strong convexity of
806:
543:on each tangent space
310:
5517:Differential geometry
3857:
3712:
3501:∖{0} satisfies
3485:
3249:
2946:
2779:
2344:with fixed endpoints
2332:functional derivative
2322:
2138:
1990:
1679:
1539:
1335:
1333:{\displaystyle (M,F)}
1292:
1169:
1129:
1088:
1002:differential one-form
987:
985:{\displaystyle (M,a)}
807:
555:. The Finsler metric
522:positive definiteness
311:
145:differential geometry
5532:Riemannian manifolds
5419:Gauss–Bonnet theorem
5326:Covariant derivative
4846:Covariant derivative
4397:Topological manifold
3952:: →
3731:
3720:In analogy with the
3577:
3551:nonlinear connection
3268:
3063:
2809:
2415:
2211:
2148:differentiable curve
2044:
1749:
1583:
1425:
1312:
1185:
1142:
1100:
1015:
964:
945:Riemannian manifolds
890:A Finsler metric is
653:
628:Here the Hessian of
498:positive homogeneity
321:Riemannian manifolds
212:
61:improve this article
5527:Riemannian geometry
5491:Poincaré conjecture
5352:Riemannian manifold
5340:Musical isomorphism
5255:Riemannian geometry
4880:Exterior derivative
4482:Atiyah–Singer index
4431:Riemannian manifold
4101:Antonelli, Peter L.
3879:Ehresmann curvature
3569:vertical projection
3047:smooth vector field
2253:
2076:
1809:
994:Riemannian manifold
939:normed vector space
244:
5481:General relativity
5424:Hopf–Rinow theorem
5371:Types of manifolds
5347:Parallel transport
5186:Secondary calculus
5140:Singularity theory
5095:Parallel transport
4863:De Rham cohomology
4502:Generalized Stokes
4196:Chern, Shiing-Shen
4133:Chern, Shiing-Shen
4048:10338.dmlcz/134230
3895:Hopf–Rinow theorem
3877:) in terms of the
3852:
3707:
3480:
3244:
3022:is a geodesic of (
2970:) with respect to
2941:
2774:
2317:
2239:
2163:reparametrizations
2133:
2062:
1985:
1795:
1674:
1625:
1534:
1330:
1287:
1164:
1124:
1083:
982:
817:fundamental tensor
815:also known as the
802:
565:, more precisely:
306:
230:
160:where a (possibly
76:"Finsler manifold"
5504:
5503:
5221:
5220:
5103:
5102:
4868:Differential form
4522:Whitney embedding
4456:Differential form
4266:978-3-642-51612-2
4116:978-1-4020-1557-1
3814:
3796:
3764:
3747:
3665:
3509:and =
3493:The vector field
3435:
3380:
3307:
3216:
3133:
2921:
2882:
2747:
2719:
2685:
2652:
2615:
2585:
2552:
2491:
2462:
2293:
2237:
2200:). Equivalently,
2109:
2025:by this formula.
1976:
1955:
1949:
1928:
1922:
1876:
1868:
1842:
1790:
1669:
1607:
1560: → is
1436:
1388:Around any point
1253:
1176:Einstein notation
1072:
956:Randers manifolds
894:if, in addition,
727:
694:
622:positive definite
335:quasimetric space
277:
137:
136:
129:
111:
16:(Redirected from
5544:
5537:Smooth manifolds
5522:Finsler geometry
5248:
5241:
5234:
5225:
5224:
5213:Stratified space
5171:Fréchet manifold
4885:Interior product
4778:
4777:
4475:
4371:
4364:
4357:
4348:
4347:
4338:
4313:
4278:
4239:
4223:
4204:
4191:
4166:
4127:
4087:Hilbert manifold
4077:
4072:Fréchet manifold
4053:
4052:
4050:
4018:
3980: >
3861:
3859:
3858:
3853:
3845:
3841:
3816:
3815:
3807:
3799:
3798:
3797:
3789:
3767:
3766:
3765:
3757:
3750:
3749:
3748:
3740:
3716:
3714:
3713:
3708:
3703:
3702:
3693:
3692:
3687:
3686:
3673:
3672:
3666:
3658:
3628:
3596:
3566:
3489:
3487:
3486:
3481:
3476:
3475:
3466:
3465:
3456:
3452:
3436:
3434:
3433:
3432:
3419:
3418:
3417:
3401:
3381:
3379:
3378:
3377:
3364:
3363:
3362:
3346:
3321:
3320:
3308:
3300:
3280:
3279:
3253:
3251:
3250:
3245:
3240:
3239:
3222:
3218:
3217:
3215:
3214:
3213:
3197:
3180:
3179:
3157:
3156:
3139:
3135:
3134:
3132:
3131:
3130:
3114:
3112:
3111:
3094:
3093:
3076:
3040:
3021:
2956:strong convexity
2950:
2948:
2947:
2942:
2937:
2933:
2932:
2931:
2926:
2922:
2920:
2919:
2918:
2902:
2893:
2892:
2887:
2883:
2881:
2880:
2879:
2863:
2852:
2851:
2824:
2823:
2783:
2781:
2780:
2775:
2755:
2754:
2749:
2748:
2740:
2727:
2726:
2721:
2720:
2712:
2708:
2704:
2703:
2702:
2687:
2686:
2678:
2660:
2659:
2653:
2651:
2650:
2649:
2636:
2635:
2634:
2618:
2616:
2608:
2603:
2602:
2587:
2586:
2578:
2560:
2559:
2553:
2551:
2550:
2549:
2536:
2535:
2534:
2518:
2499:
2498:
2493:
2492:
2484:
2480:
2479:
2464:
2463:
2455:
2437:
2436:
2430:
2429:
2371:
2357:
2343:
2326:
2324:
2323:
2318:
2309:
2305:
2295:
2294:
2286:
2263:
2262:
2252:
2247:
2238:
2230:
2142:
2140:
2139:
2134:
2125:
2121:
2111:
2110:
2102:
2075:
2070:
1994:
1992:
1991:
1986:
1981:
1977:
1974:
1953:
1947:
1926:
1920:
1892:
1891:
1874:
1873:
1869:
1866:
1858:
1854:
1844:
1843:
1835:
1808:
1803:
1788:
1761:
1760:
1738:of the original
1737:
1699:(0) =
1691:is any curve in
1683:
1681:
1680:
1675:
1670:
1665:
1627:
1624:
1543:
1541:
1540:
1535:
1437:
1429:
1342:Randers manifold
1339:
1337:
1336:
1331:
1296:
1294:
1293:
1288:
1286:
1285:
1267:
1266:
1254:
1252:
1251:
1242:
1241:
1223:
1222:
1210:
1173:
1171:
1170:
1165:
1160:
1159:
1133:
1131:
1130:
1125:
1123:
1119:
1118:
1092:
1090:
1089:
1084:
1073:
1071:
1070:
1061:
1060:
1051:
1050:
1038:
1033:
1032:
991:
989:
988:
983:
915:
882:
876:
875:
874:
863:
856:
855:
844:
836:
830:
824:
811:
809:
808:
803:
798:
797:
780:
776:
775:
771:
770:
769:
728:
726:
712:
711:
702:
695:
687:
667:
666:
661:
639:
633:
619:
613:
603:
584:
573:
560:
554:
538:
528:In other words,
519:
512:
495:
491:
487:
462:
456:
450:
440:
406:
400:
390:
376:together with a
375:
365:Finsler manifold
315:
313:
312:
307:
299:
293:
289:
279:
278:
270:
243:
238:
204:
191:
179:
159:
149:Finsler manifold
132:
125:
121:
118:
112:
110:
69:
45:
37:
21:
5552:
5551:
5547:
5546:
5545:
5543:
5542:
5541:
5507:
5506:
5505:
5500:
5469:
5448:Generalizations
5443:
5397:
5366:
5301:Exponential map
5262:
5252:
5222:
5217:
5156:Banach manifold
5149:Generalizations
5144:
5099:
5036:
4933:
4895:Ricci curvature
4851:Cotangent space
4829:
4767:
4609:
4603:
4562:Exponential map
4526:
4471:
4465:
4385:
4375:
4323:
4320:
4302:
4267:
4202:
4155:
4117:
4096:
4081:Global analysis
4075:
4066:Banach manifold
4062:
4057:
4056:
4019:
4015:
4010:
3939:integral curves
3933:) ∈ T
3891:
3867:Jacobi equation
3806:
3805:
3804:
3800:
3788:
3787:
3783:
3756:
3755:
3751:
3739:
3738:
3734:
3732:
3729:
3728:
3698:
3697:
3688:
3682:
3681:
3680:
3668:
3667:
3657:
3624:
3592:
3578:
3575:
3574:
3557:
3471:
3467:
3461:
3457:
3428:
3424:
3420:
3410:
3406:
3402:
3400:
3373:
3369:
3365:
3355:
3351:
3347:
3345:
3341:
3337:
3313:
3309:
3299:
3275:
3271:
3269:
3266:
3265:
3223:
3209:
3205:
3201:
3196:
3175:
3171:
3167:
3164:
3163:
3140:
3126:
3122:
3118:
3113:
3107:
3103:
3102:
3099:
3098:
3077:
3067:
3066:
3064:
3061:
3060:
3031:
3013:
2991:
2979:
2927:
2914:
2910:
2906:
2901:
2898:
2897:
2888:
2875:
2871:
2867:
2862:
2859:
2858:
2857:
2853:
2847:
2843:
2816:
2812:
2810:
2807:
2806:
2801:
2750:
2739:
2738:
2737:
2722:
2711:
2710:
2709:
2698:
2697:
2677:
2676:
2655:
2654:
2645:
2641:
2637:
2627:
2623:
2619:
2617:
2607:
2598:
2597:
2577:
2576:
2555:
2554:
2545:
2541:
2537:
2527:
2523:
2519:
2517:
2516:
2512:
2494:
2483:
2482:
2481:
2475:
2474:
2454:
2453:
2432:
2431:
2422:
2418:
2416:
2413:
2412:
2378:
2359:
2345:
2335:
2285:
2284:
2268:
2264:
2258:
2254:
2248:
2243:
2229:
2212:
2209:
2208:
2179:
2101:
2100:
2084:
2080:
2071:
2066:
2045:
2042:
2041:
2031:
2020:
1887:
1883:
1834:
1833:
1817:
1813:
1804:
1799:
1794:
1791:
1787:
1783:
1756:
1752:
1750:
1747:
1746:
1727:
1719:
1628:
1626:
1611:
1584:
1581:
1580:
1428:
1426:
1423:
1422:
1404:and a constant
1350:
1313:
1310:
1309:
1281:
1277:
1262:
1258:
1247:
1243:
1237:
1233:
1215:
1211:
1209:
1186:
1183:
1182:
1152:
1148:
1143:
1140:
1139:
1111:
1107:
1103:
1101:
1098:
1097:
1066:
1062:
1056:
1052:
1043:
1039:
1037:
1028:
1024:
1016:
1013:
1012:
965:
962:
961:
958:
934:
898:
878:
865:
859:
858:
846:
840:
839:
838:
832:
826:
820:
781:
765:
761:
733:
729:
713:
707:
703:
701:
700:
697:
696:
686:
662:
657:
656:
654:
651:
650:
635:
629:
615:
609:
598:
579:
569:
556:
550:
544:
541:asymmetric norm
529:
514:
503:
493:
489:
470:
458:
452:
442:
411:
402:
396:
391:defined on the
381:
371:
361:
341:Élie Cartan
295:
269:
268:
252:
248:
239:
234:
213:
210:
209:
196:
187:
181:
170:
155:
143:, particularly
133:
122:
116:
113:
70:
68:
58:
46:
35:
28:
23:
22:
15:
12:
11:
5:
5550:
5540:
5539:
5534:
5529:
5524:
5519:
5502:
5501:
5499:
5498:
5493:
5488:
5483:
5477:
5475:
5471:
5470:
5468:
5467:
5465:Sub-Riemannian
5462:
5457:
5451:
5449:
5445:
5444:
5442:
5441:
5436:
5431:
5426:
5421:
5416:
5411:
5405:
5403:
5399:
5398:
5396:
5395:
5390:
5385:
5380:
5374:
5372:
5368:
5367:
5365:
5364:
5359:
5354:
5349:
5344:
5343:
5342:
5333:
5328:
5323:
5313:
5308:
5303:
5298:
5297:
5296:
5291:
5286:
5281:
5270:
5268:
5267:Basic concepts
5264:
5263:
5251:
5250:
5243:
5236:
5228:
5219:
5218:
5216:
5215:
5210:
5205:
5200:
5195:
5194:
5193:
5183:
5178:
5173:
5168:
5163:
5158:
5152:
5150:
5146:
5145:
5143:
5142:
5137:
5132:
5127:
5122:
5117:
5111:
5109:
5105:
5104:
5101:
5100:
5098:
5097:
5092:
5087:
5082:
5077:
5072:
5067:
5062:
5057:
5052:
5046:
5044:
5038:
5037:
5035:
5034:
5029:
5024:
5019:
5014:
5009:
5004:
4994:
4989:
4984:
4974:
4969:
4964:
4959:
4954:
4949:
4943:
4941:
4935:
4934:
4932:
4931:
4926:
4921:
4920:
4919:
4909:
4904:
4903:
4902:
4892:
4887:
4882:
4877:
4876:
4875:
4865:
4860:
4859:
4858:
4848:
4843:
4837:
4835:
4831:
4830:
4828:
4827:
4822:
4817:
4812:
4811:
4810:
4800:
4795:
4790:
4784:
4782:
4775:
4769:
4768:
4766:
4765:
4760:
4750:
4745:
4731:
4726:
4721:
4716:
4711:
4709:Parallelizable
4706:
4701:
4696:
4695:
4694:
4684:
4679:
4674:
4669:
4664:
4659:
4654:
4649:
4644:
4639:
4629:
4619:
4613:
4611:
4605:
4604:
4602:
4601:
4596:
4591:
4589:Lie derivative
4586:
4584:Integral curve
4581:
4576:
4571:
4570:
4569:
4559:
4554:
4553:
4552:
4545:Diffeomorphism
4542:
4536:
4534:
4528:
4527:
4525:
4524:
4519:
4514:
4509:
4504:
4499:
4494:
4489:
4484:
4478:
4476:
4467:
4466:
4464:
4463:
4458:
4453:
4448:
4443:
4438:
4433:
4428:
4423:
4422:
4421:
4416:
4406:
4405:
4404:
4393:
4391:
4390:Basic concepts
4387:
4386:
4374:
4373:
4366:
4359:
4351:
4345:
4344:
4339:
4319:
4318:External links
4316:
4315:
4314:
4300:
4279:
4265:
4241:
4224:
4192:
4167:
4153:
4128:
4115:
4103:, ed. (2003),
4095:
4092:
4091:
4090:
4084:
4078:
4069:
4061:
4058:
4055:
4054:
4033:(2): 195–199.
4012:
4011:
4009:
4006:
3890:
3887:
3863:
3862:
3851:
3848:
3844:
3840:
3837:
3834:
3831:
3828:
3825:
3822:
3819:
3813:
3810:
3803:
3795:
3792:
3786:
3782:
3779:
3776:
3773:
3770:
3763:
3760:
3754:
3746:
3743:
3737:
3718:
3717:
3706:
3701:
3696:
3691:
3685:
3679:
3676:
3671:
3664:
3661:
3656:
3653:
3649:
3646:
3643:
3640:
3637:
3634:
3631:
3627:
3623:
3620:
3617:
3614:
3611:
3608:
3605:
3602:
3599:
3595:
3591:
3588:
3585:
3582:
3491:
3490:
3479:
3474:
3470:
3464:
3460:
3455:
3451:
3448:
3445:
3442:
3439:
3431:
3427:
3423:
3416:
3413:
3409:
3405:
3399:
3396:
3393:
3390:
3387:
3384:
3376:
3372:
3368:
3361:
3358:
3354:
3350:
3344:
3340:
3336:
3333:
3330:
3327:
3324:
3319:
3316:
3312:
3306:
3303:
3298:
3295:
3292:
3289:
3286:
3283:
3278:
3274:
3255:
3254:
3243:
3238:
3235:
3232:
3229:
3226:
3221:
3212:
3208:
3204:
3200:
3195:
3192:
3189:
3186:
3183:
3178:
3174:
3170:
3166:
3162:
3155:
3152:
3149:
3146:
3143:
3138:
3129:
3125:
3121:
3117:
3110:
3106:
3101:
3097:
3092:
3089:
3086:
3083:
3080:
3075:
3072:
3069:
3043:integral curve
2987:
2975:
2952:
2951:
2940:
2936:
2930:
2925:
2917:
2913:
2909:
2905:
2900:
2896:
2891:
2886:
2878:
2874:
2870:
2866:
2861:
2856:
2850:
2846:
2842:
2839:
2836:
2833:
2830:
2827:
2822:
2819:
2815:
2799:
2785:
2784:
2773:
2770:
2767:
2764:
2761:
2758:
2753:
2746:
2743:
2736:
2733:
2730:
2725:
2718:
2715:
2707:
2701:
2696:
2693:
2690:
2684:
2681:
2675:
2672:
2669:
2666:
2663:
2658:
2648:
2644:
2640:
2633:
2630:
2626:
2622:
2614:
2611:
2606:
2601:
2596:
2593:
2590:
2584:
2581:
2575:
2572:
2569:
2566:
2563:
2558:
2548:
2544:
2540:
2533:
2530:
2526:
2522:
2515:
2511:
2508:
2505:
2502:
2497:
2490:
2487:
2478:
2473:
2470:
2467:
2461:
2458:
2452:
2449:
2446:
2443:
2440:
2435:
2428:
2425:
2421:
2377:
2374:
2328:
2327:
2316:
2313:
2308:
2304:
2301:
2298:
2292:
2289:
2283:
2280:
2277:
2274:
2271:
2267:
2261:
2257:
2251:
2246:
2242:
2236:
2233:
2228:
2225:
2222:
2219:
2216:
2178:
2144:
2143:
2132:
2129:
2124:
2120:
2117:
2114:
2108:
2105:
2099:
2096:
2093:
2090:
2087:
2083:
2079:
2074:
2069:
2065:
2061:
2058:
2055:
2052:
2049:
2030:
2027:
2016:
1996:
1995:
1984:
1980:
1973:
1970:
1967:
1964:
1961:
1958:
1952:
1946:
1943:
1940:
1937:
1934:
1931:
1925:
1919:
1916:
1913:
1910:
1907:
1904:
1901:
1898:
1895:
1890:
1886:
1882:
1879:
1872:
1865:
1862:
1857:
1853:
1850:
1847:
1841:
1838:
1832:
1829:
1826:
1823:
1820:
1816:
1812:
1807:
1802:
1798:
1793:
1786:
1782:
1779:
1776:
1773:
1770:
1767:
1764:
1759:
1755:
1723:
1685:
1684:
1673:
1668:
1664:
1661:
1658:
1655:
1652:
1649:
1646:
1643:
1640:
1637:
1634:
1631:
1623:
1620:
1617:
1614:
1610:
1606:
1603:
1600:
1597:
1594:
1591:
1588:
1566:
1565:
1546:
1545:
1544:
1533:
1530:
1527:
1524:
1521:
1518:
1515:
1512:
1509:
1506:
1503:
1500:
1497:
1494:
1491:
1488:
1485:
1482:
1479:
1476:
1473:
1470:
1467:
1464:
1461:
1458:
1455:
1452:
1449:
1446:
1443:
1440:
1435:
1432:
1349:
1346:
1329:
1326:
1323:
1320:
1317:
1302:Randers metric
1298:
1297:
1284:
1280:
1276:
1273:
1270:
1265:
1261:
1257:
1250:
1246:
1240:
1236:
1232:
1229:
1226:
1221:
1218:
1214:
1208:
1205:
1202:
1199:
1196:
1193:
1190:
1178:is used. Then
1163:
1158:
1155:
1151:
1147:
1136:inverse matrix
1122:
1117:
1114:
1110:
1106:
1094:
1093:
1082:
1079:
1076:
1069:
1065:
1059:
1055:
1049:
1046:
1042:
1036:
1031:
1027:
1023:
1020:
981:
978:
975:
972:
969:
957:
954:
953:
952:
942:
933:
930:
922:
921:
885:Minkowski norm
813:
812:
801:
796:
793:
790:
787:
784:
779:
774:
768:
764:
760:
757:
754:
751:
748:
745:
742:
739:
736:
732:
725:
722:
719:
716:
710:
706:
699:
693:
690:
685:
682:
679:
676:
673:
670:
665:
660:
626:
625:
606:Hessian matrix
587:
586:
546:
526:
525:
501:
468:
393:tangent bundle
378:Finsler metric
360:
357:
325:inner products
317:
316:
305:
302:
298:
292:
288:
285:
282:
276:
273:
267:
264:
261:
258:
255:
251:
247:
242:
237:
233:
229:
226:
223:
220:
217:
183:
167:Minkowski norm
135:
134:
49:
47:
40:
26:
9:
6:
4:
3:
2:
5549:
5538:
5535:
5533:
5530:
5528:
5525:
5523:
5520:
5518:
5515:
5514:
5512:
5497:
5494:
5492:
5489:
5487:
5484:
5482:
5479:
5478:
5476:
5472:
5466:
5463:
5461:
5458:
5456:
5453:
5452:
5450:
5446:
5440:
5439:Schur's lemma
5437:
5435:
5432:
5430:
5427:
5425:
5422:
5420:
5417:
5415:
5414:Gauss's lemma
5412:
5410:
5407:
5406:
5404:
5400:
5394:
5391:
5389:
5386:
5384:
5381:
5379:
5376:
5375:
5373:
5369:
5363:
5360:
5358:
5355:
5353:
5350:
5348:
5345:
5341:
5337:
5334:
5332:
5329:
5327:
5324:
5322:
5319:
5318:
5317:
5316:Metric tensor
5314:
5312:
5311:Inner product
5309:
5307:
5304:
5302:
5299:
5295:
5292:
5290:
5287:
5285:
5282:
5280:
5277:
5276:
5275:
5272:
5271:
5269:
5265:
5260:
5256:
5249:
5244:
5242:
5237:
5235:
5230:
5229:
5226:
5214:
5211:
5209:
5208:Supermanifold
5206:
5204:
5201:
5199:
5196:
5192:
5189:
5188:
5187:
5184:
5182:
5179:
5177:
5174:
5172:
5169:
5167:
5164:
5162:
5159:
5157:
5154:
5153:
5151:
5147:
5141:
5138:
5136:
5133:
5131:
5128:
5126:
5123:
5121:
5118:
5116:
5113:
5112:
5110:
5106:
5096:
5093:
5091:
5088:
5086:
5083:
5081:
5078:
5076:
5073:
5071:
5068:
5066:
5063:
5061:
5058:
5056:
5053:
5051:
5048:
5047:
5045:
5043:
5039:
5033:
5030:
5028:
5025:
5023:
5020:
5018:
5015:
5013:
5010:
5008:
5005:
5003:
4999:
4995:
4993:
4990:
4988:
4985:
4983:
4979:
4975:
4973:
4970:
4968:
4965:
4963:
4960:
4958:
4955:
4953:
4950:
4948:
4945:
4944:
4942:
4940:
4936:
4930:
4929:Wedge product
4927:
4925:
4922:
4918:
4915:
4914:
4913:
4910:
4908:
4905:
4901:
4898:
4897:
4896:
4893:
4891:
4888:
4886:
4883:
4881:
4878:
4874:
4873:Vector-valued
4871:
4870:
4869:
4866:
4864:
4861:
4857:
4854:
4853:
4852:
4849:
4847:
4844:
4842:
4839:
4838:
4836:
4832:
4826:
4823:
4821:
4818:
4816:
4813:
4809:
4806:
4805:
4804:
4803:Tangent space
4801:
4799:
4796:
4794:
4791:
4789:
4786:
4785:
4783:
4779:
4776:
4774:
4770:
4764:
4761:
4759:
4755:
4751:
4749:
4746:
4744:
4740:
4736:
4732:
4730:
4727:
4725:
4722:
4720:
4717:
4715:
4712:
4710:
4707:
4705:
4702:
4700:
4697:
4693:
4690:
4689:
4688:
4685:
4683:
4680:
4678:
4675:
4673:
4670:
4668:
4665:
4663:
4660:
4658:
4655:
4653:
4650:
4648:
4645:
4643:
4640:
4638:
4634:
4630:
4628:
4624:
4620:
4618:
4615:
4614:
4612:
4606:
4600:
4597:
4595:
4592:
4590:
4587:
4585:
4582:
4580:
4577:
4575:
4572:
4568:
4567:in Lie theory
4565:
4564:
4563:
4560:
4558:
4555:
4551:
4548:
4547:
4546:
4543:
4541:
4538:
4537:
4535:
4533:
4529:
4523:
4520:
4518:
4515:
4513:
4510:
4508:
4505:
4503:
4500:
4498:
4495:
4493:
4490:
4488:
4485:
4483:
4480:
4479:
4477:
4474:
4470:Main results
4468:
4462:
4459:
4457:
4454:
4452:
4451:Tangent space
4449:
4447:
4444:
4442:
4439:
4437:
4434:
4432:
4429:
4427:
4424:
4420:
4417:
4415:
4412:
4411:
4410:
4407:
4403:
4400:
4399:
4398:
4395:
4394:
4392:
4388:
4383:
4379:
4372:
4367:
4365:
4360:
4358:
4353:
4352:
4349:
4343:
4340:
4336:
4332:
4331:
4326:
4322:
4321:
4311:
4307:
4303:
4301:981-02-4531-9
4297:
4293:
4289:
4285:
4280:
4276:
4272:
4268:
4262:
4258:
4254:
4250:
4246:
4242:
4238:
4234:
4230:
4225:
4222:
4218:
4215:(9): 959–63,
4214:
4210:
4209:
4201:
4197:
4193:
4190:
4186:
4182:
4178:
4177:
4172:
4168:
4164:
4160:
4156:
4154:0-387-98948-X
4150:
4146:
4142:
4138:
4134:
4129:
4126:
4122:
4118:
4112:
4108:
4107:
4102:
4098:
4097:
4088:
4085:
4082:
4079:
4073:
4070:
4067:
4064:
4063:
4049:
4044:
4040:
4036:
4032:
4029:
4028:
4023:
4017:
4013:
4005:
4003:
3999:
3995:
3991:
3987:
3983:
3979:
3975:
3971:
3967:
3963:
3959:
3955:
3951:
3947:
3942:
3940:
3936:
3932:
3928:
3924:
3920:
3916:
3912:
3908:
3904:
3900:
3896:
3886:
3884:
3880:
3876:
3872:
3868:
3849:
3846:
3842:
3835:
3829:
3826:
3820:
3811:
3808:
3801:
3793:
3790:
3784:
3780:
3774:
3768:
3761:
3758:
3752:
3744:
3741:
3735:
3727:
3726:
3725:
3723:
3704:
3694:
3689:
3677:
3674:
3662:
3659:
3654:
3651:
3647:
3638:
3629:
3618:
3606:
3597:
3586:
3583:
3580:
3573:
3572:
3571:
3570:
3565:
3562:∖{0} →
3561:
3556:
3552:
3548:
3544:
3540:
3536:
3532:
3528:
3524:
3520:
3516:
3512:
3508:
3505: =
3504:
3500:
3496:
3477:
3472:
3468:
3462:
3458:
3453:
3446:
3443:
3440:
3429:
3425:
3414:
3411:
3407:
3397:
3391:
3388:
3385:
3374:
3370:
3359:
3356:
3352:
3342:
3338:
3331:
3328:
3325:
3317:
3314:
3310:
3304:
3301:
3296:
3290:
3287:
3284:
3276:
3272:
3264:
3263:
3262:
3261:are given by
3260:
3241:
3233:
3230:
3227:
3219:
3210:
3206:
3190:
3187:
3184:
3176:
3172:
3168:
3160:
3150:
3147:
3144:
3136:
3127:
3123:
3108:
3104:
3095:
3087:
3084:
3081:
3073:
3070:
3059:
3058:
3057:
3055:
3051:
3048:
3044:
3038:
3034:
3029:
3025:
3020:
3016:
3011:
3007:
3003:
2999:
2995:
2990:
2986:
2983:, the matrix
2982:
2978:
2973:
2969:
2965:
2961:
2957:
2954:Assuming the
2938:
2934:
2928:
2923:
2915:
2911:
2894:
2889:
2884:
2876:
2872:
2854:
2848:
2844:
2840:
2834:
2831:
2828:
2820:
2817:
2813:
2805:
2804:
2803:
2798:
2794:
2790:
2771:
2768:
2765:
2759:
2751:
2744:
2741:
2731:
2723:
2716:
2713:
2705:
2691:
2682:
2679:
2673:
2667:
2661:
2646:
2642:
2631:
2628:
2624:
2612:
2609:
2604:
2591:
2582:
2579:
2573:
2567:
2561:
2546:
2542:
2531:
2528:
2524:
2513:
2509:
2503:
2495:
2488:
2485:
2468:
2459:
2456:
2450:
2444:
2438:
2426:
2423:
2419:
2411:
2410:
2409:
2407:
2403:
2399:
2395:
2391:
2387:
2383:
2373:
2370:
2366:
2362:
2356:
2352:
2348:
2342:
2338:
2333:
2314:
2311:
2306:
2299:
2290:
2287:
2281:
2275:
2269:
2265:
2259:
2255:
2249:
2244:
2240:
2234:
2231:
2226:
2220:
2214:
2207:
2206:
2205:
2203:
2199:
2195:
2191:
2187:
2183:
2176:
2172:
2168:
2164:
2160:
2156:
2152:
2149:
2130:
2127:
2122:
2115:
2106:
2103:
2097:
2091:
2085:
2081:
2077:
2072:
2067:
2063:
2059:
2053:
2047:
2040:
2039:
2038:
2036:
2026:
2024:
2019:
2015:
2012:
2009:
2005:
2001:
1982:
1978:
1971:
1968:
1962:
1956:
1950:
1944:
1941:
1935:
1929:
1923:
1914:
1911:
1905:
1902:
1899:
1888:
1884:
1880:
1877:
1870:
1863:
1860:
1855:
1848:
1839:
1836:
1830:
1824:
1818:
1814:
1810:
1805:
1800:
1796:
1784:
1777:
1771:
1768:
1765:
1757:
1753:
1745:
1744:
1743:
1741:
1735:
1731:
1726:
1722:
1718:
1714:
1710:
1706:
1702:
1698:
1694:
1690:
1671:
1666:
1656:
1650:
1647:
1641:
1635:
1629:
1621:
1618:
1612:
1604:
1598:
1595:
1592:
1586:
1579:
1578:
1577:
1575:
1571:
1563:
1559:
1556: ×
1555:
1551:
1548:The function
1547:
1531:
1522:
1516:
1513:
1507:
1501:
1495:
1492:
1486:
1483:
1480:
1474:
1471:
1462:
1456:
1453:
1447:
1441:
1433:
1430:
1421:
1420:
1419:
1416: ∈
1415:
1411:
1407:
1403:
1400:, φ) of
1399:
1395:
1391:
1387:
1386:
1385:
1383:
1379:
1375:
1371:
1367:
1363:
1359:
1355:
1345:
1343:
1324:
1321:
1318:
1307:
1303:
1282:
1278:
1271:
1263:
1259:
1255:
1248:
1244:
1238:
1234:
1227:
1219:
1216:
1212:
1206:
1200:
1197:
1194:
1188:
1181:
1180:
1179:
1177:
1156:
1153:
1149:
1137:
1120:
1115:
1112:
1108:
1104:
1080:
1077:
1074:
1067:
1063:
1057:
1053:
1047:
1044:
1040:
1034:
1029:
1021:
1011:
1010:
1009:
1007:
1003:
999:
995:
976:
973:
970:
950:
946:
943:
940:
936:
935:
929:
927:
919:
913:
909:
905:
901:
897:
896:
895:
893:
888:
886:
881:
872:
868:
862:
853:
849:
843:
835:
829:
823:
818:
799:
794:
791:
788:
785:
782:
777:
772:
766:
758:
755:
752:
749:
746:
743:
740:
734:
730:
723:
717:
708:
691:
688:
683:
677:
674:
671:
663:
649:
648:
647:
646:
645:bilinear form
643:
638:
632:
623:
618:
612:
607:
601:
596:
595:
594:
592:
583:
577:
572:
568:
567:
566:
564:
559:
553:
549:
542:
536:
532:
523:
517:
510:
506:
502:
499:
485:
481:
477:
473:
469:
466:
465:subadditivity
461:
455:
449:
445:
438:
434:
430:
426:
422:
418:
414:
410:
409:
408:
405:
399:
394:
388:
384:
379:
374:
370:
366:
356:
354:
350:
346:
342:
338:
336:
333:
328:
326:
322:
303:
300:
290:
283:
274:
271:
265:
259:
253:
249:
245:
240:
235:
231:
227:
221:
215:
208:
207:
206:
203:
199:
195:
190:
186:
177:
173:
169:
168:
163:
158:
154:
150:
146:
142:
131:
128:
120:
109:
106:
102:
99:
95:
92:
88:
85:
81:
78: –
77:
73:
72:Find sources:
66:
62:
56:
55:
50:This article
48:
44:
39:
38:
33:
19:
18:Finsler space
5474:Applications
5454:
5402:Main results
5135:Moving frame
5130:Morse theory
5120:Gauge theory
4912:Tensor field
4841:Closed/Exact
4820:Vector field
4788:Distribution
4729:Hypercomplex
4724:Quaternionic
4646:
4461:Vector field
4419:Smooth atlas
4328:
4292:10.1142/4619
4283:
4248:
4228:
4212:
4206:
4180:
4174:
4171:Cartan, Élie
4136:
4131:Bao, David;
4105:
4030:
4025:
4016:
4000:, as in the
3997:
3993:
3989:
3985:
3981:
3977:
3973:
3969:
3961:
3957:
3953:
3949:
3945:
3943:
3934:
3930:
3926:
3922:
3918:
3914:
3910:
3906:
3902:
3898:
3892:
3874:
3870:
3864:
3719:
3567:through the
3563:
3559:
3555:fibre bundle
3546:
3545:. The spray
3542:
3534:
3530:
3518:
3514:
3510:
3506:
3502:
3498:
3494:
3492:
3258:
3256:
3053:
3049:
3036:
3032:
3027:
3023:
3018:
3014:
3009:
3005:
3001:
2997:
2993:
2988:
2984:
2980:
2976:
2971:
2967:
2963:
2959:
2953:
2796:
2792:
2788:
2786:
2405:
2401:
2397:
2393:
2389:
2385:
2379:
2368:
2364:
2360:
2354:
2350:
2346:
2340:
2336:
2329:
2201:
2197:
2193:
2189:
2185:
2181:
2174:
2166:
2158:
2154:
2150:
2145:
2034:
2032:
2022:
2017:
2013:
2003:
1999:
1997:
1733:
1729:
1724:
1720:
1712:
1708:
1704:
1700:
1696:
1692:
1688:
1686:
1573:
1569:
1567:
1557:
1553:
1549:
1417:
1413:
1409:
1405:
1401:
1397:
1393:
1389:
1381:
1373:
1365:
1357:
1353:
1351:
1341:
1305:
1301:
1299:
1095:
1005:
997:
959:
923:
917:
911:
907:
903:
899:
891:
889:
884:
879:
870:
866:
860:
851:
847:
841:
833:
827:
821:
816:
814:
636:
630:
627:
616:
610:
599:
590:
588:
581:
570:
562:
557:
551:
547:
534:
530:
527:
515:
508:
504:
483:
479:
475:
471:
459:
453:
447:
443:
436:
432:
428:
424:
420:
416:
412:
403:
397:
386:
382:
377:
372:
364:
362:
353:Finsler 1918
349:Paul Finsler
339:
329:
318:
201:
197:
194:smooth curve
188:
184:
175:
171:
165:
156:
148:
138:
123:
114:
104:
97:
90:
83:
71:
59:Please help
54:verification
51:
32:Paul Finsler
5080:Levi-Civita
5070:Generalized
5042:Connections
4992:Lie algebra
4924:Volume form
4825:Vector flow
4798:Pushforward
4793:Lie bracket
4692:Lie algebra
4657:G-structure
4446:Pushforward
4426:Submanifold
4245:Rund, Hanno
4183:: 582–586,
4022:Randers, G.
3039:∖{0}
3017:: →
2339:: →
2037:the length
2011:quasimetric
1740:quasimetric
1576: → by
1362:quasimetric
451:tangent to
200: : →
141:mathematics
5511:Categories
5434:Ricci flow
5383:Hyperbolic
5203:Stratifold
5161:Diffeology
4957:Associated
4758:Symplectic
4743:Riemannian
4672:Hyperbolic
4599:Submersion
4507:Hopf–Rinow
4441:Submersion
4436:Smooth map
4237:46.1131.02
4189:0006.22501
4094:References
4027:Phys. Rev.
4002:Riemannian
3976:, and for
3972:(0) along
3722:Riemannian
3549:defines a
2791:= 1, ...,
1368:is also a
1300:defines a
892:reversible
359:Definition
162:asymmetric
87:newspapers
5378:Hermitian
5331:Signature
5294:Sectional
5274:Curvature
5085:Principal
5060:Ehresmann
5017:Subbundle
5007:Principal
4982:Fibration
4962:Cotangent
4834:Covectors
4687:Lie group
4667:Hermitian
4610:manifolds
4579:Immersion
4574:Foliation
4512:Noether's
4497:Frobenius
4492:De Rham's
4487:Darboux's
4378:Manifolds
4335:EMS Press
3966:conjugate
3812:˙
3809:γ
3794:˙
3791:γ
3762:˙
3759:γ
3745:˙
3742:γ
3633:∖
3616:→
3601:∖
3473:ℓ
3422:∂
3415:ℓ
3404:∂
3398:−
3375:ℓ
3367:∂
3349:∂
3203:∂
3199:∂
3161:−
3120:∂
3116:∂
2974:∈ T
2908:∂
2904:∂
2869:∂
2865:∂
2745:˙
2742:γ
2717:˙
2714:γ
2683:˙
2680:γ
2662:γ
2639:∂
2621:∂
2605:−
2583:˙
2580:γ
2562:γ
2539:∂
2521:∂
2489:¨
2486:γ
2460:˙
2457:γ
2439:γ
2291:˙
2288:γ
2270:γ
2241:∫
2221:γ
2107:˙
2104:γ
2086:γ
2064:∫
2054:γ
2029:Geodesics
2008:intrinsic
1957:γ
1930:γ
1881:∈
1878:γ
1840:˙
1837:γ
1819:γ
1797:∫
1651:γ
1636:γ
1616:→
1529:‖
1517:ϕ
1514:−
1502:ϕ
1499:‖
1493:≤
1472:≤
1469:‖
1457:ϕ
1454:−
1442:ϕ
1439:‖
1026:‖
1019:‖
947:(but not
721:∂
715:∂
705:∂
642:symmetric
494:λ < 0)
389:→ [0, +∞)
332:intrinsic
275:˙
272:γ
254:γ
232:∫
222:γ
5393:Kenmotsu
5306:Geodesic
5259:Glossary
5181:Orbifold
5176:K-theory
5166:Diffiety
4890:Pullback
4704:Oriented
4682:Kenmotsu
4662:Hadamard
4608:Types of
4557:Geodesic
4382:Glossary
4247:(1959).
4198:(1996),
4060:See also
3541:on
3525:and the
3521:are the
3513:, where
3012:). Then
2171:geodesic
2002:: T
1364:so that
1174:and the
932:Examples
511:) > 0
488:for all
117:May 2017
5460:Hilbert
5455:Finsler
5125:History
5108:Related
5022:Tangent
5000:)
4980:)
4947:Adjoint
4939:Bundles
4917:density
4815:Torsion
4781:Vectors
4773:Tensors
4756:)
4741:)
4737:,
4735:Pseudo−
4714:Poisson
4647:Finsler
4642:Fibered
4637:Contact
4635:)
4627:Complex
4625:)
4594:Section
4337:, 2001
4310:1845637
4275:0105726
4221:1400859
4163:1747675
4125:2067663
3996:) near
3988:(0) to
3929:,
3901:,
3865:of the
3553:on the
3045:of the
2400:, ...,
2392:, ...,
1572::
1552::
1412:,
1360:) be a
1134:is the
864:⁄
845:⁄
640:is the
513:unless
343: (
101:scholar
5388:Kähler
5284:Scalar
5279:tensor
5090:Vector
5075:Koszul
5055:Cartan
5050:Affine
5032:Vector
5027:Tensor
5012:Spinor
5002:Normal
4998:Stable
4952:Affine
4856:bundle
4808:bundle
4754:Almost
4677:Kähler
4633:Almost
4623:Almost
4617:Closed
4517:Sard's
4473:(list)
4308:
4298:
4273:
4263:
4235:
4219:
4187:
4161:
4151:
4123:
4113:
4004:case.
3041:is an
3035:: → T
2787:where
2404:) of T
1975:
1954:
1948:
1927:
1921:
1875:
1867:
1789:
1715:. The
1687:where
1562:smooth
1096:where
604:, the
576:smooth
563:smooth
539:is an
103:
96:
89:
82:
74:
5289:Ricci
5198:Sheaf
4972:Fiber
4748:Rizza
4719:Prime
4550:Local
4540:Curve
4402:Atlas
4203:(PDF)
4008:Notes
3917:with
3539:spray
3537:is a
2192:) to
2184:from
2169:is a
2153:: →
2146:of a
1695:with
1352:Let (
1340:is a
1008:with
992:be a
877:. If
490:λ ≥ 0
478:) = λ
367:is a
151:is a
108:JSTOR
94:books
5065:Form
4967:Dual
4900:flow
4763:Tame
4739:Sub−
4652:Flat
4532:Maps
4296:ISBN
4261:ISBN
4149:ISBN
4111:ISBN
3881:and
3529:on T
3517:and
3497:on T
3052:on T
2795:and
2380:The
2367:) =
2358:and
2353:) =
1703:and
1372:and
1308:and
1075:<
996:and
960:Let
926:norm
906:) =
537:, −)
431:) +
423:) ≤
345:1933
178:, −)
147:, a
80:news
4987:Jet
4288:doi
4253:doi
4233:JFM
4185:Zbl
4181:196
4141:doi
4043:hdl
4035:doi
3968:to
3944:If
3893:By
2958:of
2408:as
2157:in
2021:on
1781:inf
1609:lim
1392:on
1380:of
1304:on
1138:of
1004:on
825:at
819:of
634:at
620:is
614:at
608:of
602:≠ 0
574:is
518:= 0
457:at
401:of
385:: T
355:).
205:as
139:In
63:by
5513::
4978:Co
4333:,
4327:,
4306:MR
4304:.
4294:.
4271:MR
4269:.
4259:.
4217:MR
4213:43
4211:,
4205:,
4179:,
4159:MR
4157:.
4147:.
4121:MR
4119:,
4041:.
4031:59
3964:)
3941:.
3923:γ'
3885:.
3873:,
3655::=
3503:JH
3297::=
3096::=
3033:γ'
3026:,
3008:,
2996:,
2989:ij
2966:,
2841::=
2800:ij
2396:,
2372:.
2227::=
2060::=
1778::=
1736:→
1732:×
1728::
1705:γ'
1605::=
1574:TM
1356:,
1207::=
1035::=
1000:a
902:(−
857:≠
684::=
593::
524:).
500:).
474:(λ
467:).
419:+
407:,
363:A
327:.
164:)
5338:/
5261:)
5257:(
5247:e
5240:t
5233:v
4996:(
4976:(
4752:(
4733:(
4631:(
4621:(
4384:)
4380:(
4370:e
4363:t
4356:v
4312:.
4290::
4277:.
4255::
4165:.
4143::
4051:.
4045::
4037::
3998:γ
3994:t
3992:(
3990:γ
3986:γ
3982:s
3978:t
3974:γ
3970:γ
3962:s
3960:(
3958:γ
3954:M
3950:γ
3946:F
3935:M
3931:v
3927:x
3919:γ
3915:γ
3911:F
3907:E
3903:F
3899:M
3875:H
3871:M
3850:0
3847:=
3843:)
3839:)
3836:t
3833:(
3830:X
3827:,
3824:)
3821:t
3818:(
3802:(
3785:R
3781:+
3778:)
3775:t
3772:(
3769:X
3753:D
3736:D
3705:.
3700:)
3695:J
3690:H
3684:L
3678:+
3675:I
3670:(
3663:2
3660:1
3652:v
3648:;
3645:)
3642:}
3639:0
3636:{
3630:M
3626:T
3622:(
3619:T
3613:)
3610:}
3607:0
3604:{
3598:M
3594:T
3590:(
3587:T
3584::
3581:v
3564:M
3560:M
3558:T
3547:H
3543:M
3535:H
3531:M
3519:V
3515:J
3511:H
3507:V
3499:M
3495:H
3478:.
3469:v
3463:k
3459:v
3454:)
3450:)
3447:v
3444:,
3441:x
3438:(
3430:j
3426:x
3412:k
3408:g
3395:)
3392:v
3389:,
3386:x
3383:(
3371:x
3360:k
3357:j
3353:g
3343:2
3339:(
3335:)
3332:v
3329:,
3326:x
3323:(
3318:j
3315:i
3311:g
3305:4
3302:1
3294:)
3291:v
3288:,
3285:x
3282:(
3277:i
3273:G
3259:G
3242:,
3237:)
3234:v
3231:,
3228:x
3225:(
3220:|
3211:i
3207:v
3194:)
3191:v
3188:,
3185:x
3182:(
3177:i
3173:G
3169:2
3154:)
3151:v
3148:,
3145:x
3142:(
3137:|
3128:i
3124:x
3109:i
3105:v
3091:)
3088:v
3085:,
3082:x
3079:(
3074:|
3071:H
3054:M
3050:H
3037:M
3028:F
3024:M
3019:M
3015:γ
3010:v
3006:x
3004:(
3002:g
2998:v
2994:x
2992:(
2985:g
2981:M
2977:x
2972:v
2968:v
2964:x
2962:(
2960:F
2939:.
2935:)
2929:x
2924:|
2916:j
2912:x
2895:,
2890:x
2885:|
2877:i
2873:x
2855:(
2849:v
2845:g
2838:)
2835:v
2832:,
2829:x
2826:(
2821:j
2818:i
2814:g
2797:g
2793:n
2789:k
2772:,
2769:0
2766:=
2763:)
2760:t
2757:(
2752:j
2735:)
2732:t
2729:(
2724:i
2706:)
2700:)
2695:)
2692:t
2689:(
2674:,
2671:)
2668:t
2665:(
2657:(
2647:k
2643:x
2632:j
2629:i
2625:g
2613:2
2610:1
2600:)
2595:)
2592:t
2589:(
2574:,
2571:)
2568:t
2565:(
2557:(
2547:j
2543:x
2532:k
2529:i
2525:g
2514:(
2510:+
2507:)
2504:t
2501:(
2496:i
2477:)
2472:)
2469:t
2466:(
2451:,
2448:)
2445:t
2442:(
2434:(
2427:k
2424:i
2420:g
2406:M
2402:v
2398:v
2394:x
2390:x
2386:E
2369:y
2365:b
2363:(
2361:γ
2355:x
2351:a
2349:(
2347:γ
2341:M
2337:γ
2315:t
2312:d
2307:)
2303:)
2300:t
2297:(
2282:,
2279:)
2276:t
2273:(
2266:(
2260:2
2256:F
2250:b
2245:a
2235:2
2232:1
2224:]
2218:[
2215:E
2202:γ
2198:d
2196:(
2194:γ
2190:c
2188:(
2186:γ
2182:M
2177:|
2175:γ
2167:γ
2159:M
2155:M
2151:γ
2131:t
2128:d
2123:)
2119:)
2116:t
2113:(
2098:,
2095:)
2092:t
2089:(
2082:(
2078:F
2073:b
2068:a
2057:]
2051:[
2048:L
2035:F
2023:M
2018:L
2014:d
2004:M
2000:F
1983:,
1979:}
1972:y
1969:=
1966:)
1963:1
1960:(
1951:,
1945:x
1942:=
1939:)
1936:0
1933:(
1924:,
1918:)
1915:M
1912:,
1909:]
1906:1
1903:,
1900:0
1897:[
1894:(
1889:1
1885:C
1871:|
1864:t
1861:d
1856:)
1852:)
1849:t
1846:(
1831:,
1828:)
1825:t
1822:(
1815:(
1811:F
1806:1
1801:0
1785:{
1775:)
1772:y
1769:,
1766:x
1763:(
1758:L
1754:d
1734:M
1730:M
1725:L
1721:d
1713:M
1709:F
1701:x
1697:γ
1693:M
1689:γ
1672:,
1667:t
1663:)
1660:)
1657:t
1654:(
1648:,
1645:)
1642:0
1639:(
1633:(
1630:d
1622:+
1619:0
1613:t
1602:)
1599:v
1596:,
1593:x
1590:(
1587:F
1570:F
1558:M
1554:M
1550:d
1532:.
1526:)
1523:x
1520:(
1511:)
1508:y
1505:(
1496:C
1490:)
1487:y
1484:,
1481:x
1478:(
1475:d
1466:)
1463:x
1460:(
1451:)
1448:y
1445:(
1434:C
1431:1
1418:U
1414:y
1410:x
1406:C
1402:M
1398:U
1394:M
1390:z
1382:M
1374:d
1366:M
1358:d
1354:M
1328:)
1325:F
1322:,
1319:M
1316:(
1306:M
1283:i
1279:v
1275:)
1272:x
1269:(
1264:i
1260:b
1256:+
1249:j
1245:v
1239:i
1235:v
1231:)
1228:x
1225:(
1220:j
1217:i
1213:a
1204:)
1201:v
1198:,
1195:x
1192:(
1189:F
1162:)
1157:j
1154:i
1150:a
1146:(
1121:)
1116:j
1113:i
1109:a
1105:(
1081:,
1078:1
1068:j
1064:b
1058:i
1054:b
1048:j
1045:i
1041:a
1030:a
1022:b
1006:M
998:b
980:)
977:a
974:,
971:M
968:(
920:.
918:v
914:)
912:v
910:(
908:F
904:v
900:F
880:F
873:)
871:v
869:(
867:F
861:v
854:)
852:u
850:(
848:F
842:u
834:F
828:v
822:F
800:,
795:0
792:=
789:t
786:=
783:s
778:|
773:]
767:2
763:)
759:Y
756:t
753:+
750:X
747:s
744:+
741:v
738:(
735:F
731:[
724:t
718:s
709:2
692:2
689:1
681:)
678:Y
675:,
672:X
669:(
664:v
659:g
637:v
631:F
624:.
617:v
611:F
600:v
585:.
582:M
580:T
571:F
558:F
552:M
548:x
545:T
535:x
533:(
531:F
520:(
516:v
509:v
507:(
505:F
496:(
486:)
484:v
482:(
480:F
476:v
472:F
463:(
460:x
454:M
448:w
446:,
444:v
439:)
437:w
435:(
433:F
429:v
427:(
425:F
421:w
417:v
415:(
413:F
404:M
398:x
387:M
383:F
373:M
304:.
301:t
297:d
291:)
287:)
284:t
281:(
266:,
263:)
260:t
257:(
250:(
246:F
241:b
236:a
228:=
225:)
219:(
216:L
202:M
198:γ
189:M
185:x
182:T
176:x
174:(
172:F
157:M
130:)
124:(
119:)
115:(
105:·
98:·
91:·
84:·
57:.
34:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
