42:
1300:
3755:
3501:
1471:
In all of the following theorems we assume some local behavior of the space (usually formulated using curvature assumption) to derive some information about the global structure of the space, including either some information on the topological type of the manifold or on the behavior of points at
1397:
What follows is an incomplete list of the most classical theorems in
Riemannian geometry. The choice is made depending on its importance and elegance of formulation. Most of the results can be found in the classic monograph by
1929:-dimensional Riemannian manifold has nonnegative Ricci curvature and a straight line (i.e. a geodesic that minimizes distance on each interval) then it is isometric to a direct product of the real line and a complete (
1405:
The formulations given are far from being very exact or the most general. This list is oriented to those who already know the basic definitions and want to know what these definitions are about.
3204:
2052:
1769:
at any point. It implies that any two points of a simply connected complete
Riemannian manifold with nonpositive sectional curvature are joined by a unique geodesic.
3715:
2396:
3251:
3570:
3535:
3486:
3275:
3021:
2559:
2409:
2022:
1386:
1170:
1958:
3575:
2157:
Joachim
Lohkamp has shown (Annals of Mathematics, 1994) that any manifold of dimension greater than two admits a metric of negative Ricci curvature.
2269:
From
Riemann to Differential Geometry and Relativity (Lizhen Ji, Athanase Papadopoulos, and Sumio Yamada, Eds.) Springer, 2017, XXXIV, 647 p.
1307:
Riemannian geometry was first put forward in generality by
Bernhard Riemann in the 19th century. It deals with a broad range of geometries whose
3057:
3011:
2589:
2564:
3195:
2451:
2274:
1766:
270:
2679:
1381:
3690:
3303:
3680:
3665:
3585:
2932:
1255:. Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of
3725:
2937:
2424:
2389:
2259:
2240:
2191:
1246:
236:
3740:
3257:
1950:-dimensional Riemannian manifold with positive Ricci curvature has volume at most that of the volume of a ball of the same radius
1436:
2970:
3735:
3730:
3705:
3560:
3528:
3073:
1163:
1117:
723:
182:
3151:
3214:
3504:
3246:
3081:
3555:
1937:
3781:
3670:
3068:
2486:
2382:
2346:
2073:
1972:
1268:
3695:
2828:
2067:
1734:
3109:
2843:
2350:
1341:
There exists a close analogy of differential geometry with the mathematical structure of defects in regular crystals.
3720:
3700:
3396:
2838:
2791:
2507:
1327:
1138:
748:
3786:
3759:
3521:
3426:
3052:
2733:
2636:
1156:
3710:
2569:
1415:
3600:
3565:
3451:
3001:
2761:
2672:
2429:
1828:
1376:
1280:
125:
3180:
3102:
3620:
2722:
2713:
2615:
1842:
1835:
1260:
1250:
551:
231:
88:
3269:
3175:
2574:
3635:
3406:
3133:
2807:
2771:
2646:
2641:
2579:
2512:
2471:
1864:
1443:
1366:
1312:
1245:" ("On the Hypotheses on which Geometry is Based"). It is a very broad and abstract generalization of the
627:
338:
216:
101:
2112:
1787:
is a complete
Riemannian manifold with sectional curvature bounded above by a strictly negative constant
3660:
3411:
3016:
2955:
2047:
1323:
1288:
1276:
1187:
399:
360:
319:
314:
167:
1299:
1016:
763:
3316:
3242:
3086:
2688:
2476:
2313:
2141:
1903:
1544:
1428:
1308:
1067:
990:
838:
743:
265:
160:
74:
3595:
3580:
3391:
3386:
3381:
3376:
3208:
3157:
2975:
2786:
2717:
2665:
2533:
2502:
2490:
2461:
2444:
1477:
1448:
1419:
The integral of the Gauss curvature on a compact 2-dimensional
Riemannian manifold is equal to 2ÏÏ(
1361:
1191:
1072:
929:
783:
688:
578:
449:
439:
302:
177:
155:
130:
118:
70:
65:
46:
3280:
1498:-dimensional Riemannian manifold with sectional curvature strictly pinched between 1/4 and 1 then
41:
3605:
3456:
3356:
3138:
3119:
3113:
3064:
3006:
2915:
2833:
2751:
2728:
2696:
2631:
2528:
2497:
2329:
2303:
2132:
2057:
1331:
1284:
1031:
758:
598:
226:
150:
140:
111:
96:
2538:
3675:
3615:
3466:
3336:
3298:
3290:
2891:
2848:
2543:
2357:
2270:
2255:
2236:
2187:
2062:
1920:
1896:
1853:
1800:
1435:. This theorem has a generalization to any compact even-dimensional Riemannian manifold, see
1319:
1102:
1092:
1021:
890:
868:
818:
793:
728:
652:
307:
199:
145:
106:
3610:
3590:
3544:
3431:
3371:
3361:
3308:
3285:
2863:
2610:
2605:
2481:
2434:
2321:
2096:
1815:
1739:
1335:
1238:
1082:
823:
533:
411:
346:
204:
189:
54:
3655:
3630:
3446:
3421:
3346:
3341:
3224:
3185:
3147:
3091:
2965:
2901:
2439:
2218:
1849:
1751:
1456:
1264:
1211:
505:
378:
368:
211:
194:
135:
3229:
1077:
1046:
980:
950:
828:
773:
768:
708:
2361:
2317:
2186:, University Lecture Series, vol. 17, Rhode Island: American Mathematical Society,
2145:
3645:
3640:
3471:
3143:
3127:
3123:
3026:
2996:
2853:
2756:
2291:
2180:
2128:
2108:
2078:
1990:
1963:
The set of all
Riemannian manifolds with positive Ricci curvature and diameter at most
1676:-dimensional Riemannian manifold with positive sectional curvature then the sum of its
1485:
1133:
1107:
1041:
985:
858:
738:
718:
698:
603:
3775:
3461:
3441:
3436:
3351:
3219:
3047:
2991:
2823:
2776:
2466:
2287:
2175:
2029:-dimensional Riemannian manifold is â„ Ï then the average scalar curvature is at most
1890:
1860:
1773:
1747:
1631:
1356:
1207:
1203:
1112:
1097:
1026:
843:
803:
753:
528:
491:
458:
296:
292:
2333:
3625:
3481:
3401:
3366:
2896:
2858:
2204:
1968:
1912:-manifold has non-negative Ricci curvature, then its first Betti number is at most
1792:
1677:
1585:
1570:
1399:
1346:
1272:
1223:
1051:
1000:
813:
668:
583:
373:
17:
2374:
3650:
3265:
3234:
2781:
1639:
1342:
1087:
960:
778:
713:
641:
613:
588:
3476:
3042:
2886:
2881:
2584:
2325:
1219:
945:
924:
914:
904:
863:
808:
703:
693:
593:
444:
2873:
2366:
2199:(Provides a historical review and survey, including hundreds of references.)
1452:
1371:
955:
673:
636:
500:
472:
2294:(2008), "Classification of manifolds with weakly 1/4-pinched curvatures",
2018:-dimensional torus does not admit a metric with positive scalar curvature.
1642:
in 1994 gave an astonishingly elegant/short proof of the Soul
Conjecture:
3416:
2766:
2456:
2134:
Multivalued Fields in
Condensed Matter, Electromagnetism, and Gravitation
1986:
1933:-1)-dimensional Riemannian manifold that has nonnegative Ricci curvature.
1326:. It also serves as an entry level for the more complicated structure of
1256:
1231:
1195:
1036:
995:
965:
853:
848:
798:
523:
482:
430:
324:
287:
33:
1895:
If a complete Riemannian manifold has positive Ricci curvature then its
1818:. This has many implications for the structure of the fundamental group:
1776:
of any compact Riemannian manifold with negative sectional curvature is
1559:-dimensional Riemannian manifold has a metric with sectional curvature |
1916:, with equality if and only if the Riemannian manifold is a flat torus.
1777:
970:
683:
477:
421:
221:
3096:
1311:
properties vary from point to point, including the standard types of
1227:
919:
909:
788:
733:
608:
571:
559:
514:
467:
385:
50:
3513:
2182:
Riemannian Geometry During the Second Half of the Twentieth Century
2308:
1989:
of a compact Riemannian manifold with negative Ricci curvature is
1352:
The following articles provide some useful introductory material:
1298:
1215:
1214:
from point to point). This gives, in particular, local notions of
975:
899:
833:
678:
282:
277:
3077:
566:
416:
3517:
2661:
2378:
2000:â„ 3 admits a Riemannian metric with negative Ricci curvature. (
1520:, there are only finitely many (up to diffeomorphism) compact
2657:
1569:
and diameter †1 then its finite cover is diffeomorphic to a
1259:
on them, with techniques that can be applied to the study of
1230:. From those, some other global quantities can be derived by
1524:-dimensional Riemannian manifolds with sectional curvature |
1706:-dimensional Riemannian manifolds with sectional curvature
1243:
Ueber die Hypothesen, welche der Geometrie zu Grunde liegen
1702:, there are only finitely many homotopy types of compact
1330:, which (in four dimensions) are the main objects of the
1867:, so that it does not contain a subgroup isomorphic to
1630:
has strictly positive curvature everywhere, then it is
1334:. Other generalizations of Riemannian geometry include
2225:, Universitext (3rd ed.), Berlin: Springer-Verlag
2053:
Introduction to the mathematics of general relativity
1263:
of higher dimensions. It enabled the formulation of
3329:
3194:
3166:
3035:
2984:
2946:
2925:
2914:
2872:
2816:
2800:
2742:
2706:
2695:
2624:
2598:
2552:
2521:
2417:
2179:
1237:Riemannian geometry originated with the vision of
1602:contains a compact, totally geodesic submanifold
1594:is a non-compact complete non-negatively curved
1650:if it has positive curvature at only one point.
3529:
2673:
2390:
1164:
8:
3571:GrothendieckâHirzebruchâRiemannâRoch theorem
3536:
3522:
3514:
2922:
2703:
2680:
2666:
2658:
2560:Fundamental theorem of Riemannian geometry
2397:
2383:
2375:
2233:Riemannian Geometry and Geometric Analysis
2209:Comparison theorems in Riemannian geometry
1387:Glossary of Riemannian and metric geometry
1171:
1157:
886:
405:
40:
29:
3716:RiemannâRoch theorem for smooth manifolds
2307:
1610:is diffeomorphic to the normal bundle of
1322:, which often helps to solve problems of
2211:, Providence, RI: AMS Chelsea Publishing
1746:with nonpositive sectional curvature is
2213:; Revised reprint of the 1975 original.
2114:Gauge Fields in Condensed Matter Vol II
2089:
1598:-dimensional Riemannian manifold, then
1125:
1059:
1008:
937:
889:
651:
513:
490:
457:
429:
32:
2070:in EinsteinâCartan theory (motivation)
1942:The volume of a metric ball of radius
271:Straightedge and compass constructions
2117:, World Scientific, pp. 743â1440
7:
2140:, World Scientific, pp. 1â496,
1688:GroveâPetersen's finiteness theorem.
1382:List of differential geometry topics
1241:expressed in his inaugural lecture "
3252:TolmanâOppenheimerâVolkoff equation
3205:FriedmannâLemaĂźtreâRobertsonâWalker
3681:Riemannian connection on a surface
3586:Measurable Riemann mapping theorem
25:
3022:HamiltonâJacobiâEinstein equation
1996:Any smooth manifold of dimension
1727:Sectional curvature bounded above
1578:Sectional curvature bounded below
1283:, and spurred the development of
1247:differential geometry of surfaces
237:Noncommutative algebraic geometry
3754:
3753:
3500:
3499:
1472:"sufficiently large" distances.
1437:generalized Gauss-Bonnet theorem
1349:produce torsions and curvature.
3666:Riemann's differential equation
3576:HirzebruchâRiemannâRoch theorem
1848:it contains only finitely many
1841:the group Î has finite virtual
1318:Every smooth manifold admits a
27:Branch of differential geometry
3691:RiemannâHilbert correspondence
3561:Generalized Riemann hypothesis
2829:Massâenergy equivalence (E=mc)
2221:; Lafontaine, Jacques (2004),
1838:for Î has a positive solution;
1654:Gromov's Betti number theorem.
1545:Gromov's almost flat manifolds
1494:is a simply connected compact
630:- / other-dimensional
1:
3726:RiemannâSiegel theta function
2002:This is not true for surfaces
1884:Ricci curvature bounded below
1506:Cheeger's finiteness theorem.
1502:is diffeomorphic to a sphere.
3741:Riemannâvon Mangoldt formula
2487:Raising and lowering indices
1959:Gromov's compactness theorem
1555:> 0 such that if an
1332:theory of general relativity
1269:general theory of relativity
2844:Relativistic Doppler effect
2351:Encyclopedia of Mathematics
2254:, Berlin: Springer-Verlag,
2235:, Berlin: Springer-Verlag,
1328:pseudo-Riemannian manifolds
3803:
3736:RiemannâStieltjes integral
3731:RiemannâSilberstein vector
3706:RiemannâLiouville integral
3315:In computational physics:
2839:Relativity of simultaneity
2508:Pseudo-Riemannian manifold
2349:by V. A. Toponogov at the
1271:, made profound impact on
1210:at each point that varies
3749:
3671:Riemann's minimal surface
3551:
3497:
3152:LenseâThirring precession
2734:Doubly special relativity
2637:Geometrization conjecture
2326:10.1007/s11511-008-0022-7
2207:; Ebin, David G. (2008),
2074:Riemann's minimal surface
2009:Positive scalar curvature
1402:and D. Ebin (see below).
3696:RiemannâHilbert problems
3601:Riemann curvature tensor
3566:Grand Riemann hypothesis
3556:CauchyâRiemann equations
3012:Post-Newtonian formalism
3002:Einstein field equations
2938:Mathematical formulation
2762:Hyperbolic orthogonality
2250:Petersen, Peter (2006),
1980:Negative Ricci curvature
1938:BishopâGromov inequality
1908:If a compact Riemannian
1854:elements of finite order
1447:. They state that every
1377:Riemann curvature tensor
1261:differentiable manifolds
126:Non-Archimedean geometry
3621:Riemann mapping theorem
2723:Galilean transformation
2714:Principle of relativity
2068:RiemannâCartan geometry
1973:Gromov-Hausdorff metric
1843:cohomological dimension
1738:states that a complete
1735:CartanâHadamard theorem
1672:is a compact connected
1444:Nash embedding theorems
232:Noncommutative geometry
3721:RiemannâSiegel formula
3701:RiemannâLebesgue lemma
3636:Riemann series theorem
2808:Lorentz transformation
2647:Uniformization theorem
2580:Nash embedding theorem
2513:Riemannian volume form
2472:Levi-Civita connection
1367:Levi-Civita connection
1313:non-Euclidean geometry
1304:
200:Discrete/Combinatorial
3661:Riemann zeta function
3276:WeylâLewisâPapapetrou
3017:Raychaudhuri equation
2956:Equivalence principle
2362:"Riemannian Geometry"
2231:Jost, JĂŒrgen (2002),
2048:Shape of the universe
1626:.) In particular, if
1451:can be isometrically
1324:differential topology
1302:
1289:differential topology
1277:representation theory
1234:local contributions.
1188:differential geometry
183:Discrete differential
3711:RiemannâRoch theorem
3317:Numerical relativity
3158:pulsar timing arrays
2570:GaussâBonnet theorem
2477:Covariant derivative
1799:. Consequently, its
1742:Riemannian manifold
1656:There is a constant
1646:is diffeomorphic to
1429:Euler characteristic
1416:GaussâBonnet theorem
1192:Riemannian manifolds
3782:Riemannian geometry
3686:Riemannian geometry
3596:Riemann Xi function
3581:Local zeta function
3209:Friedmann equations
3103:HulseâTaylor binary
3065:Gravitational waves
2961:Riemannian geometry
2787:Proper acceleration
2772:Maxwell's equations
2718:Galilean relativity
2642:Poincaré conjecture
2503:Riemannian manifold
2491:Musical isomorphism
2406:Riemannian geometry
2347:Riemannian geometry
2318:2007arXiv0705.3963B
2252:Riemannian Geometry
2223:Riemannian geometry
2217:Gallot, Sylvestre;
2146:2008mfcm.book.....K
1954:in Euclidean space.
1863:subgroups of Î are
1478:sectional curvature
1449:Riemannian manifold
1362:Riemannian manifold
1184:Riemannian geometry
450:Pythagorean theorem
18:Riemannian Geometry
3606:Riemann hypothesis
3258:ReissnerâNordström
3176:BransâDicke theory
3007:Linearized gravity
2834:Length contraction
2752:Frame of reference
2729:Special relativity
2632:General relativity
2575:HopfâRinow theorem
2522:Types of manifolds
2498:Parallel transport
2358:Weisstein, Eric W.
2292:Schoen, Richard M.
2058:Normal coordinates
2023:injectivity radius
1829:finitely presented
1584:CheegerâGromoll's
1393:Classical theorems
1305:
3769:
3768:
3676:Riemannian circle
3616:Riemann invariant
3511:
3510:
3325:
3324:
3304:OzsvĂĄthâSchĂŒcking
2910:
2909:
2892:Minkowski diagram
2849:Thomas precession
2792:Relativistic mass
2655:
2654:
2275:978-3-319-60039-0
2063:Systolic geometry
1921:Splitting theorem
1904:Bochner's formula
1897:fundamental group
1850:conjugacy classes
1816:Gromov hyperbolic
1801:fundamental group
1467:Geometry in large
1320:Riemannian metric
1200:Riemannian metric
1186:is the branch of
1181:
1180:
1146:
1145:
869:List of geometers
552:Three-dimensional
541:
540:
16:(Redirected from
3794:
3787:Bernhard Riemann
3757:
3756:
3611:Riemann integral
3591:Riemann (crater)
3545:Bernhard Riemann
3538:
3531:
3524:
3515:
3503:
3502:
3286:van Stockum dust
3058:Two-body problem
2976:Mach's principle
2923:
2864:Terrell rotation
2704:
2682:
2675:
2668:
2659:
2399:
2392:
2385:
2376:
2371:
2370:
2336:
2311:
2264:
2245:
2226:
2219:Hulin, Dominique
2212:
2196:
2185:
2158:
2155:
2149:
2148:
2139:
2125:
2119:
2118:
2105:
2099:
2094:
1865:virtually cyclic
1806:
1740:simply connected
1690:Given constants
1508:Given constants
1409:General theorems
1336:Finsler geometry
1303:Bernhard Riemann
1239:Bernhard Riemann
1220:length of curves
1196:smooth manifolds
1173:
1166:
1159:
887:
406:
339:Zero-dimensional
44:
30:
21:
3802:
3801:
3797:
3796:
3795:
3793:
3792:
3791:
3772:
3771:
3770:
3765:
3745:
3656:Riemann surface
3631:Riemann problem
3547:
3542:
3512:
3507:
3493:
3321:
3225:BKL singularity
3215:LemaĂźtreâTolman
3190:
3186:Quantum gravity
3168:
3162:
3148:geodetic effect
3122:(together with
3092:LISA Pathfinder
3031:
2980:
2966:Penrose diagram
2948:
2942:
2917:
2906:
2902:Minkowski space
2868:
2812:
2796:
2744:
2738:
2698:
2691:
2686:
2656:
2651:
2620:
2599:Generalizations
2594:
2548:
2517:
2452:Exponential map
2413:
2403:
2356:
2355:
2343:
2286:
2262:
2249:
2243:
2230:
2216:
2203:
2194:
2174:
2166:
2161:
2156:
2152:
2137:
2129:Kleinert, Hagen
2127:
2126:
2122:
2109:Kleinert, Hagen
2107:
2106:
2102:
2095:
2091:
2087:
2044:
2011:
1982:
1886:
1809:
1804:
1767:exponential map
1752:Euclidean space
1729:
1668:) such that if
1580:
1568:
1554:
1481:
1469:
1457:Euclidean space
1411:
1395:
1297:
1177:
1148:
1147:
884:
883:
874:
873:
664:
663:
647:
646:
632:
631:
619:
618:
555:
554:
543:
542:
403:
402:
400:Two-dimensional
391:
390:
364:
363:
361:One-dimensional
352:
351:
342:
341:
330:
329:
263:
262:
261:
244:
243:
92:
91:
80:
57:
28:
23:
22:
15:
12:
11:
5:
3800:
3798:
3790:
3789:
3784:
3774:
3773:
3767:
3766:
3764:
3763:
3750:
3747:
3746:
3744:
3743:
3738:
3733:
3728:
3723:
3718:
3713:
3708:
3703:
3698:
3693:
3688:
3683:
3678:
3673:
3668:
3663:
3658:
3653:
3648:
3646:Riemann sphere
3643:
3641:Riemann solver
3638:
3633:
3628:
3623:
3618:
3613:
3608:
3603:
3598:
3593:
3588:
3583:
3578:
3573:
3568:
3563:
3558:
3552:
3549:
3548:
3543:
3541:
3540:
3533:
3526:
3518:
3509:
3508:
3498:
3495:
3494:
3492:
3491:
3484:
3479:
3474:
3469:
3464:
3459:
3454:
3449:
3444:
3439:
3434:
3429:
3424:
3419:
3414:
3412:Choquet-Bruhat
3409:
3404:
3399:
3394:
3389:
3384:
3379:
3374:
3369:
3364:
3359:
3354:
3349:
3344:
3339:
3333:
3331:
3327:
3326:
3323:
3322:
3320:
3319:
3312:
3311:
3306:
3301:
3294:
3293:
3288:
3283:
3278:
3273:
3264:Axisymmetric:
3261:
3260:
3255:
3249:
3238:
3237:
3232:
3227:
3222:
3217:
3212:
3203:Cosmological:
3200:
3198:
3192:
3191:
3189:
3188:
3183:
3178:
3172:
3170:
3164:
3163:
3161:
3160:
3155:
3144:frame-dragging
3141:
3136:
3131:
3128:Einstein rings
3124:Einstein cross
3117:
3106:
3105:
3100:
3094:
3089:
3084:
3071:
3061:
3060:
3055:
3050:
3045:
3039:
3037:
3033:
3032:
3030:
3029:
3027:Ernst equation
3024:
3019:
3014:
3009:
3004:
2999:
2997:BSSN formalism
2994:
2988:
2986:
2982:
2981:
2979:
2978:
2973:
2968:
2963:
2958:
2952:
2950:
2944:
2943:
2941:
2940:
2935:
2929:
2927:
2920:
2912:
2911:
2908:
2907:
2905:
2904:
2899:
2894:
2889:
2884:
2878:
2876:
2870:
2869:
2867:
2866:
2861:
2856:
2854:Ladder paradox
2851:
2846:
2841:
2836:
2831:
2826:
2820:
2818:
2814:
2813:
2811:
2810:
2804:
2802:
2798:
2797:
2795:
2794:
2789:
2784:
2779:
2774:
2769:
2764:
2759:
2757:Speed of light
2754:
2748:
2746:
2740:
2739:
2737:
2736:
2731:
2726:
2720:
2710:
2708:
2701:
2693:
2692:
2687:
2685:
2684:
2677:
2670:
2662:
2653:
2652:
2650:
2649:
2644:
2639:
2634:
2628:
2626:
2622:
2621:
2619:
2618:
2616:Sub-Riemannian
2613:
2608:
2602:
2600:
2596:
2595:
2593:
2592:
2587:
2582:
2577:
2572:
2567:
2562:
2556:
2554:
2550:
2549:
2547:
2546:
2541:
2536:
2531:
2525:
2523:
2519:
2518:
2516:
2515:
2510:
2505:
2500:
2495:
2494:
2493:
2484:
2479:
2474:
2464:
2459:
2454:
2449:
2448:
2447:
2442:
2437:
2432:
2421:
2419:
2418:Basic concepts
2415:
2414:
2404:
2402:
2401:
2394:
2387:
2379:
2373:
2372:
2353:
2342:
2341:External links
2339:
2338:
2337:
2288:Brendle, Simon
2283:
2282:
2278:
2277:
2266:
2265:
2260:
2247:
2241:
2228:
2214:
2201:
2192:
2176:Berger, Marcel
2171:
2170:
2165:
2162:
2160:
2159:
2150:
2120:
2100:
2088:
2086:
2083:
2082:
2081:
2079:Reilly formula
2076:
2071:
2065:
2060:
2055:
2050:
2043:
2040:
2039:
2038:
2019:
2010:
2007:
2006:
2005:
1994:
1987:isometry group
1981:
1978:
1977:
1976:
1955:
1946:in a complete
1934:
1925:If a complete
1917:
1900:
1885:
1882:
1881:
1880:
1879:
1878:
1877:
1876:
1857:
1846:
1839:
1832:
1820:
1819:
1807:
1781:
1770:
1728:
1725:
1724:
1723:
1685:
1651:
1618:is called the
1579:
1576:
1575:
1574:
1564:
1550:
1541:
1503:
1486:Sphere theorem
1480:
1474:
1468:
1465:
1464:
1463:
1440:
1427:) denotes the
1410:
1407:
1394:
1391:
1390:
1389:
1384:
1379:
1374:
1369:
1364:
1359:
1296:
1293:
1179:
1178:
1176:
1175:
1168:
1161:
1153:
1150:
1149:
1144:
1143:
1142:
1141:
1136:
1128:
1127:
1123:
1122:
1121:
1120:
1115:
1110:
1105:
1100:
1095:
1090:
1085:
1080:
1075:
1070:
1062:
1061:
1057:
1056:
1055:
1054:
1049:
1044:
1039:
1034:
1029:
1024:
1019:
1011:
1010:
1006:
1005:
1004:
1003:
998:
993:
988:
983:
978:
973:
968:
963:
958:
953:
948:
940:
939:
935:
934:
933:
932:
927:
922:
917:
912:
907:
902:
894:
893:
885:
881:
880:
879:
876:
875:
872:
871:
866:
861:
856:
851:
846:
841:
836:
831:
826:
821:
816:
811:
806:
801:
796:
791:
786:
781:
776:
771:
766:
761:
756:
751:
746:
741:
736:
731:
726:
721:
716:
711:
706:
701:
696:
691:
686:
681:
676:
671:
665:
661:
660:
659:
656:
655:
649:
648:
645:
644:
639:
633:
626:
625:
624:
621:
620:
617:
616:
611:
606:
604:Platonic Solid
601:
596:
591:
586:
581:
576:
575:
574:
563:
562:
556:
550:
549:
548:
545:
544:
539:
538:
537:
536:
531:
526:
518:
517:
511:
510:
509:
508:
503:
495:
494:
488:
487:
486:
485:
480:
475:
470:
462:
461:
455:
454:
453:
452:
447:
442:
434:
433:
427:
426:
425:
424:
419:
414:
404:
398:
397:
396:
393:
392:
389:
388:
383:
382:
381:
376:
365:
359:
358:
357:
354:
353:
350:
349:
343:
337:
336:
335:
332:
331:
328:
327:
322:
317:
311:
310:
305:
300:
290:
285:
280:
274:
273:
264:
260:
259:
256:
252:
251:
250:
249:
246:
245:
242:
241:
240:
239:
229:
224:
219:
214:
209:
208:
207:
197:
192:
187:
186:
185:
180:
175:
165:
164:
163:
158:
148:
143:
138:
133:
128:
123:
122:
121:
116:
115:
114:
99:
93:
87:
86:
85:
82:
81:
79:
78:
68:
62:
59:
58:
45:
37:
36:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3799:
3788:
3785:
3783:
3780:
3779:
3777:
3762:
3761:
3752:
3751:
3748:
3742:
3739:
3737:
3734:
3732:
3729:
3727:
3724:
3722:
3719:
3717:
3714:
3712:
3709:
3707:
3704:
3702:
3699:
3697:
3694:
3692:
3689:
3687:
3684:
3682:
3679:
3677:
3674:
3672:
3669:
3667:
3664:
3662:
3659:
3657:
3654:
3652:
3649:
3647:
3644:
3642:
3639:
3637:
3634:
3632:
3629:
3627:
3624:
3622:
3619:
3617:
3614:
3612:
3609:
3607:
3604:
3602:
3599:
3597:
3594:
3592:
3589:
3587:
3584:
3582:
3579:
3577:
3574:
3572:
3569:
3567:
3564:
3562:
3559:
3557:
3554:
3553:
3550:
3546:
3539:
3534:
3532:
3527:
3525:
3520:
3519:
3516:
3506:
3496:
3490:
3489:
3485:
3483:
3480:
3478:
3475:
3473:
3470:
3468:
3465:
3463:
3460:
3458:
3455:
3453:
3450:
3448:
3445:
3443:
3440:
3438:
3435:
3433:
3430:
3428:
3425:
3423:
3420:
3418:
3415:
3413:
3410:
3408:
3405:
3403:
3400:
3398:
3397:Chandrasekhar
3395:
3393:
3390:
3388:
3385:
3383:
3380:
3378:
3375:
3373:
3370:
3368:
3365:
3363:
3360:
3358:
3357:Schwarzschild
3355:
3353:
3350:
3348:
3345:
3343:
3340:
3338:
3335:
3334:
3332:
3328:
3318:
3314:
3313:
3310:
3307:
3305:
3302:
3300:
3296:
3295:
3292:
3289:
3287:
3284:
3282:
3279:
3277:
3274:
3271:
3267:
3263:
3262:
3259:
3256:
3253:
3250:
3248:
3244:
3243:Schwarzschild
3240:
3239:
3236:
3233:
3231:
3228:
3226:
3223:
3221:
3218:
3216:
3213:
3210:
3206:
3202:
3201:
3199:
3197:
3193:
3187:
3184:
3182:
3179:
3177:
3174:
3173:
3171:
3165:
3159:
3156:
3153:
3149:
3145:
3142:
3140:
3139:Shapiro delay
3137:
3135:
3132:
3129:
3125:
3121:
3118:
3115:
3111:
3108:
3107:
3104:
3101:
3098:
3095:
3093:
3090:
3088:
3085:
3083:
3082:collaboration
3079:
3075:
3072:
3070:
3066:
3063:
3062:
3059:
3056:
3054:
3051:
3049:
3048:Event horizon
3046:
3044:
3041:
3040:
3038:
3034:
3028:
3025:
3023:
3020:
3018:
3015:
3013:
3010:
3008:
3005:
3003:
3000:
2998:
2995:
2993:
2992:ADM formalism
2990:
2989:
2987:
2983:
2977:
2974:
2972:
2969:
2967:
2964:
2962:
2959:
2957:
2954:
2953:
2951:
2945:
2939:
2936:
2934:
2931:
2930:
2928:
2924:
2921:
2919:
2913:
2903:
2900:
2898:
2897:Biquaternions
2895:
2893:
2890:
2888:
2885:
2883:
2880:
2879:
2877:
2875:
2871:
2865:
2862:
2860:
2857:
2855:
2852:
2850:
2847:
2845:
2842:
2840:
2837:
2835:
2832:
2830:
2827:
2825:
2824:Time dilation
2822:
2821:
2819:
2815:
2809:
2806:
2805:
2803:
2799:
2793:
2790:
2788:
2785:
2783:
2780:
2778:
2777:Proper length
2775:
2773:
2770:
2768:
2765:
2763:
2760:
2758:
2755:
2753:
2750:
2749:
2747:
2741:
2735:
2732:
2730:
2727:
2724:
2721:
2719:
2715:
2712:
2711:
2709:
2705:
2702:
2700:
2694:
2690:
2683:
2678:
2676:
2671:
2669:
2664:
2663:
2660:
2648:
2645:
2643:
2640:
2638:
2635:
2633:
2630:
2629:
2627:
2623:
2617:
2614:
2612:
2609:
2607:
2604:
2603:
2601:
2597:
2591:
2590:Schur's lemma
2588:
2586:
2583:
2581:
2578:
2576:
2573:
2571:
2568:
2566:
2565:Gauss's lemma
2563:
2561:
2558:
2557:
2555:
2551:
2545:
2542:
2540:
2537:
2535:
2532:
2530:
2527:
2526:
2524:
2520:
2514:
2511:
2509:
2506:
2504:
2501:
2499:
2496:
2492:
2488:
2485:
2483:
2480:
2478:
2475:
2473:
2470:
2469:
2468:
2467:Metric tensor
2465:
2463:
2462:Inner product
2460:
2458:
2455:
2453:
2450:
2446:
2443:
2441:
2438:
2436:
2433:
2431:
2428:
2427:
2426:
2423:
2422:
2420:
2416:
2411:
2407:
2400:
2395:
2393:
2388:
2386:
2381:
2380:
2377:
2369:
2368:
2363:
2359:
2354:
2352:
2348:
2345:
2344:
2340:
2335:
2331:
2327:
2323:
2319:
2315:
2310:
2305:
2301:
2297:
2293:
2289:
2285:
2284:
2280:
2279:
2276:
2272:
2268:
2267:
2263:
2261:0-387-98212-4
2257:
2253:
2248:
2244:
2242:3-540-42627-2
2238:
2234:
2229:
2224:
2220:
2215:
2210:
2206:
2205:Cheeger, Jeff
2202:
2200:
2195:
2193:0-8218-2052-4
2189:
2184:
2183:
2177:
2173:
2172:
2168:
2167:
2163:
2154:
2151:
2147:
2143:
2136:
2135:
2130:
2124:
2121:
2116:
2115:
2110:
2104:
2101:
2098:
2093:
2090:
2084:
2080:
2077:
2075:
2072:
2069:
2066:
2064:
2061:
2059:
2056:
2054:
2051:
2049:
2046:
2045:
2041:
2036:
2032:
2028:
2025:of a compact
2024:
2020:
2017:
2013:
2012:
2008:
2003:
1999:
1995:
1992:
1988:
1984:
1983:
1979:
1974:
1970:
1966:
1962:
1960:
1956:
1953:
1949:
1945:
1941:
1939:
1935:
1932:
1928:
1924:
1922:
1918:
1915:
1911:
1907:
1905:
1901:
1898:
1894:
1892:
1891:Myers theorem
1888:
1887:
1883:
1874:
1870:
1866:
1862:
1858:
1855:
1851:
1847:
1844:
1840:
1837:
1833:
1830:
1826:
1825:
1824:
1823:
1822:
1821:
1817:
1813:
1802:
1798:
1796:
1791:then it is a
1790:
1786:
1782:
1779:
1775:
1774:geodesic flow
1771:
1768:
1764:
1760:
1756:
1753:
1749:
1748:diffeomorphic
1745:
1741:
1737:
1736:
1731:
1730:
1726:
1721:
1718:and volume â„
1717:
1714:, diameter â€
1713:
1709:
1705:
1701:
1697:
1693:
1689:
1686:
1683:
1679:
1678:Betti numbers
1675:
1671:
1667:
1663:
1659:
1655:
1652:
1649:
1645:
1641:
1637:
1633:
1632:diffeomorphic
1629:
1625:
1621:
1617:
1613:
1609:
1605:
1601:
1597:
1593:
1589:
1587:
1582:
1581:
1577:
1572:
1567:
1562:
1558:
1553:
1549:There is an Δ
1548:
1546:
1542:
1539:
1536:and volume â„
1535:
1532:, diameter â€
1531:
1527:
1523:
1519:
1515:
1511:
1507:
1504:
1501:
1497:
1493:
1489:
1487:
1483:
1482:
1479:
1475:
1473:
1466:
1461:
1458:
1454:
1450:
1446:
1445:
1441:
1438:
1434:
1430:
1426:
1422:
1418:
1417:
1413:
1412:
1408:
1406:
1403:
1401:
1392:
1388:
1385:
1383:
1380:
1378:
1375:
1373:
1370:
1368:
1365:
1363:
1360:
1358:
1357:Metric tensor
1355:
1354:
1353:
1350:
1348:
1347:disclinations
1344:
1339:
1337:
1333:
1329:
1325:
1321:
1316:
1314:
1310:
1301:
1294:
1292:
1290:
1286:
1282:
1279:, as well as
1278:
1274:
1270:
1266:
1262:
1258:
1254:
1253:
1248:
1244:
1240:
1235:
1233:
1229:
1225:
1221:
1217:
1213:
1209:
1208:tangent space
1205:
1204:inner product
1201:
1197:
1194:, defined as
1193:
1190:that studies
1189:
1185:
1174:
1169:
1167:
1162:
1160:
1155:
1154:
1152:
1151:
1140:
1137:
1135:
1132:
1131:
1130:
1129:
1124:
1119:
1116:
1114:
1111:
1109:
1106:
1104:
1101:
1099:
1096:
1094:
1091:
1089:
1086:
1084:
1081:
1079:
1076:
1074:
1071:
1069:
1066:
1065:
1064:
1063:
1058:
1053:
1050:
1048:
1045:
1043:
1040:
1038:
1035:
1033:
1030:
1028:
1025:
1023:
1020:
1018:
1015:
1014:
1013:
1012:
1007:
1002:
999:
997:
994:
992:
989:
987:
984:
982:
979:
977:
974:
972:
969:
967:
964:
962:
959:
957:
954:
952:
949:
947:
944:
943:
942:
941:
936:
931:
928:
926:
923:
921:
918:
916:
913:
911:
908:
906:
903:
901:
898:
897:
896:
895:
892:
888:
878:
877:
870:
867:
865:
862:
860:
857:
855:
852:
850:
847:
845:
842:
840:
837:
835:
832:
830:
827:
825:
822:
820:
817:
815:
812:
810:
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529:Circumference
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3758:
3685:
3626:Riemann form
3487:
3181:KaluzaâKlein
2960:
2933:Introduction
2859:Twin paradox
2625:Applications
2553:Main results
2405:
2365:
2299:
2295:
2251:
2232:
2222:
2208:
2198:
2181:
2153:
2133:
2123:
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2103:
2097:maths.tcd.ie
2092:
2034:
2030:
2026:
2015:
2001:
1997:
1964:
1957:
1951:
1947:
1943:
1936:
1930:
1926:
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1913:
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1889:
1872:
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1836:word problem
1811:
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1586:soul theorem
1583:
1571:nil manifold
1565:
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1470:
1459:
1442:
1432:
1424:
1420:
1414:
1404:
1400:Jeff Cheeger
1396:
1351:
1343:Dislocations
1340:
1317:
1306:
1295:Introduction
1273:group theory
1251:
1242:
1236:
1224:surface area
1199:
1183:
1182:
1001:Parameshvara
814:Parameshvara
584:Dodecahedron
172:
168:Differential
3651:Riemann sum
3270:KerrâNewman
3241:Spherical:
3110:Other tests
3053:Singularity
2985:Formulation
2947:Fundamental
2801:Formulation
2782:Proper time
2743:Fundamental
1969:pre-compact
1680:is at most
1640:G. Perelman
1232:integrating
1126:Present day
1073:Lobachevsky
1060:1700sâ1900s
1017:JyeáčŁáčhadeva
1009:1400sâ1700s
961:Brahmagupta
784:Lobachevsky
764:JyeáčŁáčhadeva
714:Brahmagupta
642:Hypersphere
614:Tetrahedron
589:Icosahedron
161:Diophantine
3776:Categories
3422:Zel'dovich
3330:Scientists
3309:Alcubierre
3116:of Mercury
3114:precession
3043:Black hole
2926:Background
2918:relativity
2887:World line
2882:Light cone
2707:Background
2699:relativity
2689:Relativity
2585:Ricci flow
2534:Hyperbolic
2164:References
1899:is finite.
1606:such that
1423:) where Ï(
986:al-Yasamin
930:Apollonius
925:Archimedes
915:Pythagoras
905:Baudhayana
859:al-Yasamin
809:Pythagoras
704:Baudhayana
694:Archimedes
689:Apollonius
594:Octahedron
445:Hypotenuse
320:Similarity
315:Congruence
227:Incidence
178:Symplectic
173:Riemannian
156:Arithmetic
131:Projective
119:Hyperbolic
47:Projecting
3392:Robertson
3377:Friedmann
3372:Eddington
3362:de Sitter
3196:Solutions
3074:detectors
3069:astronomy
3036:Phenomena
2971:Geodesics
2874:Spacetime
2817:Phenomena
2529:Hermitian
2482:Signature
2445:Sectional
2425:Curvature
2367:MathWorld
2309:0705.3963
2296:Acta Math
1803:Î =
1372:Curvature
1285:algebraic
1257:geodesics
1103:Minkowski
1022:Descartes
956:Aryabhata
951:KÄtyÄyana
882:by period
794:Minkowski
769:KÄtyÄyana
729:Descartes
674:Aryabhata
653:Geometers
637:Tesseract
501:Trapezoid
473:Rectangle
266:Dimension
151:Algebraic
141:Synthetic
112:Spherical
97:Euclidean
3760:Category
3505:Category
3382:LemaĂźtre
3347:Einstein
3337:Poincaré
3297:Others:
3281:TaubâNUT
3247:interior
3169:theories
3167:Advanced
3134:redshift
2949:concepts
2767:Rapidity
2745:concepts
2544:Kenmotsu
2457:Geodesic
2410:Glossary
2334:15463483
2302:: 1â13,
2178:(2000),
2131:(2008),
2111:(1989),
2042:See also
1991:discrete
1765:via the
1476:Pinched
1453:embedded
1281:analysis
1265:Einstein
1212:smoothly
1093:Poincaré
1037:Minggatu
996:Yang Hui
966:Virasena
854:Yang Hui
849:Virasena
819:Poincaré
799:Minggatu
579:Cylinder
524:Diameter
483:Rhomboid
440:Altitude
431:Triangle
325:Symmetry
303:Parallel
288:Diagonal
258:Features
255:Concepts
146:Analytic
107:Elliptic
89:Branches
75:Timeline
34:Geometry
3447:Hawking
3442:Penrose
3427:Novikov
3407:Wheeler
3352:Hilbert
3342:Lorentz
3299:pp-wave
3120:lensing
2916:General
2697:Special
2611:Hilbert
2606:Finsler
2314:Bibcode
2142:Bibcode
2021:If the
1971:in the
1861:abelian
1797:) space
1778:ergodic
1750:to the
1206:on the
1198:with a
1118:Coxeter
1098:Hilbert
1083:Riemann
1032:Huygens
991:al-Tusi
981:KhayyĂĄm
971:Alhazen
938:1â1400s
839:al-Tusi
824:Riemann
774:KhayyĂĄm
759:Huygens
754:Hilbert
724:Coxeter
684:Alhazen
662:by name
599:Pyramid
478:Rhombus
422:Polygon
374:segment
222:Fractal
205:Digital
190:Complex
71:History
66:Outline
3488:others
3477:Thorne
3467:Misner
3452:Taylor
3437:Geroch
3432:Ehlers
3402:Zwicky
3220:Kasner
2539:KĂ€hler
2435:Scalar
2430:tensor
2332:
2281:Papers
2273:
2258:
2239:
2190:
1827:it is
1761:= dim
1563:| †Δ
1309:metric
1228:volume
1139:Gromov
1134:Atiyah
1113:Veblen
1108:Cartan
1078:Bolyai
1047:Sakabe
1027:Pascal
920:Euclid
910:Manava
844:Veblen
829:Sakabe
804:Pascal
789:Manava
749:Gromov
734:Euclid
719:Cartan
709:Bolyai
699:Atiyah
609:Sphere
572:cuboid
560:Volume
515:Circle
468:Square
386:Length
308:Vertex
212:Convex
195:Finite
136:Affine
51:sphere
3482:Weiss
3462:Bondi
3457:Hulse
3387:Milne
3291:discs
3235:Milne
3230:Gödel
3087:Virgo
2440:Ricci
2330:S2CID
2304:arXiv
2169:Books
2138:(PDF)
2085:Notes
1814:) is
1757:with
1455:in a
1216:angle
1088:Klein
1068:Gauss
1042:Euler
976:Sijzi
946:Zhang
900:Ahmes
864:Zhang
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3080:and
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669:Aida
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369:Line
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3097:GEO
2322:doi
2300:200
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1852:of
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1634:to
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