49:
2046:
811:
3002:
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3023:). Note that a quasi-isometry is not assumed to be continuous. For example, any map between compact metric spaces is a quasi isometry. If there exists a quasi-isometry from X to Y, then X and Y are said to be
695:
3191:
830:
via the exponential map; for metric spaces, the statement that a connected, simply connected complete geodesic metric space with non-positive curvature in the sense of
Alexandrov is a (globally)
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1004:
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1765:
Riemannian manifold the injectivity radius is either half the minimal length of a closed geodesic or the minimal distance between conjugate points on a geodesic.
1306:
168:
The following articles may also be useful; they either contain specialised vocabulary or provide more detailed expositions of the definitions given below.
121:
93:
100:
202:
3631:
107:
826:
is the statement that a connected, simply connected complete
Riemannian manifold with non-positive sectional curvature is diffeomorphic to
721:
2884:
2658:
3794:
89:
1113:
is conformally flat if it is locally conformally equivalent to a
Euclidean space, for example standard sphere is conformally flat.
480:
a generalization of
Riemannian manifolds with upper, lower or integral curvature bounds (the last one works only in dimension 2)
584:
3824:
3799:
3686:
3132:
1703:
1473:
3196:
It can be also generalized to arbitrary codimension, in which case it is a quadratic form with values in the normal space.
207:
114:
3907:
17:
3659:
3624:
1311:
140:
78:
3122:
is a quadratic form on the tangent space of hypersurface, usually denoted by II, an equivalent way to describe the
2265:: An element of the minimal set of manifolds which includes a point, and has the following property: any oriented
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172:
59:
2041:{\displaystyle J(t)=\left.{\frac {\partial \gamma _{\tau }(t)}{\partial \tau }}\right|_{\tau =0}.}
1012:
3850:
3475:
3453:
3118:
855:
63:
2292:-bundle over a nilmanifold is a nilmanifold. It also can be defined as a factor of a connected
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1789:
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3814:
3809:
3747:
3706:
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2100:
1491:
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1894:γ which can be obtained on the following way: Take a smooth one parameter family of geodesics
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of metric space is the infimum of radii of metric balls which contain the space completely.
2384:
3711:
2268:
2136:
2078:
2052:
1454:
1406:
850:, using Riemannian-Cartan geometry instead of Riemannian geometry. This extension provides
484:
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8:
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37:
2483:. Equivalently, if every closed bounded subset is compact. Every proper metric space is
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2448:
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with a metric such that each simplex with induced metric is isometric to a simplex in
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and others, do not have exactly the same meaning as in general mathematical usage.
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is a metric space where any two points are the endpoints of a unique minimizing
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1711:
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707:
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33:
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acting on itself by left multiplication and a finite group of automorphisms
854:, which allows for non-symmetric curvature tensors and the incorporation of
3604:
3326:
1887:
1882:
1559:
831:
182:
3307:(there is no standard agreement whether to use + or − in the definition).
3599:
2261:
2104:
is a natural way to differentiate vector fields on
Riemannian manifolds.
1855:
1658:
is a metric space where any two points are the endpoints of a minimizing
1427:
of a metric space is the supremum of distances between pairs of points.
32:"Radius of convexity" redirects here. For the anatomical feature of the
3819:
1679:
1563:
2463:
is the maximum and minimum normal curvatures at a point on a surface.
310:
3311:
3058:
is a one side infinite geodesic which is minimizing on each interval
2296:
1496:
1413:
806:{\displaystyle B_{\gamma }(p)=\lim _{t\to \infty }(|\gamma (t)-p|-t)}
177:
3048:
of a
Riemannian manifold is the largest radius of a ball which is a
48:
3691:
3575:
in the submanifold are also geodesics of the surrounding manifold.
3571:
3342:
2997:{\displaystyle {1 \over K}d(x,y)-C\leq d(f(x),f(y))\leq Kd(x,y)+C.}
2771:{\displaystyle {1 \over K}d(x,y)-C\leq d(f(x),f(y))\leq Kd(x,y)+C.}
2436:
1891:
1862:
1762:
1642:
1532:
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2781:
Note that a quasigeodesic is not necessarily a continuous curve.
2346:, the normal bundle is a vector bundle whose fiber at each point
1675:
is a complete simply connected space with nonpositive curvature.
1718:
is the infimum of the injectivity radii at all points. See also
1702:
of a
Riemannian manifold is the largest radius for which the
1528:
3350:
between metric spaces is called a submetry if there exists
1981:
690:{\displaystyle c|xy|_{X}\leq |f(x)f(y)|_{Y}\leq C|xy|_{X}}
1445:
of a map between metric spaces is the infimum of numbers
1194:
on a
Riemannian manifold is a convex if for any geodesic
299:: many terms in Riemannian and metric geometry, such as
3186:{\displaystyle {\text{II}}(v,w)=\langle S(v),w\rangle }
2177:
is a submanifold with (vector of) mean curvature zero.
559:
is called bi-Lipschitz if there are positive constants
2505:
has two meanings; here we give the most common. A map
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For complete manifolds, if the injectivity radius at
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931:if it is a point of global minimum of the function
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3547:is called totally convex if for any two points in
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2116:between metric spaces is the infimum of numbers
1854:. An infranilmanifold is finitely covered by a
748:
3551:any geodesic connecting them lies entirely in
3094:is a map between Riemannian manifolds which is
1352:{\displaystyle f\circ \gamma (t)-\lambda t^{2}}
2469:is the direction of the principal curvatures.
203:Glossary of differential geometry and topology
3625:
2124:map between these spaces with constants exp(-
2110:the convergence defined by Lipschitz metric.
1771:Given a simply connected nilpotent Lie group
1733:, then either there is a geodesic of length 2
3180:
3159:
161:— it doesn't cover the terminology of
90:"Glossary of Riemannian and metric geometry"
2311:: associated to an imbedding of a manifold
1716:injectivity radius of a Riemannian manifold
872:is called the center of mass of the points
293:denotes a self-reference to this glossary.
77:. Unsourced material may be challenged and
3795:Fundamental theorem of Riemannian geometry
3632:
3618:
1379:is called convex if for any two points in
999:{\displaystyle f(x)=\sum _{i}|p_{i}x|^{2}}
2551:
2360:
2325:
153:This is a glossary of some terms used in
141:Learn how and when to remove this message
3208:is a linear operator on tangent spaces,
1954:, then the Jacobi field is described by
1614:{\displaystyle (\gamma (t),\gamma '(t))}
1009:Such a point is unique if all distances
924:{\displaystyle p_{1},p_{2},\dots ,p_{k}}
3607:constructed using a set of generators.
3442:{\displaystyle f(B_{r}(x))=B_{r}(f(x))}
2558:{\displaystyle I\subseteq \mathbb {R} }
2145:is a right inverse of Exponential map.
1387:connecting them which lies entirely in
14:
3895:
715:, the Busemann function is defined by
3613:
3297:{\displaystyle S(v)=\pm \nabla _{v}n}
1474:Exponential map (Riemannian geometry)
1866:is a map which preserves distances.
277:denotes the distance between points
208:List of differential geometry topics
75:adding citations to reliable sources
42:
1947:{\displaystyle \gamma _{0}=\gamma }
24:
3315:is a distance non increasing map.
3282:
2374:{\displaystyle {\mathbb {R} }^{N}}
2339:{\displaystyle {\mathbb {R} }^{N}}
2188:is the parametrization by length.
2010:
1986:
1698:The injectivity radius at a point
758:
25:
3929:
2475:is a metric space in which every
2350:is the orthogonal complement (in
1103:is a map which preserves angles.
213:Unless stated otherwise, letters
2315:into an ambient Euclidean space
1783:one can define an action of the
47:
2120:such that there is a bijective
1914:{\displaystyle \gamma _{\tau }}
313:
3603:on a group is a metric of the
3584:Uniquely geodesic metric space
3521:-cycle nonhomologous to zero.
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2521:
2246:{\displaystyle \leq \epsilon }
2005:
1999:
1973:
1967:
1608:
1605:
1599:
1585:
1579:
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1330:
1324:
1233:{\displaystyle f\circ \gamma }
1153:if there is a Jacobi field on
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233:denote Riemannian manifolds, |
13:
1:
3362:we have that image of metric
2565:is a subsegment) is called a
3722:Raising and lowering indices
1649:Gromov-Hausdorff convergence
1470:Exponential map (Lie theory)
1268:-convex if for any geodesic
1049:{\displaystyle |p_{i}p_{j}|}
312:
225:below denote metric spaces,
198:Glossary of general topology
7:
3517:, is the minimal volume of
3510:{\displaystyle syst_{k}(M)}
3330:is a factor of a connected
1753:above) and on the distance
1449:such that the given map is
10:
3934:
3743:Pseudo-Riemannian manifold
3253:is a tangent vector then
3245:is a unit normal field to
2492:Pseudo-Riemannian manifold
1839:{\displaystyle N\rtimes F}
1820:by a discrete subgroup of
1805:{\displaystyle N\rtimes F}
31:
3908:Glossaries of mathematics
3872:Geometrization conjecture
3859:
3833:
3787:
3756:
3652:
3543:of a Riemannian manifold
2826:if there are constants
2257:which generalize limits.
2253:. This is distinct from
2222:-net if for any point in
2215:{\displaystyle \epsilon }
1737:which starts and ends at
1375:of a Riemannian manifold
3354:such that for any point
3077:Riemann curvature tensor
2815:{\displaystyle f:X\to Y}
2646:{\displaystyle x,y\in I}
2530:{\displaystyle f:I\to Y}
1531:which locally minimizes
1261:{\displaystyle \lambda }
552:{\displaystyle f:X\to Y}
325:
320:
270:{\displaystyle |xy|_{X}}
3454:Sub-Riemannian manifold
3119:Second fundamental form
2871:{\displaystyle C\geq 0}
2845:{\displaystyle K\geq 1}
2614:{\displaystyle C\geq 0}
2588:{\displaystyle K\geq 1}
2569:if there are constants
2381:) of the tangent space
2186:Natural parametrization
1634:{\displaystyle \gamma }
1288:with natural parameter
1281:{\displaystyle \gamma }
1207:{\displaystyle \gamma }
1166:{\displaystyle \gamma }
1142:{\displaystyle \gamma }
823:Cartan–Hadamard theorem
3882:Uniformization theorem
3815:Nash embedding theorem
3748:Riemannian volume form
3707:Levi-Civita connection
3511:
3443:
3298:
3187:
2998:
2872:
2846:
2816:
2772:
2647:
2615:
2589:
2559:
2531:
2405:
2404:{\displaystyle T_{p}M}
2375:
2340:
2286:
2247:
2216:
2101:Levi-Civita connection
2065:Kähler-Einstein metric
2042:
1948:
1915:
1840:
1806:
1635:
1615:
1497:embedding or immersion
1492:First fundamental form
1353:
1302:
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1234:
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848:Einstein–Cartan theory
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271:
3903:Differential geometry
3512:
3444:
3299:
3188:
3091:Riemannian submersion
3011:has distance at most
2999:
2873:
2847:
2817:
2773:
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2616:
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2560:
2532:
2406:
2376:
2341:
2287:
2285:{\displaystyle S^{1}}
2248:
2217:
2108:Lipschitz convergence
2043:
1949:
1916:
1846:which acts freely on
1841:
1816:. An orbit space of
1807:
1741:or there is a point
1655:Geodesic metric space
1636:
1616:
1354:
1303:
1283:
1263:
1235:
1209:
1168:
1144:
1086:Complete metric space
1051:
1001:
926:
808:
692:
554:
272:
163:differential topology
3805:Gauss–Bonnet theorem
3712:Covariant derivative
3593:
3578:
3524:
3476:
3377:
3260:
3133:
3126:of a hypersurface,
3105:
3030:
2885:
2856:
2830:
2794:
2659:
2625:
2621:such that for every
2599:
2573:
2541:
2509:
2497:
2423:
2385:
2354:
2319:
2269:
2234:
2226:there is a point in
2206:
2180:
2148:
2084:
2079:Killing vector field
2058:
1961:
1925:
1898:
1886:A Jacobi field is a
1876:
1824:
1790:
1690:
1665:
1625:
1570:
1517:
1478:
1460:
1419:
1407:Covariant derivative
1312:
1292:
1272:
1252:
1218:
1198:
1173:which has a zero at
1157:
1133:
1013:
938:
876:
842:extended Einstein's
816:
722:
585:
531:
509:
485:Almost flat manifold
470:
241:
71:improve this article
29:Mathematics glossary
3918:Riemannian geometry
3877:Poincaré conjecture
3738:Riemannian manifold
3726:Musical isomorphism
3641:Riemannian geometry
3204:for a hypersurface
3084:Riemannian manifold
3042:Radius of convexity
3015:from some point of
3007:and every point in
2473:Proper metric space
2467:Principal direction
2460:Principal curvature
1729:is a finite number
1432:Developable surface
1072:Collapsing manifold
1058:radius of convexity
856:spin–orbit coupling
314:Contents:
188:Riemannian manifold
155:Riemannian geometry
38:Convexity of radius
3867:General relativity
3810:Hopf–Rinow theorem
3757:Types of manifolds
3733:Parallel transport
3507:
3439:
3332:solvable Lie group
3294:
3183:
3102:at the same time.
2994:
2868:
2842:
2812:
2768:
2643:
2611:
2585:
2555:
2527:
2449:simplicial complex
2430:Parallel transport
2401:
2371:
2336:
2282:
2243:
2212:
2198:of a metric space
2114:Lipschitz distance
2038:
1944:
1911:
1836:
1802:
1785:semidirect product
1696:Injectivity radius
1631:
1611:
1349:
1298:
1278:
1258:
1230:
1204:
1163:
1139:
1065:Christoffel symbol
1046:
996:
965:
921:
844:General relativity
803:
762:
711:, γ : [0, ∞)→
687:
567:such that for any
549:
267:
18:Injectivity radius
3890:
3889:
3565:submanifold is a
3139:
2896:
2670:
2017:
1685:Busemann function
1558:, generated by a
1301:{\displaystyle t}
1079:Complete manifold
956:
747:
702:Busemann function
525:bi-Lipschitz map.
491:Arc-wise isometry
151:
150:
143:
125:
16:(Redirected from
3925:
3634:
3627:
3620:
3611:
3610:
3563:Totally geodesic
3516:
3514:
3513:
3508:
3497:
3496:
3448:
3446:
3445:
3440:
3420:
3419:
3395:
3394:
3303:
3301:
3300:
3295:
3290:
3289:
3192:
3190:
3189:
3184:
3140:
3137:
3112:Scalar curvature
3003:
3001:
3000:
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2889:
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2874:
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2562:
2561:
2556:
2554:
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2534:
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2444:Polyhedral space
2415:Nonexpanding map
2410:
2408:
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2337:
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2334:
2329:
2328:
2291:
2289:
2288:
2283:
2281:
2280:
2255:topological nets
2252:
2250:
2249:
2244:
2230:on the distance
2221:
2219:
2218:
2213:
2094:intrinsic metric
2047:
2045:
2044:
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2033:
2022:
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2016:
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1984:
1953:
1951:
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1918:
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1909:
1871:Intrinsic metric
1852:infranilmanifold
1845:
1843:
1842:
1837:
1811:
1809:
1808:
1803:
1769:Infranilmanifold
1640:
1638:
1637:
1632:
1620:
1618:
1617:
1612:
1598:
1566:are of the form
1400:Cotangent bundle
1358:
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1307:
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1118:Conjugate points
1107:Conformally flat
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505:totally geodesic
477:Alexandrov space
315:
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3913:Metric geometry
3893:
3892:
3891:
3886:
3855:
3834:Generalizations
3829:
3783:
3752:
3687:Exponential map
3648:
3638:
3596:
3581:
3537:Totally convex.
3527:
3492:
3488:
3477:
3474:
3473:
3415:
3411:
3390:
3386:
3378:
3375:
3374:
3320:Smooth manifold
3285:
3281:
3261:
3258:
3257:
3237:
3225:
3216:
3136:
3134:
3131:
3130:
3108:
3063:Ricci curvature
3033:
3025:quasi-isometric
2888:
2886:
2883:
2882:
2857:
2854:
2853:
2831:
2828:
2827:
2795:
2792:
2791:
2662:
2660:
2657:
2656:
2626:
2623:
2622:
2600:
2597:
2596:
2574:
2571:
2570:
2550:
2542:
2539:
2538:
2510:
2507:
2506:
2500:
2453:Euclidean space
2426:
2392:
2388:
2386:
2383:
2382:
2365:
2359:
2358:
2357:
2355:
2352:
2351:
2330:
2324:
2323:
2322:
2320:
2317:
2316:
2276:
2272:
2270:
2267:
2266:
2235:
2232:
2231:
2207:
2204:
2203:
2183:
2174:Minimal surface
2151:
2143:Logarithmic map
2087:
2061:
2023:
2009:
1993:
1989:
1985:
1983:
1980:
1979:
1962:
1959:
1958:
1932:
1928:
1926:
1923:
1922:
1905:
1901:
1899:
1896:
1895:
1879:
1825:
1822:
1821:
1791:
1788:
1787:
1751:conjugate point
1704:exponential map
1693:
1683:a level set of
1668:
1626:
1623:
1622:
1591:
1571:
1568:
1567:
1520:
1481:
1466:Exponential map
1463:
1422:
1343:
1339:
1313:
1310:
1309:
1308:, the function
1293:
1290:
1289:
1273:
1270:
1269:
1253:
1250:
1249:
1219:
1216:
1215:
1199:
1196:
1195:
1186:Convex function
1158:
1155:
1154:
1134:
1131:
1130:
1041:
1035:
1031:
1025:
1021:
1016:
1014:
1011:
1010:
990:
985:
984:
975:
971:
966:
960:
939:
936:
935:
915:
911:
896:
892:
883:
879:
877:
874:
873:
819:
789:
766:
751:
729:
725:
723:
720:
719:
681:
676:
675:
664:
652:
647:
646:
617:
608:
603:
602:
591:
586:
583:
582:
532:
529:
528:
512:
473:
467:
465:
464:
463:
462:
316:
301:convex function
261:
256:
255:
244:
242:
239:
238:
159:metric geometry
147:
136:
130:
127:
84:
82:
68:
52:
41:
30:
23:
22:
15:
12:
11:
5:
3931:
3921:
3920:
3915:
3910:
3905:
3888:
3887:
3885:
3884:
3879:
3874:
3869:
3863:
3861:
3857:
3856:
3854:
3853:
3851:Sub-Riemannian
3848:
3843:
3837:
3835:
3831:
3830:
3828:
3827:
3822:
3817:
3812:
3807:
3802:
3797:
3791:
3789:
3785:
3784:
3782:
3781:
3776:
3771:
3766:
3760:
3758:
3754:
3753:
3751:
3750:
3745:
3740:
3735:
3730:
3729:
3728:
3719:
3714:
3709:
3699:
3694:
3689:
3684:
3683:
3682:
3677:
3672:
3667:
3656:
3654:
3653:Basic concepts
3650:
3649:
3637:
3636:
3629:
3622:
3614:
3595:
3592:
3580:
3577:
3569:such that all
3531:Tangent bundle
3526:
3523:
3506:
3503:
3500:
3495:
3491:
3487:
3484:
3481:
3450:
3449:
3438:
3435:
3432:
3429:
3426:
3423:
3418:
3414:
3410:
3407:
3404:
3401:
3398:
3393:
3389:
3385:
3382:
3305:
3304:
3293:
3288:
3284:
3280:
3277:
3274:
3271:
3268:
3265:
3233:
3221:
3212:
3201:Shape operator
3194:
3193:
3182:
3179:
3176:
3173:
3170:
3167:
3164:
3161:
3158:
3155:
3152:
3149:
3146:
3143:
3124:shape operator
3107:
3104:
3032:
3029:
3005:
3004:
2993:
2990:
2987:
2984:
2981:
2978:
2975:
2972:
2969:
2966:
2963:
2960:
2957:
2954:
2951:
2948:
2945:
2942:
2939:
2936:
2933:
2930:
2927:
2924:
2921:
2918:
2915:
2912:
2909:
2906:
2903:
2900:
2895:
2892:
2867:
2864:
2861:
2841:
2838:
2835:
2824:quasi-isometry
2811:
2808:
2805:
2802:
2799:
2786:Quasi-isometry
2779:
2778:
2767:
2764:
2761:
2758:
2755:
2752:
2749:
2746:
2743:
2740:
2737:
2734:
2731:
2728:
2725:
2722:
2719:
2716:
2713:
2710:
2707:
2704:
2701:
2698:
2695:
2692:
2689:
2686:
2683:
2680:
2677:
2674:
2669:
2666:
2642:
2639:
2636:
2633:
2630:
2610:
2607:
2604:
2584:
2581:
2578:
2553:
2549:
2546:
2526:
2523:
2520:
2517:
2514:
2499:
2496:
2425:
2422:
2400:
2395:
2391:
2368:
2362:
2333:
2327:
2279:
2275:
2242:
2239:
2211:
2182:
2179:
2155:Mean curvature
2150:
2147:
2086:
2083:
2060:
2057:
2049:
2048:
2037:
2032:
2029:
2026:
2021:
2015:
2012:
2007:
2004:
2001:
1996:
1992:
1988:
1982:
1978:
1975:
1972:
1969:
1966:
1943:
1940:
1935:
1931:
1908:
1904:
1878:
1875:
1835:
1832:
1829:
1801:
1798:
1795:
1745:conjugate to
1712:diffeomorphism
1692:
1689:
1672:Hadamard space
1667:
1664:
1630:
1610:
1607:
1604:
1601:
1597:
1594:
1590:
1587:
1584:
1581:
1578:
1575:
1554:of a manifold
1549:tangent bundle
1519:
1516:
1485:Finsler metric
1480:
1477:
1462:
1459:
1439:to the plane.
1421:
1418:
1393:totally convex
1346:
1342:
1338:
1335:
1332:
1329:
1326:
1323:
1320:
1317:
1297:
1277:
1257:
1229:
1226:
1223:
1203:
1162:
1138:
1129:on a geodesic
1056:are less than
1044:
1038:
1034:
1028:
1024:
1019:
1007:
1006:
993:
988:
983:
978:
974:
969:
963:
959:
955:
952:
949:
946:
943:
918:
914:
910:
907:
904:
899:
895:
891:
886:
882:
862:Center of mass
852:affine torsion
818:
815:
814:
813:
802:
799:
796:
792:
788:
785:
782:
779:
776:
773:
769:
765:
760:
757:
754:
750:
746:
743:
740:
737:
732:
728:
698:
697:
684:
679:
674:
671:
667:
663:
660:
655:
650:
645:
642:
639:
636:
633:
630:
627:
624:
620:
616:
611:
606:
601:
598:
594:
590:
548:
545:
542:
539:
536:
519:center of mass
511:
508:
472:
469:
459:
458:
453:
448:
443:
438:
433:
428:
423:
418:
413:
408:
403:
398:
393:
388:
383:
378:
373:
368:
363:
358:
353:
348:
343:
338:
333:
328:
323:
317:
311:
309:
264:
259:
254:
251:
247:
211:
210:
205:
200:
191:
190:
185:
180:
175:
149:
148:
55:
53:
46:
28:
9:
6:
4:
3:
2:
3930:
3919:
3916:
3914:
3911:
3909:
3906:
3904:
3901:
3900:
3898:
3883:
3880:
3878:
3875:
3873:
3870:
3868:
3865:
3864:
3862:
3858:
3852:
3849:
3847:
3844:
3842:
3839:
3838:
3836:
3832:
3826:
3825:Schur's lemma
3823:
3821:
3818:
3816:
3813:
3811:
3808:
3806:
3803:
3801:
3800:Gauss's lemma
3798:
3796:
3793:
3792:
3790:
3786:
3780:
3777:
3775:
3772:
3770:
3767:
3765:
3762:
3761:
3759:
3755:
3749:
3746:
3744:
3741:
3739:
3736:
3734:
3731:
3727:
3723:
3720:
3718:
3715:
3713:
3710:
3708:
3705:
3704:
3703:
3702:Metric tensor
3700:
3698:
3697:Inner product
3695:
3693:
3690:
3688:
3685:
3681:
3678:
3676:
3673:
3671:
3668:
3666:
3663:
3662:
3661:
3658:
3657:
3655:
3651:
3646:
3642:
3635:
3630:
3628:
3623:
3621:
3616:
3615:
3612:
3608:
3606:
3602:
3601:
3591:
3589:
3585:
3576:
3574:
3573:
3568:
3564:
3560:
3558:
3554:
3550:
3546:
3542:
3538:
3534:
3533:
3532:
3522:
3520:
3501:
3493:
3489:
3485:
3482:
3479:
3471:
3467:
3463:
3462:
3457:
3456:
3455:
3430:
3424:
3416:
3412:
3408:
3399:
3391:
3387:
3380:
3373:
3372:
3371:
3369:
3365:
3361:
3357:
3353:
3349:
3345:
3344:
3339:
3337:
3333:
3329:
3328:
3323:
3322:
3321:
3316:
3314:
3313:
3308:
3291:
3286:
3278:
3275:
3269:
3263:
3256:
3255:
3254:
3252:
3248:
3244:
3240:
3236:
3232:
3228:
3224:
3220:
3215:
3211:
3207:
3203:
3202:
3197:
3177:
3174:
3168:
3162:
3156:
3150:
3147:
3144:
3129:
3128:
3127:
3125:
3121:
3120:
3115:
3114:
3113:
3103:
3101:
3097:
3093:
3092:
3087:
3086:
3085:
3080:
3079:
3078:
3073:
3072:
3071:
3066:
3065:
3064:
3059:
3057:
3053:
3051:
3047:
3043:
3039:
3037:
3028:
3026:
3022:
3018:
3014:
3010:
2991:
2988:
2985:
2979:
2976:
2973:
2967:
2964:
2961:
2952:
2946:
2943:
2937:
2931:
2925:
2922:
2919:
2916:
2910:
2907:
2904:
2898:
2893:
2890:
2881:
2880:
2879:
2865:
2862:
2859:
2839:
2836:
2833:
2825:
2809:
2803:
2800:
2797:
2789:
2787:
2782:
2765:
2762:
2759:
2753:
2750:
2747:
2741:
2738:
2735:
2726:
2720:
2717:
2711:
2705:
2699:
2696:
2693:
2690:
2684:
2681:
2678:
2672:
2667:
2664:
2655:
2654:
2653:
2640:
2637:
2634:
2631:
2628:
2608:
2605:
2602:
2582:
2579:
2576:
2568:
2567:quasigeodesic
2547:
2544:
2524:
2518:
2515:
2512:
2504:
2503:Quasigeodesic
2495:
2494:
2493:
2488:
2486:
2482:
2478:
2474:
2470:
2468:
2464:
2462:
2461:
2456:
2454:
2450:
2446:
2445:
2440:
2439:
2438:
2437:Path isometry
2433:
2432:
2431:
2421:
2420:
2416:
2412:
2398:
2393:
2389:
2366:
2349:
2331:
2314:
2310:
2309:
2308:Normal bundle
2304:
2302:
2298:
2295:
2277:
2273:
2264:
2263:
2258:
2256:
2240:
2237:
2229:
2225:
2209:
2201:
2197:
2193:
2189:
2187:
2178:
2176:
2175:
2170:
2169:
2168:
2167:Metric tensor
2163:
2162:
2158:
2157:
2156:
2146:
2144:
2140:
2139:
2138:
2137:Lipschitz map
2133:
2131:
2127:
2123:
2119:
2115:
2111:
2109:
2105:
2103:
2102:
2097:
2095:
2091:
2090:Length metric
2082:
2081:
2080:
2075:
2074:
2073:
2072:Kähler metric
2068:
2067:
2066:
2056:
2055:
2054:
2035:
2030:
2027:
2024:
2019:
2013:
2002:
1994:
1990:
1976:
1970:
1964:
1957:
1956:
1955:
1941:
1938:
1933:
1929:
1906:
1902:
1893:
1889:
1885:
1884:
1874:
1873:
1872:
1867:
1865:
1864:
1859:
1857:
1853:
1850:is called an
1849:
1833:
1830:
1827:
1819:
1815:
1799:
1796:
1793:
1786:
1782:
1778:
1774:
1770:
1766:
1764:
1760:
1756:
1752:
1748:
1744:
1740:
1736:
1732:
1728:
1723:
1721:
1717:
1713:
1709:
1705:
1701:
1697:
1688:
1686:
1682:
1681:
1676:
1674:
1673:
1663:
1661:
1657:
1656:
1651:
1650:
1646:
1644:
1628:
1602:
1595:
1592:
1588:
1582:
1576:
1565:
1561:
1557:
1553:
1550:
1546:
1542:
1541:
1540:Geodesic flow
1536:
1534:
1530:
1526:
1525:
1515:
1514:
1513:
1512:Flat manifold
1508:
1506:
1505:metric tensor
1502:
1498:
1494:
1493:
1488:
1487:
1486:
1476:
1475:
1471:
1467:
1458:
1456:
1452:
1448:
1444:
1440:
1438:
1435:is a surface
1434:
1433:
1428:
1426:
1417:
1416:
1415:
1410:
1409:
1408:
1403:
1402:
1401:
1396:
1394:
1390:
1386:
1385:shortest path
1382:
1378:
1374:
1370:
1369:
1364:
1362:
1344:
1340:
1336:
1333:
1327:
1321:
1318:
1315:
1295:
1275:
1255:
1247:
1244:. A function
1243:
1227:
1224:
1221:
1214:the function
1201:
1193:
1189:
1187:
1182:
1180:
1176:
1160:
1152:
1136:
1128:
1124:
1120:
1119:
1114:
1112:
1108:
1104:
1102:
1101:
1100:Conformal map
1096:
1095:
1094:
1089:
1088:
1087:
1082:
1081:
1080:
1075:
1074:
1073:
1068:
1067:
1066:
1061:
1059:
1036:
1032:
1026:
1022:
991:
981:
976:
972:
961:
957:
953:
947:
941:
934:
933:
932:
916:
912:
908:
905:
902:
897:
893:
889:
884:
880:
871:
868: ∈
867:
863:
859:
857:
853:
849:
845:
841:
840:
835:
833:
829:
825:
824:
797:
794:
786:
783:
777:
771:
752:
744:
738:
730:
726:
718:
717:
716:
714:
710:
709:
704:
703:
682:
672:
669:
661:
658:
653:
640:
634:
628:
622:
614:
609:
599:
596:
588:
581:
580:
579:
578:
574:
570:
566:
562:
546:
540:
537:
534:
526:
522:
520:
516:
507:
506:
502:
498:
496:
495:path isometry
492:
488:
487:
486:
481:
479:
478:
468:
461:
457:
454:
452:
449:
447:
444:
442:
439:
437:
434:
432:
429:
427:
424:
422:
419:
417:
414:
412:
409:
407:
404:
402:
399:
397:
394:
392:
389:
387:
384:
382:
379:
377:
374:
372:
369:
367:
364:
362:
359:
357:
354:
352:
349:
347:
344:
342:
339:
337:
334:
332:
329:
327:
324:
322:
319:
318:
308:
306:
302:
298:
294:
292:
288:
284:
280:
262:
252:
249:
236:
232:
228:
224:
220:
216:
209:
206:
204:
201:
199:
196:
195:
194:
189:
186:
184:
181:
179:
176:
174:
171:
170:
169:
166:
164:
160:
156:
145:
142:
134:
131:December 2009
123:
120:
116:
113:
109:
106:
102:
99:
95:
92: –
91:
87:
86:Find sources:
80:
76:
72:
66:
65:
61:
56:This article
54:
50:
45:
44:
39:
35:
27:
19:
3860:Applications
3788:Main results
3644:
3605:Cayley graph
3598:
3597:
3583:
3582:
3570:
3566:
3562:
3561:
3556:
3552:
3548:
3544:
3540:
3536:
3535:
3529:
3528:
3518:
3469:
3468:-systole of
3465:
3459:
3458:
3452:
3451:
3370:-ball, i.e.
3367:
3366:-ball is an
3363:
3359:
3355:
3351:
3347:
3346:a short map
3341:
3340:
3327:Sol manifold
3325:
3324:
3318:
3317:
3310:
3309:
3306:
3250:
3246:
3242:
3238:
3234:
3230:
3226:
3222:
3218:
3213:
3209:
3205:
3199:
3198:
3195:
3123:
3117:
3116:
3110:
3109:
3099:
3089:
3088:
3082:
3081:
3075:
3074:
3068:
3067:
3061:
3060:
3055:
3054:
3049:
3045:
3041:
3040:
3035:
3034:
3024:
3020:
3016:
3012:
3008:
3006:
2823:
2822:is called a
2784:
2783:
2780:
2566:
2502:
2501:
2490:
2489:
2472:
2471:
2466:
2465:
2458:
2457:
2442:
2441:
2435:
2434:
2428:
2427:
2418:
2414:
2413:
2347:
2312:
2306:
2305:
2260:
2259:
2227:
2223:
2199:
2195:
2191:
2190:
2185:
2184:
2172:
2171:
2165:
2164:
2160:
2159:
2153:
2152:
2142:
2141:
2135:
2134:
2129:
2125:
2122:bi-Lipschitz
2121:
2117:
2113:
2112:
2107:
2106:
2099:
2098:
2093:
2092:the same as
2089:
2088:
2077:
2076:
2070:
2069:
2063:
2062:
2053:Jordan curve
2051:
2050:
1888:vector field
1883:Jacobi field
1881:
1880:
1869:
1868:
1861:
1860:
1851:
1847:
1817:
1813:
1780:
1776:
1772:
1768:
1767:
1758:
1754:
1750:
1746:
1742:
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1734:
1730:
1726:
1724:
1715:
1707:
1699:
1695:
1694:
1684:
1678:
1677:
1670:
1669:
1653:
1652:
1647:
1564:trajectories
1560:vector field
1555:
1551:
1538:
1537:
1522:
1521:
1510:
1509:
1490:
1489:
1483:
1482:
1465:
1464:
1450:
1446:
1442:
1441:
1430:
1429:
1424:
1423:
1412:
1411:
1405:
1404:
1398:
1397:
1392:
1388:
1384:
1380:
1376:
1372:
1366:
1365:
1245:
1191:
1184:
1183:
1178:
1174:
1150:
1126:
1122:
1116:
1115:
1110:
1106:
1105:
1098:
1097:
1091:
1090:
1084:
1083:
1077:
1076:
1070:
1069:
1063:
1062:
1057:
1008:
869:
865:
861:
860:
837:
836:
832:CAT(0) space
827:
821:
820:
712:
706:
700:
699:
576:
572:
568:
564:
560:
524:
523:
518:
514:
513:
504:
503:the same as
501:Autoparallel
500:
499:
494:
493:the same as
490:
489:
483:
482:
475:
474:
466:
460:
304:
300:
296:
295:
290:
286:
282:
278:
234:
230:
226:
222:
218:
214:
212:
192:
183:Metric space
167:
152:
137:
128:
118:
111:
104:
97:
85:
69:Please help
57:
26:
3600:Word metric
3567:submanifold
3555:, see also
3358:and radius
3044:at a point
2878:such that
2477:closed ball
2262:Nilmanifold
2194:. A subset
2161:Metric ball
1856:nilmanifold
1391:, see also
1383:there is a
1190:A function
1149:are called
1121:two points
1109:a manifold
34:radius bone
3897:Categories
3820:Ricci flow
3769:Hyperbolic
3096:submersion
2202:is called
1680:Horosphere
1248:is called
1093:Completion
864:. A point
515:Barycenter
305:convex set
193:See also:
173:Connection
101:newspapers
3764:Hermitian
3717:Signature
3680:Sectional
3660:Curvature
3572:geodesics
3539:A subset
3312:Short map
3283:∇
3279:±
3181:⟩
3160:⟨
2962:≤
2923:≤
2917:−
2863:≥
2837:≥
2807:→
2736:≤
2697:≤
2691:−
2638:∈
2606:≥
2580:≥
2548:⊆
2522:→
2419:short map
2297:Lie group
2294:nilpotent
2241:ϵ
2238:≤
2210:ϵ
2025:τ
2014:τ
2011:∂
1995:τ
1991:γ
1987:∂
1942:γ
1930:γ
1907:τ
1903:γ
1831:⋊
1797:⋊
1720:cut locus
1629:γ
1593:γ
1577:γ
1455:Lipschitz
1437:isometric
1414:Cut locus
1371:A subset
1337:λ
1334:−
1322:γ
1319:∘
1276:γ
1256:λ
1228:γ
1225:∘
1202:γ
1161:γ
1151:conjugate
1137:γ
958:∑
906:…
795:−
784:−
772:γ
759:∞
756:→
731:γ
659:≤
615:≤
544:→
289:. Italic
178:Curvature
58:does not
3779:Kenmotsu
3692:Geodesic
3645:Glossary
3588:geodesic
3360:r < R
3352:R > 0
3343:Submetry
3100:submetry
3052:subset.
2485:complete
2417:same as
1892:geodesic
1863:Isometry
1761:. For a
1660:geodesic
1643:geodesic
1596:′
1533:distance
1524:Geodesic
1501:pullback
1443:Dilation
1425:Diameter
705:given a
297:A caveat
3846:Hilbert
3841:Finsler
3464:. The
3461:Systole
3336:lattice
3217::
3070:Riemann
2537:(where
2481:compact
2301:lattice
2128:), exp(
1503:of the
1499:is the
1495:for an
115:scholar
79:removed
64:sources
3774:Kähler
3670:Scalar
3665:tensor
3557:convex
3050:convex
3036:Radius
2790:A map
1763:closed
1714:. The
1621:where
1562:whose
1368:Convex
1361:convex
1242:convex
839:Cartan
527:A map
517:, see
117:
110:
103:
96:
88:
36:, see
3675:Ricci
3334:by a
3241:. If
2299:by a
1921:with
1890:on a
1757:from
1749:(see
1710:is a
1641:is a
1547:on a
1543:is a
1529:curve
1527:is a
237:| or
122:JSTOR
108:books
3249:and
3098:and
2852:and
2595:and
1545:flow
1177:and
1125:and
571:and
563:and
291:word
281:and
157:and
94:news
62:any
60:cite
3056:Ray
2479:is
2192:Net
2132:).
1812:on
1779:of
1706:at
1359:is
1240:is
846:to
749:lim
708:ray
575:in
326:0–9
321:Top
285:in
73:by
3899::
3590:.
3559:.
3472:,
3338:.
3138:II
3027:.
2487:.
2455:.
2447:a
2411:.
2303:.
2096:.
1858:.
1722:.
1687:.
1662:.
1645:.
1552:TM
1535:.
1507:.
1472:,
1468::
1457:.
1395:.
1363:.
1181:.
1060:.
858:.
834:.
521:.
497:.
303:,
235:xy
229:,
221:,
217:,
165:.
3724:/
3647:)
3643:(
3633:e
3626:t
3619:v
3594:W
3579:U
3553:K
3549:K
3545:M
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3525:T
3519:k
3505:)
3502:M
3499:(
3494:k
3490:t
3486:s
3483:y
3480:s
3470:M
3466:k
3437:)
3434:)
3431:x
3428:(
3425:f
3422:(
3417:r
3413:B
3409:=
3406:)
3403:)
3400:x
3397:(
3392:r
3388:B
3384:(
3381:f
3368:r
3364:r
3356:x
3348:f
3292:n
3287:v
3276:=
3273:)
3270:v
3267:(
3264:S
3251:v
3247:M
3243:n
3239:M
3235:p
3231:T
3229:→
3227:M
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3214:p
3210:S
3206:M
3178:w
3175:,
3172:)
3169:v
3166:(
3163:S
3157:=
3154:)
3151:w
3148:,
3145:v
3142:(
3106:S
3046:p
3031:R
3021:X
3019:(
3017:f
3013:C
3009:Y
2992:.
2989:C
2986:+
2983:)
2980:y
2977:,
2974:x
2971:(
2968:d
2965:K
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2953:y
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2911:y
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2905:x
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2899:d
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2891:1
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2860:C
2840:1
2834:K
2810:Y
2804:X
2801::
2798:f
2788:.
2766:.
2763:C
2760:+
2757:)
2754:y
2751:,
2748:x
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2516::
2513:f
2498:Q
2424:P
2399:M
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2367:N
2361:R
2348:p
2332:N
2326:R
2313:M
2278:1
2274:S
2228:S
2224:X
2200:X
2196:S
2181:N
2149:M
2130:r
2126:r
2118:r
2085:L
2059:K
2036:.
2031:0
2028:=
2020:|
2006:)
2003:t
2000:(
1977:=
1974:)
1971:t
1968:(
1965:J
1939:=
1934:0
1877:J
1848:N
1834:F
1828:N
1818:N
1814:N
1800:F
1794:N
1781:N
1777:F
1773:N
1759:p
1755:r
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1739:p
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1600:(
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992:2
987:|
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968:|
962:i
954:=
951:)
948:x
945:(
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913:p
909:,
903:,
898:2
894:p
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885:1
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791:|
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778:t
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768:|
764:(
753:t
745:=
742:)
739:p
736:(
727:B
713:X
683:X
678:|
673:y
670:x
666:|
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635:f
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629:x
626:(
623:f
619:|
610:X
605:|
600:y
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593:|
589:c
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569:x
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547:Y
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538::
535:f
510:B
471:A
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446:X
441:W
436:V
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426:T
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416:R
411:Q
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401:O
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376:J
371:I
366:H
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356:F
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346:D
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231:N
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223:Z
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138:(
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129:(
119:·
112:·
105:·
98:·
81:.
67:.
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