2612:
1786:
2130:
2281:
1976:
123:
A basic result of differential geometry states that normal coordinates at a point always exist on a manifold with a symmetric affine connection. In such coordinates the covariant derivative reduces to a partial derivative (at
538:
1413:
2481:
2500:
234:
1638:
345:
1518:
968:
1987:
1583:
2369:
Polar coordinates provide a number of fundamental tools in
Riemannian geometry. The radial coordinate is the most significant: geometrically it represents the geodesic distance to
1215:
882:
1292:
2417:
2736:
837:
1463:
2141:
1160:
791:
1075:
1610:
1797:
1322:
1130:
737:
378:
291:
764:
663:
1630:
1538:
1242:
1097:
1031:
1008:
988:
707:
683:
636:
616:
596:
257:
465:
1327:
2701:
2374:
176:
73:
2425:
2665:
2607:{\displaystyle g={\begin{bmatrix}1&0&\cdots \ 0\\0&&\\\vdots &&g_{\phi \phi }(r,\phi )\\0&&\end{bmatrix}}.}
2678:
1781:{\displaystyle g_{\mu \nu }(x)=\delta _{\mu \nu }-{\tfrac {1}{3}}R_{\mu \sigma \nu \tau }(0)x^{\sigma }x^{\tau }+O(|x|^{3}).}
184:
299:
2731:
1468:
887:
141:
2125:{\displaystyle e_{\mu }^{*a}(x)=\delta _{a\mu }-{\tfrac {1}{6}}R_{a\sigma \mu \tau }(0)x^{\sigma }x^{\tau }+O(x^{2}),}
65:
2716:
1543:
1165:
351:
61:
50:
1247:
2391:
93:
2711:
2626:
2296:
145:
38:
842:
800:
2276:{\displaystyle {\omega ^{a}}_{b\mu }(x)=-{\tfrac {1}{2}}{R^{a}}_{b\mu \tau }(0)x^{\tau }+O(|x|^{2}).}
1427:
1221:
97:
85:
1135:
2670:
2386:
769:
113:
1971:{\displaystyle {\Gamma ^{\lambda }}_{\mu \nu }(x)=-{\tfrac {1}{3}}{\bigl }x^{\tau }+O(|x|^{2}).}
578:
The properties of normal coordinates often simplify computations. In the following, assume that
1036:
2706:
2674:
2643:
1588:
563:
385:
153:
57:
2635:
1297:
1105:
712:
567:
380:. If the additional structure of a Riemannian metric is imposed, then the basis defined by
357:
266:
157:
2655:
742:
641:
17:
2651:
137:
105:
2307:
obtained by introducing the standard spherical coordinate system on the
Euclidean space
2491:
1615:
1523:
1227:
1082:
1016:
993:
973:
692:
668:
621:
601:
581:
436:
242:
149:
54:
175:
are local coordinates on a manifold with an affine connection defined by means of the
92:, thus often simplifying local calculations. In normal coordinates associated to the
2725:
1100:
419:
152:. Normal coordinates always exist for the Levi-Civita connection of a Riemannian or
101:
77:
156:
manifold. By contrast, in general there is no way to define normal coordinates for
533:{\displaystyle \varphi :=E^{-1}\circ \exp _{p}^{-1}:U\rightarrow \mathbb {R} ^{n}}
30:
This article is about
Differential geometry. For use in classical mechanics, see
31:
1408:{\displaystyle {\frac {\partial g_{ij}}{\partial x^{k}}}(p)=0,\,\forall i,j,k}
2647:
990:. Thus radial paths in normal coordinates are exactly the geodesics through
136:(the affine parameter). This idea was implemented in a fundamental way by
2378:
2351:
794:
2639:
2476:{\displaystyle \langle df,dr\rangle ={\frac {\partial f}{\partial r}}}
2624:
Busemann, Herbert (1955), "On normal coordinates in
Finsler spaces",
1465:
equipped with a locally orthonormal coordinate system in which
1294:. In the Riemannian case, so do the first partial derivatives of
1791:
The corresponding Levi-Civita connection
Christoffel symbols are
562:. The existence of these sort of open neighborhoods (they form a
395:
Normal coordinates exist on a normal neighborhood of a point
160:
in a way that the exponential map are twice-differentiable (
2339: ≥ 0 is the radial parameter and φ = (φ
2490:. As a result, the metric in polar coordinates assumes a
1612:
so that the components of the metric tensor away from
388:, and the resulting coordinate system is then known as a
2291:
On a
Riemannian manifold, a normal coordinate system at
2515:
2183:
2038:
1839:
1684:
229:{\displaystyle \exp _{p}:T_{p}M\supset V\rightarrow M}
2503:
2428:
2394:
2144:
2135:
and the spin-connection coefficients take the values
1990:
1800:
1641:
1618:
1591:
1546:
1526:
1471:
1430:
1330:
1300:
1250:
1230:
1168:
1138:
1108:
1085:
1039:
1019:
996:
976:
890:
845:
803:
772:
745:
715:
695:
671:
644:
624:
604:
584:
468:
360:
302:
269:
245:
187:
340:{\displaystyle E:\mathbb {R} ^{n}\rightarrow T_{p}M}
1981:Similarly we can construct local coframes in which
2606:
2475:
2411:
2275:
2124:
1970:
1780:
1624:
1604:
1577:
1532:
1513:{\displaystyle g_{\mu \nu }(0)=\delta _{\mu \nu }}
1512:
1457:
1407:
1316:
1286:
1236:
1209:
1154:
1124:
1091:
1069:
1025:
1002:
982:
963:{\displaystyle \gamma _{V}(t)=(tV^{1},...,tV^{n})}
962:
876:
831:
785:
758:
731:
701:
677:
657:
630:
610:
590:
532:
372:
339:
285:
251:
228:
27:Special coordinate system in Differential Geometry
547:and therefore the chart, is in no way unique. A
2362:,φ) with the inverse of the exponential map at
2663:Kobayashi, Shoshichi; Nomizu, Katsumi (1996),
1918:
1852:
598:is a normal neighborhood centered at a point
8:
2447:
2429:
2295:facilitates the introduction of a system of
1079:In Riemannian normal coordinates at a point
354:of the tangent space at the fixed basepoint
2737:Coordinate systems in differential geometry
1578:{\displaystyle R_{\mu \sigma \nu \tau }(0)}
2331:the standard spherical coordinate system (
2560:
2510:
2502:
2453:
2427:
2398:
2393:
2261:
2256:
2247:
2232:
2207:
2200:
2195:
2182:
2158:
2151:
2146:
2143:
2110:
2091:
2081:
2053:
2037:
2025:
2000:
1995:
1989:
1956:
1951:
1942:
1927:
1917:
1916:
1892:
1861:
1851:
1850:
1838:
1814:
1807:
1802:
1799:
1766:
1761:
1752:
1737:
1727:
1699:
1683:
1671:
1646:
1640:
1617:
1596:
1590:
1551:
1545:
1525:
1501:
1476:
1470:
1429:
1386:
1359:
1341:
1331:
1329:
1305:
1299:
1263:
1255:
1249:
1229:
1198:
1173:
1167:
1143:
1137:
1113:
1107:
1084:
1038:
1018:
995:
975:
951:
923:
895:
889:
850:
844:
808:
802:
777:
771:
750:
744:
720:
714:
694:
670:
649:
643:
623:
603:
583:
524:
520:
519:
500:
495:
479:
467:
414:such that there is a proper neighborhood
359:
328:
315:
311:
310:
301:
274:
268:
244:
205:
192:
186:
161:
100:, one can additionally arrange that the
2687:Chern, S. S.; Chen, W. H.; Lam, K. S.;
1210:{\displaystyle g_{ij}(p)=\delta _{ij}}
88:of the connection vanish at the point
84:. In a normal coordinate system, the
1287:{\displaystyle \Gamma _{ij}^{k}(p)=0}
7:
2666:Foundations of Differential Geometry
2412:{\displaystyle \partial /\partial r}
1424:In the neighbourhood of any point
390:Riemannian normal coordinate system
2464:
2456:
2403:
2395:
1804:
1387:
1352:
1334:
1252:
570:for symmetric affine connections.
554:is a normal neighborhood of every
384:may be required in addition to be
25:
2689:Lectures on Differential Geometry
877:{\displaystyle \gamma _{V}'(0)=V}
128:only), and the geodesics through
2303:. These are the coordinates on
1585:we can adjust the coordinates
832:{\displaystyle \gamma _{V}(0)=p}
132:are locally linear functions of
2350:) is a parameterization of the
2669:, vol. 1 (New ed.),
2581:
2569:
2366:is a polar coordinate system.
2319:. That is, one introduces on
2267:
2257:
2248:
2244:
2225:
2219:
2173:
2167:
2116:
2103:
2074:
2068:
2015:
2009:
1962:
1952:
1943:
1939:
1913:
1907:
1882:
1876:
1829:
1823:
1772:
1762:
1753:
1749:
1720:
1714:
1661:
1655:
1572:
1566:
1520:and the Riemann tensor at
1491:
1485:
1458:{\displaystyle p=(0,\ldots 0)}
1452:
1437:
1374:
1368:
1275:
1269:
1188:
1182:
1064:
1040:
957:
913:
907:
901:
884:. Then in normal coordinates,
865:
859:
820:
814:
515:
321:
220:
1:
1013:The coordinates of the point
1155:{\displaystyle \delta _{ij}}
566:) has been established by
148:uses normal coordinates via
142:general theory of relativity
786:{\displaystyle \gamma _{V}}
447:. On a normal neighborhood
173:Geodesic normal coordinates
168:Geodesic normal coordinates
18:Geodesic normal coordinates
2753:
766:in local coordinates, and
665:are normal coordinates on
549:convex normal neighborhood
29:
1070:{\displaystyle (0,...,0)}
459:, the chart is given by:
72:obtained by applying the
2691:, World Scientific, 2000
2486:for any smooth function
1605:{\displaystyle x^{\mu }}
259:an open neighborhood of
62:local coordinate system
51:differentiable manifold
2717:Synge's world function
2608:
2477:
2413:
2277:
2126:
1972:
1782:
1626:
1606:
1579:
1534:
1514:
1459:
1409:
1318:
1317:{\displaystyle g_{ij}}
1288:
1238:
1211:
1156:
1126:
1125:{\displaystyle g_{ij}}
1099:the components of the
1093:
1071:
1027:
1004:
984:
964:
878:
833:
787:
760:
733:
732:{\displaystyle T_{p}M}
703:
679:
659:
632:
612:
592:
534:
374:
373:{\displaystyle p\in M}
341:
287:
286:{\displaystyle T_{p}M}
253:
230:
94:Levi-Civita connection
2712:Local reference frame
2627:Mathematische Annalen
2609:
2478:
2414:
2297:spherical coordinates
2278:
2127:
1973:
1783:
1627:
1607:
1580:
1535:
1515:
1460:
1410:
1319:
1289:
1239:
1212:
1157:
1127:
1094:
1072:
1028:
1005:
985:
965:
879:
834:
788:
761:
759:{\displaystyle V^{i}}
734:
704:
680:
660:
658:{\displaystyle x^{i}}
633:
613:
593:
535:
418:of the origin in the
410:is an open subset of
375:
342:
293:, and an isomorphism
288:
254:
231:
146:equivalence principle
112:, and that the first
39:differential geometry
2501:
2426:
2392:
2142:
1988:
1798:
1639:
1616:
1589:
1544:
1524:
1469:
1428:
1328:
1298:
1248:
1228:
1166:
1136:
1106:
1083:
1037:
1017:
994:
974:
970:as long as it is in
888:
843:
801:
770:
743:
713:
709:be some vector from
693:
669:
642:
622:
602:
582:
466:
358:
300:
267:
243:
185:
2732:Riemannian geometry
2373:of nearby points.
2358:. Composition of (
2008:
1268:
1222:Christoffel symbols
858:
508:
405:normal neighborhood
114:partial derivatives
98:Riemannian manifold
86:Christoffel symbols
2671:Wiley Interscience
2640:10.1007/BF01362381
2604:
2595:
2473:
2409:
2387:partial derivative
2273:
2192:
2122:
2047:
1991:
1968:
1848:
1778:
1693:
1622:
1602:
1575:
1530:
1510:
1455:
1405:
1314:
1284:
1251:
1234:
1207:
1152:
1122:
1089:
1067:
1023:
1000:
980:
960:
874:
846:
829:
783:
756:
729:
699:
675:
655:
628:
608:
588:
530:
491:
370:
337:
283:
249:
226:
43:normal coordinates
2707:Fermi coordinates
2533:
2471:
2377:asserts that the
2301:polar coordinates
2287:Polar coordinates
2191:
2046:
1847:
1692:
1625:{\displaystyle p}
1533:{\displaystyle p}
1420:Explicit formulae
1366:
1237:{\displaystyle p}
1101:Riemannian metric
1092:{\displaystyle p}
1026:{\displaystyle p}
1003:{\displaystyle p}
983:{\displaystyle U}
702:{\displaystyle V}
678:{\displaystyle U}
631:{\displaystyle M}
611:{\displaystyle p}
591:{\displaystyle U}
564:topological basis
252:{\displaystyle V}
158:Finsler manifolds
154:Pseudo-Riemannian
116:of the metric at
58:affine connection
16:(Redirected from
2744:
2683:
2658:
2613:
2611:
2610:
2605:
2600:
2599:
2593:
2592:
2568:
2567:
2554:
2546:
2545:
2531:
2482:
2480:
2479:
2474:
2472:
2470:
2462:
2454:
2418:
2416:
2415:
2410:
2402:
2356:−1)-sphere
2282:
2280:
2279:
2274:
2266:
2265:
2260:
2251:
2237:
2236:
2218:
2217:
2206:
2205:
2204:
2193:
2184:
2166:
2165:
2157:
2156:
2155:
2131:
2129:
2128:
2123:
2115:
2114:
2096:
2095:
2086:
2085:
2067:
2066:
2048:
2039:
2033:
2032:
2007:
1999:
1977:
1975:
1974:
1969:
1961:
1960:
1955:
1946:
1932:
1931:
1922:
1921:
1906:
1905:
1875:
1874:
1856:
1855:
1849:
1840:
1822:
1821:
1813:
1812:
1811:
1787:
1785:
1784:
1779:
1771:
1770:
1765:
1756:
1742:
1741:
1732:
1731:
1713:
1712:
1694:
1685:
1679:
1678:
1654:
1653:
1631:
1629:
1628:
1623:
1611:
1609:
1608:
1603:
1601:
1600:
1584:
1582:
1581:
1576:
1565:
1564:
1540:takes the value
1539:
1537:
1536:
1531:
1519:
1517:
1516:
1511:
1509:
1508:
1484:
1483:
1464:
1462:
1461:
1456:
1414:
1412:
1411:
1406:
1367:
1365:
1364:
1363:
1350:
1349:
1348:
1332:
1323:
1321:
1320:
1315:
1313:
1312:
1293:
1291:
1290:
1285:
1267:
1262:
1243:
1241:
1240:
1235:
1216:
1214:
1213:
1208:
1206:
1205:
1181:
1180:
1161:
1159:
1158:
1153:
1151:
1150:
1131:
1129:
1128:
1123:
1121:
1120:
1098:
1096:
1095:
1090:
1076:
1074:
1073:
1068:
1032:
1030:
1029:
1024:
1009:
1007:
1006:
1001:
989:
987:
986:
981:
969:
967:
966:
961:
956:
955:
928:
927:
900:
899:
883:
881:
880:
875:
854:
838:
836:
835:
830:
813:
812:
792:
790:
789:
784:
782:
781:
765:
763:
762:
757:
755:
754:
739:with components
738:
736:
735:
730:
725:
724:
708:
706:
705:
700:
684:
682:
681:
676:
664:
662:
661:
656:
654:
653:
637:
635:
634:
629:
617:
615:
614:
609:
597:
595:
594:
589:
568:J.H.C. Whitehead
543:The isomorphism
539:
537:
536:
531:
529:
528:
523:
507:
499:
487:
486:
379:
377:
376:
371:
346:
344:
343:
338:
333:
332:
320:
319:
314:
292:
290:
289:
284:
279:
278:
258:
256:
255:
250:
235:
233:
232:
227:
210:
209:
197:
196:
53:equipped with a
21:
2752:
2751:
2747:
2746:
2745:
2743:
2742:
2741:
2722:
2721:
2698:
2681:
2662:
2623:
2620:
2594:
2591:
2585:
2584:
2556:
2553:
2547:
2544:
2538:
2537:
2526:
2521:
2511:
2499:
2498:
2463:
2455:
2424:
2423:
2390:
2389:
2349:
2342:
2327:
2315:
2289:
2255:
2228:
2196:
2194:
2147:
2145:
2140:
2139:
2106:
2087:
2077:
2049:
2021:
1986:
1985:
1950:
1923:
1888:
1857:
1803:
1801:
1796:
1795:
1760:
1733:
1723:
1695:
1667:
1642:
1637:
1636:
1614:
1613:
1592:
1587:
1586:
1547:
1542:
1541:
1522:
1521:
1497:
1472:
1467:
1466:
1426:
1425:
1422:
1355:
1351:
1337:
1333:
1326:
1325:
1301:
1296:
1295:
1246:
1245:
1226:
1225:
1194:
1169:
1164:
1163:
1139:
1134:
1133:
1109:
1104:
1103:
1081:
1080:
1035:
1034:
1015:
1014:
992:
991:
972:
971:
947:
919:
891:
886:
885:
841:
840:
804:
799:
798:
773:
768:
767:
746:
741:
740:
716:
711:
710:
691:
690:
667:
666:
645:
640:
639:
620:
619:
600:
599:
580:
579:
576:
518:
475:
464:
463:
434:
426:
356:
355:
324:
309:
298:
297:
270:
265:
264:
241:
240:
201:
188:
183:
182:
177:exponential map
170:
150:inertial frames
138:Albert Einstein
106:Kronecker delta
74:exponential map
35:
28:
23:
22:
15:
12:
11:
5:
2750:
2748:
2740:
2739:
2734:
2724:
2723:
2720:
2719:
2714:
2709:
2704:
2697:
2694:
2693:
2692:
2685:
2679:
2660:
2619:
2616:
2615:
2614:
2603:
2598:
2590:
2587:
2586:
2583:
2580:
2577:
2574:
2571:
2566:
2563:
2559:
2555:
2552:
2549:
2548:
2543:
2540:
2539:
2536:
2530:
2527:
2525:
2522:
2520:
2517:
2516:
2514:
2509:
2506:
2492:block diagonal
2484:
2483:
2469:
2466:
2461:
2458:
2452:
2449:
2446:
2443:
2440:
2437:
2434:
2431:
2408:
2405:
2401:
2397:
2385:is simply the
2344:
2340:
2323:
2311:
2288:
2285:
2284:
2283:
2272:
2269:
2264:
2259:
2254:
2250:
2246:
2243:
2240:
2235:
2231:
2227:
2224:
2221:
2216:
2213:
2210:
2203:
2199:
2190:
2187:
2181:
2178:
2175:
2172:
2169:
2164:
2161:
2154:
2150:
2133:
2132:
2121:
2118:
2113:
2109:
2105:
2102:
2099:
2094:
2090:
2084:
2080:
2076:
2073:
2070:
2065:
2062:
2059:
2056:
2052:
2045:
2042:
2036:
2031:
2028:
2024:
2020:
2017:
2014:
2011:
2006:
2003:
1998:
1994:
1979:
1978:
1967:
1964:
1959:
1954:
1949:
1945:
1941:
1938:
1935:
1930:
1926:
1920:
1915:
1912:
1909:
1904:
1901:
1898:
1895:
1891:
1887:
1884:
1881:
1878:
1873:
1870:
1867:
1864:
1860:
1854:
1846:
1843:
1837:
1834:
1831:
1828:
1825:
1820:
1817:
1810:
1806:
1789:
1788:
1777:
1774:
1769:
1764:
1759:
1755:
1751:
1748:
1745:
1740:
1736:
1730:
1726:
1722:
1719:
1716:
1711:
1708:
1705:
1702:
1698:
1691:
1688:
1682:
1677:
1674:
1670:
1666:
1663:
1660:
1657:
1652:
1649:
1645:
1621:
1599:
1595:
1574:
1571:
1568:
1563:
1560:
1557:
1554:
1550:
1529:
1507:
1504:
1500:
1496:
1493:
1490:
1487:
1482:
1479:
1475:
1454:
1451:
1448:
1445:
1442:
1439:
1436:
1433:
1421:
1418:
1417:
1416:
1404:
1401:
1398:
1395:
1392:
1389:
1385:
1382:
1379:
1376:
1373:
1370:
1362:
1358:
1354:
1347:
1344:
1340:
1336:
1311:
1308:
1304:
1283:
1280:
1277:
1274:
1271:
1266:
1261:
1258:
1254:
1233:
1218:
1204:
1201:
1197:
1193:
1190:
1187:
1184:
1179:
1176:
1172:
1149:
1146:
1142:
1119:
1116:
1112:
1088:
1077:
1066:
1063:
1060:
1057:
1054:
1051:
1048:
1045:
1042:
1022:
1011:
999:
979:
959:
954:
950:
946:
943:
940:
937:
934:
931:
926:
922:
918:
915:
912:
909:
906:
903:
898:
894:
873:
870:
867:
864:
861:
857:
853:
849:
828:
825:
822:
819:
816:
811:
807:
780:
776:
753:
749:
728:
723:
719:
698:
674:
652:
648:
627:
607:
587:
575:
572:
541:
540:
527:
522:
517:
514:
511:
506:
503:
498:
494:
490:
485:
482:
478:
474:
471:
437:diffeomorphism
430:
424:
369:
366:
363:
348:
347:
336:
331:
327:
323:
318:
313:
308:
305:
282:
277:
273:
248:
237:
236:
225:
222:
219:
216:
213:
208:
204:
200:
195:
191:
169:
166:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2749:
2738:
2735:
2733:
2730:
2729:
2727:
2718:
2715:
2713:
2710:
2708:
2705:
2703:
2700:
2699:
2695:
2690:
2686:
2682:
2680:0-471-15733-3
2676:
2672:
2668:
2667:
2661:
2657:
2653:
2649:
2645:
2641:
2637:
2633:
2629:
2628:
2622:
2621:
2617:
2601:
2596:
2588:
2578:
2575:
2572:
2564:
2561:
2557:
2550:
2541:
2534:
2528:
2523:
2518:
2512:
2507:
2504:
2497:
2496:
2495:
2493:
2489:
2467:
2459:
2450:
2444:
2441:
2438:
2435:
2432:
2422:
2421:
2420:
2406:
2399:
2388:
2384:
2380:
2376:
2375:Gauss's lemma
2372:
2367:
2365:
2361:
2357:
2355:
2347:
2338:
2334:
2330:
2326:
2322:
2318:
2314:
2310:
2306:
2302:
2298:
2294:
2286:
2270:
2262:
2252:
2241:
2238:
2233:
2229:
2222:
2214:
2211:
2208:
2201:
2197:
2188:
2185:
2179:
2176:
2170:
2162:
2159:
2152:
2148:
2138:
2137:
2136:
2119:
2111:
2107:
2100:
2097:
2092:
2088:
2082:
2078:
2071:
2063:
2060:
2057:
2054:
2050:
2043:
2040:
2034:
2029:
2026:
2022:
2018:
2012:
2004:
2001:
1996:
1992:
1984:
1983:
1982:
1965:
1957:
1947:
1936:
1933:
1928:
1924:
1910:
1902:
1899:
1896:
1893:
1889:
1885:
1879:
1871:
1868:
1865:
1862:
1858:
1844:
1841:
1835:
1832:
1826:
1818:
1815:
1808:
1794:
1793:
1792:
1775:
1767:
1757:
1746:
1743:
1738:
1734:
1728:
1724:
1717:
1709:
1706:
1703:
1700:
1696:
1689:
1686:
1680:
1675:
1672:
1668:
1664:
1658:
1650:
1647:
1643:
1635:
1634:
1633:
1619:
1597:
1593:
1569:
1561:
1558:
1555:
1552:
1548:
1527:
1505:
1502:
1498:
1494:
1488:
1480:
1477:
1473:
1449:
1446:
1443:
1440:
1434:
1431:
1419:
1402:
1399:
1396:
1393:
1390:
1383:
1380:
1377:
1371:
1360:
1356:
1345:
1342:
1338:
1309:
1306:
1302:
1281:
1278:
1272:
1264:
1259:
1256:
1231:
1223:
1219:
1202:
1199:
1195:
1191:
1185:
1177:
1174:
1170:
1147:
1144:
1140:
1117:
1114:
1110:
1102:
1086:
1078:
1061:
1058:
1055:
1052:
1049:
1046:
1043:
1020:
1012:
997:
977:
952:
948:
944:
941:
938:
935:
932:
929:
924:
920:
916:
910:
904:
896:
892:
871:
868:
862:
855:
851:
847:
826:
823:
817:
809:
805:
796:
778:
774:
751:
747:
726:
721:
717:
696:
688:
687:
686:
672:
650:
646:
625:
605:
585:
573:
571:
569:
565:
561:
557:
553:
550:
546:
525:
512:
509:
504:
501:
496:
492:
488:
483:
480:
476:
472:
469:
462:
461:
460:
458:
454:
450:
446:
442:
438:
433:
428:
421:
420:tangent space
417:
413:
409:
406:
402:
398:
393:
391:
387:
383:
367:
364:
361:
353:
350:given by any
334:
329:
325:
316:
306:
303:
296:
295:
294:
280:
275:
271:
262:
246:
223:
217:
214:
211:
206:
202:
198:
193:
189:
181:
180:
179:
178:
174:
167:
165:
163:
162:Busemann 1955
159:
155:
151:
147:
143:
139:
135:
131:
127:
121:
119:
115:
111:
108:at the point
107:
103:
102:metric tensor
99:
95:
91:
87:
83:
79:
78:tangent space
75:
71:
67:
63:
59:
56:
52:
48:
44:
40:
33:
19:
2688:
2664:
2631:
2625:
2487:
2485:
2419:. That is,
2382:
2370:
2368:
2363:
2359:
2353:
2345:
2336:
2332:
2328:
2324:
2320:
2316:
2312:
2308:
2304:
2300:
2292:
2290:
2134:
1980:
1790:
1423:
1132:simplify to
577:
559:
555:
551:
548:
544:
542:
456:
452:
448:
444:
440:
431:
422:
415:
411:
407:
404:
400:
396:
394:
389:
381:
349:
260:
238:
172:
171:
133:
129:
125:
122:
117:
109:
89:
81:
69:
66:neighborhood
46:
42:
36:
2702:Gauss Lemma
2634:: 417–423,
2299:, known as
386:orthonormal
45:at a point
32:Normal mode
2726:Categories
2618:References
2335:,φ) where
1224:vanish at
574:Properties
435:acts as a
2648:0025-5831
2579:ϕ
2565:ϕ
2562:ϕ
2551:⋮
2529:⋯
2465:∂
2457:∂
2448:⟩
2430:⟨
2404:∂
2396:∂
2234:τ
2215:τ
2212:μ
2180:−
2163:μ
2149:ω
2093:τ
2083:σ
2064:τ
2061:μ
2058:σ
2035:−
2030:μ
2023:δ
2002:∗
1997:μ
1929:τ
1903:τ
1900:ν
1897:μ
1894:λ
1872:τ
1869:μ
1866:ν
1863:λ
1836:−
1819:ν
1816:μ
1809:λ
1805:Γ
1739:τ
1729:σ
1710:τ
1707:ν
1704:σ
1701:μ
1681:−
1676:ν
1673:μ
1669:δ
1651:ν
1648:μ
1598:μ
1562:τ
1559:ν
1556:σ
1553:μ
1506:ν
1503:μ
1499:δ
1481:ν
1478:μ
1447:…
1388:∀
1353:∂
1335:∂
1253:Γ
1196:δ
1141:δ
893:γ
848:γ
806:γ
775:γ
516:→
502:−
489:∘
481:−
470:φ
429:, and exp
365:∈
322:→
221:→
215:⊃
55:symmetric
2696:See also
2379:gradient
2348:−1
1324:, i.e.,
1244:, i.e.,
1162:, i.e.,
856:′
795:geodesic
439:between
120:vanish.
2656:0071075
1632:become
793:be the
144:: the
140:in the
104:is the
76:to the
2677:
2654:
2646:
2532:
2488:ƒ
2343:,...,φ
60:are a
2494:form
797:with
352:basis
239:with
96:of a
64:in a
49:in a
2675:ISBN
2644:ISSN
1220:The
1033:are
839:and
689:Let
638:and
443:and
403:. A
2636:doi
2632:129
2381:of
618:in
558:in
493:exp
455:in
451:of
399:in
263:in
190:exp
164:).
80:at
68:of
37:In
2728::
2673:,
2652:MR
2650:,
2642:,
2630:,
685:.
545:E,
473::=
392:.
41:,
2684:.
2659:.
2638::
2602:.
2597:]
2589:0
2582:)
2576:,
2573:r
2570:(
2558:g
2542:0
2535:0
2524:0
2519:1
2513:[
2508:=
2505:g
2468:r
2460:f
2451:=
2445:r
2442:d
2439:,
2436:f
2433:d
2407:r
2400:/
2383:r
2371:p
2364:p
2360:r
2354:n
2352:(
2346:n
2341:1
2337:r
2333:r
2329:M
2325:p
2321:T
2317:M
2313:p
2309:T
2305:M
2293:p
2271:.
2268:)
2263:2
2258:|
2253:x
2249:|
2245:(
2242:O
2239:+
2230:x
2226:)
2223:0
2220:(
2209:b
2202:a
2198:R
2189:2
2186:1
2177:=
2174:)
2171:x
2168:(
2160:b
2153:a
2120:,
2117:)
2112:2
2108:x
2104:(
2101:O
2098:+
2089:x
2079:x
2075:)
2072:0
2069:(
2055:a
2051:R
2044:6
2041:1
2027:a
2019:=
2016:)
2013:x
2010:(
2005:a
1993:e
1966:.
1963:)
1958:2
1953:|
1948:x
1944:|
1940:(
1937:O
1934:+
1925:x
1919:]
1914:)
1911:0
1908:(
1890:R
1886:+
1883:)
1880:0
1877:(
1859:R
1853:[
1845:3
1842:1
1833:=
1830:)
1827:x
1824:(
1776:.
1773:)
1768:3
1763:|
1758:x
1754:|
1750:(
1747:O
1744:+
1735:x
1725:x
1721:)
1718:0
1715:(
1697:R
1690:3
1687:1
1665:=
1662:)
1659:x
1656:(
1644:g
1620:p
1594:x
1573:)
1570:0
1567:(
1549:R
1528:p
1495:=
1492:)
1489:0
1486:(
1474:g
1453:)
1450:0
1444:,
1441:0
1438:(
1435:=
1432:p
1415:.
1403:k
1400:,
1397:j
1394:,
1391:i
1384:,
1381:0
1378:=
1375:)
1372:p
1369:(
1361:k
1357:x
1346:j
1343:i
1339:g
1310:j
1307:i
1303:g
1282:0
1279:=
1276:)
1273:p
1270:(
1265:k
1260:j
1257:i
1232:p
1217:.
1203:j
1200:i
1192:=
1189:)
1186:p
1183:(
1178:j
1175:i
1171:g
1148:j
1145:i
1118:j
1115:i
1111:g
1087:p
1065:)
1062:0
1059:,
1056:.
1053:.
1050:.
1047:,
1044:0
1041:(
1021:p
1010:.
998:p
978:U
958:)
953:n
949:V
945:t
942:,
939:.
936:.
933:.
930:,
925:1
921:V
917:t
914:(
911:=
908:)
905:t
902:(
897:V
872:V
869:=
866:)
863:0
860:(
852:V
827:p
824:=
821:)
818:0
815:(
810:V
779:V
752:i
748:V
727:M
722:p
718:T
697:V
673:U
651:i
647:x
626:M
606:p
586:U
560:U
556:p
552:U
526:n
521:R
513:U
510::
505:1
497:p
484:1
477:E
457:M
453:p
449:U
445:V
441:U
432:p
427:M
425:p
423:T
416:V
412:M
408:U
401:M
397:p
382:E
368:M
362:p
335:M
330:p
326:T
317:n
312:R
307::
304:E
281:M
276:p
272:T
261:0
247:V
224:M
218:V
212:M
207:p
203:T
199::
194:p
134:t
130:p
126:p
118:p
110:p
90:p
82:p
70:p
47:p
34:.
20:)
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