662:
861:(with its induced metric). Geodesics going to the origin cannot be defined on the entire real line. By the HopfâRinow theorem, we can alternatively observe that it is not a complete metric space: any sequence in the plane converging to the origin is a non-converging Cauchy sequence in the punctured plane.
859:
716:
388:
146:
632:
600:
568:
814:
105:
470:
792:
754:
253:
199:
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501:
427:
327:
1071:
1935:
913:
528:
277:
223:
173:
70:
1875:
2098:
1948:
1162:
1186:
1381:
884:
2128:
2103:
915:
is geodesically complete, then it is not isometric to an open proper submanifold of any other
Riemannian manifold. The converse does not hold.
1990:
1251:
202:
1477:
875:, which describes gravity in terms of a pseudo-Riemannian geometry, many important examples of geodesically incomplete spaces arise, e.g.
1530:
1058:
1814:
823:
1963:
1928:
1027:
992:
1579:
1171:
2211:
1562:
2206:
1774:
864:
There exist non-geodesically complete compact pseudo-Riemannian (but not
Riemannian) manifolds. An example of this is the
668:
2221:
2025:
1921:
1759:
1482:
1256:
1804:
661:
336:
2046:
1809:
1779:
1487:
1443:
1424:
1191:
1135:
36:
2175:
1346:
1211:
2108:
1968:
1731:
1596:
1288:
1130:
2154:
1428:
1398:
1322:
1312:
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1098:
1051:
2113:
1196:
865:
294:
2185:
2180:
2118:
2051:
2010:
1769:
1388:
1283:
1103:
110:
1418:
1413:
473:
608:
576:
544:
2015:
1749:
1687:
1535:
1239:
1229:
1201:
1176:
1086:
876:
2072:
2041:
2029:
2000:
1983:
1944:
1887:
1860:
1569:
1447:
1432:
1361:
1120:
429:
is (geodesically) complete if and only if it satisfies one of the following equivalent conditions:
391:
40:
797:
78:
2216:
2170:
2067:
2036:
1829:
1784:
1681:
1552:
1356:
1181:
1044:
872:
288:
2077:
1366:
436:
2082:
1764:
1744:
1739:
1646:
1557:
1371:
1351:
1206:
1145:
1023:
988:
980:
646:
635:
759:
721:
2149:
2144:
2020:
1973:
1902:
1696:
1651:
1574:
1545:
1403:
1336:
1331:
1326:
1316:
1108:
1091:
883:. The fact that such incompleteness is fairly generic in general relativity is shown in the
228:
178:
479:
400:
300:
1978:
1845:
1754:
1584:
1540:
1306:
650:
539:
504:
1711:
1636:
1606:
1504:
1497:
1437:
1408:
1278:
1273:
1234:
1019:
898:
513:
262:
208:
158:
55:
2200:
2005:
1897:
1721:
1716:
1701:
1691:
1641:
1618:
1492:
1452:
1393:
1341:
1140:
642:
256:
1824:
1819:
1661:
1628:
1601:
1509:
1150:
718:
is not geodesically complete because the maximal geodesic with initial conditions
1913:
1667:
1656:
1613:
1514:
1115:
20:
2123:
1892:
1850:
1676:
1589:
1221:
1125:
987:, Mathematics: theory and applications, Boston: BirkhÀuser, pp. xvi+300,
1706:
1671:
1376:
1263:
820:
A simple example of a non-complete manifold is given by the punctured plane
16:
Riemannian manifold in which geodesics extend infinitely in all directions
1995:
1870:
1865:
1855:
1246:
1067:
880:
571:
73:
1036:
1007:. Graduate Texts in Mathematics. Springer International Publishing AG.
1462:
49:, there are straight paths extending infinitely in all directions.
660:
603:
854:{\displaystyle \mathbb {R} ^{2}\smallsetminus \lbrace 0\rbrace }
1917:
1040:
684:
297:
gives alternative characterizations of completeness. Let
901:
826:
800:
762:
724:
671:
611:
579:
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516:
482:
439:
403:
339:
303:
265:
231:
211:
181:
161:
113:
81:
58:
935:
933:
711:{\displaystyle \mathbb {R} ^{2}\backslash \{(0,0)\}}
2163:
2137:
2091:
2060:
1956:
1838:
1797:
1730:
1627:
1523:
1470:
1461:
1297:
1220:
1159:
1079:
907:
853:
808:
786:
748:
710:
626:
594:
562:
522:
495:
464:
421:
382:
321:
271:
247:
217:
193:
167:
140:
99:
64:
383:{\displaystyle d_{g}:M\times M\to [0,\infty )}
72:is (geodesically) complete if for any maximal
1929:
1052:
175:is (geodesically) complete if for all points
8:
848:
842:
705:
687:
2099:Fundamental theorem of Riemannian geometry
1936:
1922:
1914:
1467:
1059:
1045:
1037:
649:manifolds are geodesically complete. All
900:
833:
829:
828:
825:
802:
801:
799:
761:
723:
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487:
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438:
402:
344:
338:
302:
264:
236:
230:
210:
180:
160:
112:
80:
57:
963:
951:
929:
7:
1005:Introduction to Riemannian Manifolds
885:PenroseâHawking singularity theorems
141:{\displaystyle I=(-\infty ,\infty )}
939:
397:The HopfâRinow theorem states that
877:non-rotating uncharged black-holes
510:All closed and bounded subsets of
374:
152:if its domain cannot be extended.
132:
126:
14:
43:for which, starting at any point
627:{\displaystyle \mathbb {T} ^{n}}
595:{\displaystyle \mathbb {S} ^{n}}
563:{\displaystyle \mathbb {R} ^{n}}
1099:Differentiable/Smooth manifold
781:
769:
743:
731:
702:
690:
638:) are all complete manifolds.
459:
440:
416:
404:
377:
365:
362:
316:
304:
135:
120:
91:
29:geodesically complete manifold
1:
645:Riemannian manifolds and all
2026:Raising and lowering indices
809:{\displaystyle \mathbb {R} }
392:Riemannian distance function
333:Riemannian manifold and let
100:{\displaystyle \ell :I\to M}
1805:Classification of manifolds
981:do Carmo, Manfredo PerdigĂŁo
653:are geodesically complete.
2238:
2047:Pseudo-Riemannian manifold
286:
2176:Geometrization conjecture
1881:over commutative algebras
1014:O'Neill, Barrett (1983).
535:Examples and non-examples
465:{\displaystyle (M,d_{g})}
1597:Riemann curvature tensor
1016:Semi-Riemannian Geometry
787:{\displaystyle v=(1,1)}
749:{\displaystyle p=(1,1)}
2212:Geodesic (mathematics)
2186:Uniformization theorem
2119:Nash embedding theorem
2052:Riemannian volume form
2011:Levi-Civita connection
1389:Manifold with boundary
1104:Differential structure
909:
879:or cosmologies with a
855:
817:
810:
788:
750:
712:
628:
596:
564:
524:
497:
466:
423:
384:
323:
273:
249:
248:{\displaystyle T_{p}M}
219:
195:
194:{\displaystyle p\in M}
169:
142:
101:
66:
2207:Differential geometry
910:
856:
811:
794:does not have domain
789:
751:
713:
664:
629:
597:
565:
525:
498:
496:{\displaystyle d_{g}}
467:
424:
422:{\displaystyle (M,g)}
385:
324:
322:{\displaystyle (M,g)}
274:
250:
220:
196:
170:
143:
102:
67:
52:Formally, a manifold
2109:GaussâBonnet theorem
2016:Covariant derivative
1536:Covariant derivative
1087:Topological manifold
899:
824:
798:
760:
722:
669:
665:The punctured plane
634:(with their natural
609:
577:
545:
514:
480:
437:
401:
337:
301:
263:
229:
209:
179:
159:
111:
79:
56:
2222:Riemannian geometry
2181:Poincaré conjecture
2042:Riemannian manifold
2030:Musical isomorphism
1945:Riemannian geometry
1570:Exterior derivative
1172:AtiyahâSinger index
1121:Riemannian manifold
985:Riemannian geometry
41:Riemannian manifold
2171:General relativity
2114:HopfâRinow theorem
2061:Types of manifolds
2037:Parallel transport
1876:Secondary calculus
1830:Singularity theory
1785:Parallel transport
1553:De Rham cohomology
1192:Generalized Stokes
1003:Lee, John (2018).
954:, p. 146-147.
905:
873:general relativity
866:CliftonâPohl torus
851:
818:
806:
784:
746:
708:
636:Riemannian metrics
624:
592:
560:
520:
493:
462:
419:
380:
319:
295:HopfâRinow theorem
289:Hopf-Rinow theorem
283:Hopf-Rinow theorem
269:
245:
215:
191:
165:
138:
97:
62:
2194:
2193:
1911:
1910:
1793:
1792:
1558:Differential form
1212:Whitney embedding
1146:Differential form
908:{\displaystyle M}
871:In the theory of
523:{\displaystyle M}
433:The metric space
272:{\displaystyle p}
218:{\displaystyle p}
168:{\displaystyle M}
65:{\displaystyle M}
25:complete manifold
2229:
1938:
1931:
1924:
1915:
1903:Stratified space
1861:Fréchet manifold
1575:Interior product
1468:
1165:
1061:
1054:
1047:
1038:
1033:
1008:
997:
967:
961:
955:
949:
943:
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914:
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911:
906:
860:
858:
857:
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838:
837:
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815:
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793:
791:
790:
785:
755:
753:
752:
747:
717:
715:
714:
709:
683:
682:
677:
651:symmetric spaces
633:
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625:
623:
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463:
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457:
428:
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389:
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349:
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328:
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278:
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254:
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241:
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224:
222:
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216:
200:
198:
197:
192:
174:
172:
171:
166:
148:. A geodesic is
147:
145:
144:
139:
107:, it holds that
106:
104:
103:
98:
71:
69:
68:
63:
48:
34:
2237:
2236:
2232:
2231:
2230:
2228:
2227:
2226:
2197:
2196:
2195:
2190:
2159:
2138:Generalizations
2133:
2087:
2056:
1991:Exponential map
1952:
1942:
1912:
1907:
1846:Banach manifold
1839:Generalizations
1834:
1789:
1726:
1623:
1585:Ricci curvature
1541:Cotangent space
1519:
1457:
1299:
1293:
1252:Exponential map
1216:
1161:
1155:
1075:
1065:
1030:
1013:
1002:
995:
979:
976:
971:
970:
962:
958:
950:
946:
938:
931:
926:
921:
897:
896:
893:
827:
822:
821:
796:
795:
758:
757:
720:
719:
672:
667:
666:
659:
612:
607:
606:
580:
575:
574:
548:
543:
542:
540:Euclidean space
537:
512:
511:
505:Cauchy sequence
483:
478:
477:
449:
435:
434:
399:
398:
340:
335:
334:
299:
298:
291:
285:
261:
260:
232:
227:
226:
207:
206:
203:exponential map
177:
176:
157:
156:
109:
108:
77:
76:
54:
53:
44:
32:
17:
12:
11:
5:
2235:
2233:
2225:
2224:
2219:
2214:
2209:
2199:
2198:
2192:
2191:
2189:
2188:
2183:
2178:
2173:
2167:
2165:
2161:
2160:
2158:
2157:
2155:Sub-Riemannian
2152:
2147:
2141:
2139:
2135:
2134:
2132:
2131:
2126:
2121:
2116:
2111:
2106:
2101:
2095:
2093:
2089:
2088:
2086:
2085:
2080:
2075:
2070:
2064:
2062:
2058:
2057:
2055:
2054:
2049:
2044:
2039:
2034:
2033:
2032:
2023:
2018:
2013:
2003:
1998:
1993:
1988:
1987:
1986:
1981:
1976:
1971:
1960:
1958:
1957:Basic concepts
1954:
1953:
1943:
1941:
1940:
1933:
1926:
1918:
1909:
1908:
1906:
1905:
1900:
1895:
1890:
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1883:
1873:
1868:
1863:
1858:
1853:
1848:
1842:
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1836:
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1807:
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1555:
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1459:
1458:
1456:
1455:
1450:
1440:
1435:
1421:
1416:
1411:
1406:
1401:
1399:Parallelizable
1396:
1391:
1386:
1385:
1384:
1374:
1369:
1364:
1359:
1354:
1349:
1344:
1339:
1334:
1329:
1319:
1309:
1303:
1301:
1295:
1294:
1292:
1291:
1286:
1281:
1279:Lie derivative
1276:
1274:Integral curve
1271:
1266:
1261:
1260:
1259:
1249:
1244:
1243:
1242:
1235:Diffeomorphism
1232:
1226:
1224:
1218:
1217:
1215:
1214:
1209:
1204:
1199:
1194:
1189:
1184:
1179:
1174:
1168:
1166:
1157:
1156:
1154:
1153:
1148:
1143:
1138:
1133:
1128:
1123:
1118:
1113:
1112:
1111:
1106:
1096:
1095:
1094:
1083:
1081:
1080:Basic concepts
1077:
1076:
1066:
1064:
1063:
1056:
1049:
1041:
1035:
1034:
1028:
1020:Academic Press
1010:
1009:
999:
998:
993:
975:
972:
969:
968:
966:, p. 145.
956:
944:
942:, p. 131.
928:
927:
925:
922:
920:
917:
904:
892:
889:
850:
847:
844:
841:
836:
831:
804:
783:
780:
777:
774:
771:
768:
765:
745:
742:
739:
736:
733:
730:
727:
707:
704:
701:
698:
695:
692:
689:
686:
681:
676:
658:
655:
621:
616:
589:
584:
557:
552:
536:
533:
532:
531:
519:
508:
490:
486:
461:
456:
452:
448:
445:
442:
418:
415:
412:
409:
406:
379:
376:
373:
370:
367:
364:
361:
358:
355:
352:
347:
343:
318:
315:
312:
309:
306:
287:Main article:
284:
281:
268:
244:
239:
235:
225:is defined on
214:
190:
187:
184:
164:
155:Equivalently,
137:
134:
131:
128:
125:
122:
119:
116:
96:
93:
90:
87:
84:
61:
15:
13:
10:
9:
6:
4:
3:
2:
2234:
2223:
2220:
2218:
2215:
2213:
2210:
2208:
2205:
2204:
2202:
2187:
2184:
2182:
2179:
2177:
2174:
2172:
2169:
2168:
2166:
2162:
2156:
2153:
2151:
2148:
2146:
2143:
2142:
2140:
2136:
2130:
2129:Schur's lemma
2127:
2125:
2122:
2120:
2117:
2115:
2112:
2110:
2107:
2105:
2104:Gauss's lemma
2102:
2100:
2097:
2096:
2094:
2090:
2084:
2081:
2079:
2076:
2074:
2071:
2069:
2066:
2065:
2063:
2059:
2053:
2050:
2048:
2045:
2043:
2040:
2038:
2035:
2031:
2027:
2024:
2022:
2019:
2017:
2014:
2012:
2009:
2008:
2007:
2006:Metric tensor
2004:
2002:
2001:Inner product
1999:
1997:
1994:
1992:
1989:
1985:
1982:
1980:
1977:
1975:
1972:
1970:
1967:
1966:
1965:
1962:
1961:
1959:
1955:
1950:
1946:
1939:
1934:
1932:
1927:
1925:
1920:
1919:
1916:
1904:
1901:
1899:
1898:Supermanifold
1896:
1894:
1891:
1889:
1886:
1882:
1879:
1878:
1877:
1874:
1872:
1869:
1867:
1864:
1862:
1859:
1857:
1854:
1852:
1849:
1847:
1844:
1843:
1841:
1837:
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1821:
1818:
1816:
1813:
1811:
1808:
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1800:
1796:
1786:
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1778:
1776:
1773:
1771:
1768:
1766:
1763:
1761:
1758:
1756:
1753:
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1748:
1746:
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1741:
1738:
1737:
1735:
1733:
1729:
1723:
1720:
1718:
1715:
1713:
1710:
1708:
1705:
1703:
1700:
1698:
1695:
1693:
1689:
1685:
1683:
1680:
1678:
1675:
1673:
1669:
1665:
1663:
1660:
1658:
1655:
1653:
1650:
1648:
1645:
1643:
1640:
1638:
1635:
1634:
1632:
1630:
1626:
1620:
1619:Wedge product
1617:
1615:
1612:
1608:
1605:
1604:
1603:
1600:
1598:
1595:
1591:
1588:
1587:
1586:
1583:
1581:
1578:
1576:
1573:
1571:
1568:
1564:
1563:Vector-valued
1561:
1560:
1559:
1556:
1554:
1551:
1547:
1544:
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1539:
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1526:
1522:
1516:
1513:
1511:
1508:
1506:
1503:
1499:
1496:
1495:
1494:
1493:Tangent space
1491:
1489:
1486:
1484:
1481:
1479:
1476:
1475:
1473:
1469:
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1460:
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1296:
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1287:
1285:
1282:
1280:
1277:
1275:
1272:
1270:
1267:
1265:
1262:
1258:
1257:in Lie theory
1255:
1254:
1253:
1250:
1248:
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1241:
1238:
1237:
1236:
1233:
1231:
1228:
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1200:
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1190:
1188:
1185:
1183:
1180:
1178:
1175:
1173:
1170:
1169:
1167:
1164:
1160:Main results
1158:
1152:
1149:
1147:
1144:
1142:
1141:Tangent space
1139:
1137:
1134:
1132:
1129:
1127:
1124:
1122:
1119:
1117:
1114:
1110:
1107:
1105:
1102:
1101:
1100:
1097:
1093:
1090:
1089:
1088:
1085:
1084:
1082:
1078:
1073:
1069:
1062:
1057:
1055:
1050:
1048:
1043:
1042:
1039:
1031:
1029:0-12-526740-1
1025:
1022:. Chapter 3.
1021:
1017:
1012:
1011:
1006:
1001:
1000:
996:
994:0-8176-3490-8
990:
986:
982:
978:
977:
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965:
964:do Carmo 1992
960:
957:
953:
952:do Carmo 1992
948:
945:
941:
936:
934:
930:
923:
918:
916:
902:
891:Extendibility
890:
888:
886:
882:
878:
874:
869:
867:
862:
845:
839:
834:
778:
775:
772:
766:
763:
740:
737:
734:
728:
725:
699:
696:
693:
679:
663:
656:
654:
652:
648:
644:
639:
637:
619:
605:
587:
573:
555:
541:
534:
517:
509:
506:
488:
484:
475:
454:
450:
446:
443:
432:
431:
430:
413:
410:
407:
395:
393:
371:
368:
359:
356:
353:
350:
345:
341:
332:
313:
310:
307:
296:
290:
282:
280:
266:
258:
257:tangent space
255:, the entire
242:
237:
233:
212:
204:
188:
185:
182:
162:
153:
151:
129:
123:
117:
114:
94:
88:
85:
82:
75:
59:
50:
47:
42:
38:
30:
26:
22:
2164:Applications
2092:Main results
1825:Moving frame
1820:Morse theory
1810:Gauge theory
1602:Tensor field
1531:Closed/Exact
1510:Vector field
1478:Distribution
1419:Hypercomplex
1414:Quaternionic
1151:Vector field
1109:Smooth atlas
1015:
1004:
984:
959:
947:
894:
870:
863:
819:
657:Non-examples
640:
538:
530:are compact.
396:
330:
292:
154:
149:
51:
45:
28:
24:
18:
1770:Levi-Civita
1760:Generalized
1732:Connections
1682:Lie algebra
1614:Volume form
1515:Vector flow
1488:Pushforward
1483:Lie bracket
1382:Lie algebra
1347:G-structure
1136:Pushforward
1116:Submanifold
647:homogeneous
507:converges),
21:mathematics
2201:Categories
2124:Ricci flow
2073:Hyperbolic
1893:Stratifold
1851:Diffeology
1647:Associated
1448:Symplectic
1433:Riemannian
1362:Hyperbolic
1289:Submersion
1197:HopfâRinow
1131:Submersion
1126:Smooth map
919:References
602:, and the
2217:Manifolds
2068:Hermitian
2021:Signature
1984:Sectional
1964:Curvature
1775:Principal
1750:Ehresmann
1707:Subbundle
1697:Principal
1672:Fibration
1652:Cotangent
1524:Covectors
1377:Lie group
1357:Hermitian
1300:manifolds
1269:Immersion
1264:Foliation
1202:Noether's
1187:Frobenius
1182:De Rham's
1177:Darboux's
1068:Manifolds
840:∖
685:∖
375:∞
363:→
357:×
331:connected
186:∈
133:∞
127:∞
124:−
92:→
83:ℓ
2083:Kenmotsu
1996:Geodesic
1949:Glossary
1871:Orbifold
1866:K-theory
1856:Diffiety
1580:Pullback
1394:Oriented
1372:Kenmotsu
1352:Hadamard
1298:Types of
1247:Geodesic
1072:Glossary
983:(1992),
940:Lee 2018
881:Big Bang
474:complete
74:geodesic
2150:Hilbert
2145:Finsler
1815:History
1798:Related
1712:Tangent
1690:)
1670:)
1637:Adjoint
1629:Bundles
1607:density
1505:Torsion
1471:Vectors
1463:Tensors
1446:)
1431:)
1427:,
1425:Pseudoâ
1404:Poisson
1337:Finsler
1332:Fibered
1327:Contact
1325:)
1317:Complex
1315:)
1284:Section
974:Sources
643:compact
476:(every
390:be its
150:maximal
2078:KĂ€hler
1974:Scalar
1969:tensor
1780:Vector
1765:Koszul
1745:Cartan
1740:Affine
1722:Vector
1717:Tensor
1702:Spinor
1692:Normal
1688:Stable
1642:Affine
1546:bundle
1498:bundle
1444:Almost
1367:KĂ€hler
1323:Almost
1313:Almost
1307:Closed
1207:Sard's
1163:(list)
1026:
991:
572:sphere
570:, the
201:, the
37:pseudo
35:is a (
1979:Ricci
1888:Sheaf
1662:Fiber
1438:Rizza
1409:Prime
1240:Local
1230:Curve
1092:Atlas
924:Notes
329:be a
1755:Form
1657:Dual
1590:flow
1453:Tame
1429:Subâ
1342:Flat
1222:Maps
1024:ISBN
989:ISBN
641:All
604:tori
293:The
27:(or
23:, a
1677:Jet
895:If
472:is
259:at
205:at
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115:I
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