1483:
1266:
is a stronger condition than "to be locally diffeomophic." Indeed, although locally-defined diffeomorphisms preserve differentiable structure locally, one must be able to "patch up" these (local) diffeomorphisms to ensure that the domain is the entire smooth manifold.
836:
693:
301:
1361:
1330:
1301:
495:
939:
1264:
1192:
798:
619:
343:
125:
1080:
1054:
969:
900:
870:
723:
369:
155:
544:
438:
230:
1427:
1232:
1212:
1160:
1140:
1120:
1100:
1028:
1008:
763:
743:
564:
515:
409:
389:
250:
198:
178:
86:
66:
2557:
1748:
2552:
1839:
1863:
2058:
1391:
If a local diffeomorphism between two manifolds exists then their dimensions must be equal. Every local diffeomorphism is also a
2600:
803:
1928:
1697:
1659:
1632:
2154:
255:
2207:
1735:
2491:
1375:, although they do indeed have the same local differentiable structure. This is because all local diffeomorphisms are
982:
guarantees that any continuous injective function between manifolds of equal dimensions is necessarily an open map.
2256:
2610:
2239:
1848:
1620:
1449:
627:
1376:
2451:
1858:
1624:
2436:
2159:
1933:
1493:
1403:
971:
is a linear isomorphism if and only if it is injective, or equivalently, if and only if it is surjective.
2481:
2620:
2486:
2456:
2164:
2120:
2101:
1868:
1812:
1456:
1396:
975:
622:
1337:
1306:
1277:
447:
2023:
1888:
2605:
2408:
2273:
1965:
1807:
1271:
843:
586:
37:
2615:
2105:
2075:
1999:
1989:
1945:
1775:
1728:
839:
575:
311:
89:
2446:
2065:
1960:
1873:
1780:
905:
441:
93:
29:
1237:
1165:
771:
592:
316:
98:
2095:
2090:
21:
1059:
1033:
990:
All manifolds of the same dimension are "locally diffeomorphic," in the following sense: if
944:
875:
849:
702:
348:
134:
2426:
2364:
2212:
1906:
1878:
1853:
1763:
1707:
1669:
1654:, Graduate Texts in Mathematics, vol. 218 (Second ed.), New York, NY.: Springer,
1642:
1548:
1536:
1530:
1234:, let alone extend to a local diffeomorphism. Thus the existence of a local diffeomorphism
979:
974:
Here is an alternative argument for the case of an immersion: every smooth immersion is a
520:
414:
206:
8:
2564:
2246:
2124:
2109:
2038:
1797:
1542:
1392:
201:
2537:
1409:
2506:
2461:
2358:
2229:
2033:
1721:
1473:
1332:
into a differentiable manifold, but both structures are not locally diffeomorphic (see
1217:
1197:
1145:
1125:
1105:
1085:
1013:
993:
748:
728:
696:
549:
500:
394:
374:
235:
183:
163:
71:
51:
2043:
2441:
2421:
2416:
2323:
2234:
2048:
2028:
1883:
1822:
1693:
1655:
1628:
2579:
2373:
2328:
2251:
2222:
2080:
2013:
2008:
2003:
1993:
1785:
1768:
1683:
2522:
2431:
2261:
2217:
1983:
1703:
1665:
1638:
1372:
33:
2388:
2313:
2283:
2181:
2174:
2114:
2085:
1955:
1950:
1911:
1518:
1438:
304:
1688:
2594:
2574:
2398:
2393:
2378:
2368:
2318:
2295:
2169:
2129:
2070:
2018:
1817:
1524:
1380:
1677:
2501:
2496:
2338:
2305:
2278:
2186:
1827:
1527: – Mapping which preserves all topological properties of a given space
1452:
1383:
is compact, and the 2-sphere is compact whereas
Euclidean 2-space is not.
2344:
2333:
2290:
2191:
1792:
1533: – Theorem in topology about homeomorphic subsets of Euclidean space
842:(smooth local embedding), or equivalently, if and only if it is a smooth
17:
1682:, Undergraduate Texts in Mathematics (Fourth ed.), Springer, Cham,
1482:
2569:
2527:
2353:
2266:
1898:
1802:
1713:
2383:
2348:
2053:
1940:
1442:
1333:
1455:
is a local diffeomorphism such that every point in the target has a
2547:
2542:
2532:
1923:
1744:
1399:
579:
158:
1367:
As another example, there can be no local diffeomorphism from the
40:. The formal definition of a local diffeomorphism is given below.
546:
have the same dimension, which may be less than the dimension of
2139:
1368:
574:
A map is a local diffeomorphism if and only if it is a smooth
831:{\displaystyle \operatorname {dim} X=\operatorname {dim} Y}
838:) is a local diffeomorphism if and only if it is a smooth
1545: – Mathematical function revertible near each point
1551: – features of space-time representing symmetries
1412:
1340:
1309:
1280:
1240:
1220:
1200:
1168:
1148:
1128:
1108:
1088:
1062:
1036:
1016:
996:
947:
908:
878:
852:
806:
774:
751:
731:
705:
630:
595:
552:
523:
503:
450:
417:
397:
377:
351:
319:
258:
238:
209:
186:
166:
137:
101:
74:
54:
1553:
Pages displaying wikidata descriptions as a fallback
1421:
1355:
1324:
1295:
1258:
1226:
1214:need not extend to a smooth map defined on all of
1206:
1186:
1154:
1134:
1114:
1094:
1074:
1048:
1022:
1002:
963:
933:
894:
864:
830:
792:
757:
737:
717:
687:
613:
558:
538:
509:
489:
432:
403:
383:
363:
337:
295:
244:
224:
192:
172:
149:
119:
80:
60:
2592:
1521: – Isomorphism of differentiable manifolds
310:A local diffeomorphism is a special case of an
1729:
1467:
621:is a local diffeomorphism if and only if the
263:
1736:
1722:
1270:For example, one can impose two different
800:between two manifolds of equal dimension (
688:{\displaystyle Df_{x}:T_{x}X\to T_{f(x)}Y}
1687:
1343:
1312:
1283:
1743:
1402:. A local diffeomorphism has constant
1082:, then there exist open neighbourhoods
2593:
1614:
1717:
1675:
296:{\displaystyle f\vert _{U}:U\to f(U)}
1477:
569:
43:
1649:
13:
1623:, vol. 93, Providence, R.I.:
14:
2632:
1652:Introduction to smooth manifolds
1586:Introduction to smooth manifolds
1573:Introduction to smooth manifolds
1481:
1356:{\displaystyle \mathbb {R} ^{4}}
1325:{\displaystyle \mathbb {R} ^{4}}
1296:{\displaystyle \mathbb {R} ^{4}}
578:(smooth local embedding) and an
490:{\displaystyle f|_{U}:U\to f(U)}
1621:Graduate Studies in Mathematics
1617:Topics in differential geometry
1539: – Class of diffeomorphism
2601:Theory of continuous functions
1776:Differentiable/Smooth manifold
1591:
1578:
1565:
1250:
1178:
941:have the same dimension, thus
923:
917:
784:
677:
671:
660:
605:
533:
527:
484:
478:
472:
456:
427:
421:
329:
290:
284:
278:
219:
213:
111:
1:
1625:American Mathematical Society
1608:
1386:
1030:have the same dimension, and
985:
1379:, the continuous image of a
7:
2482:Classification of manifolds
1512:
1431:
846:. This is because, for any
371:, there exists an open set
10:
2637:
1471:
1468:Local flow diffeomorphisms
976:locally injective function
589:implies that a smooth map
497:is a diffeomorphism. Here
2558:over commutative algebras
2515:
2474:
2407:
2304:
2200:
2147:
2138:
1974:
1897:
1836:
1756:
1689:10.1007/978-3-031-41026-0
1679:Linear algebra done right
1615:Michor, Peter W. (2008),
1599:Linear algebra done right
1272:differentiable structures
934:{\displaystyle T_{f(x)}Y}
765:have the same dimension.
345:. In this case, for each
36:that preserves the local
2274:Riemann curvature tensor
1558:
1259:{\displaystyle f:X\to Y}
1187:{\displaystyle f:U\to V}
793:{\displaystyle f:X\to Y}
614:{\displaystyle f:X\to Y}
587:inverse function theorem
338:{\displaystyle f:X\to Y}
120:{\displaystyle f:X\to Y}
90:differentiable manifolds
38:differentiable structure
1676:Axler, Sheldon (2024),
2611:Functions and mappings
2066:Manifold with boundary
1781:Differential structure
1490:This section is empty.
1423:
1357:
1326:
1297:
1260:
1228:
1208:
1188:
1156:
1136:
1116:
1096:
1076:
1075:{\displaystyle y\in Y}
1050:
1049:{\displaystyle x\in X}
1024:
1004:
965:
964:{\displaystyle Df_{x}}
935:
896:
895:{\displaystyle T_{x}X}
866:
865:{\displaystyle x\in X}
832:
794:
768:It follows that a map
759:
739:
719:
718:{\displaystyle x\in X}
689:
615:
560:
540:
511:
491:
434:
405:
385:
365:
364:{\displaystyle x\in X}
339:
297:
246:
226:
194:
174:
151:
150:{\displaystyle x\in X}
121:
82:
62:
1650:Lee, John M. (2013),
1445:local diffeomorphism.
1424:
1358:
1327:
1298:
1261:
1229:
1209:
1189:
1162:and a diffeomorphism
1157:
1137:
1117:
1097:
1077:
1051:
1025:
1005:
966:
936:
897:
867:
833:
795:
760:
740:
720:
690:
616:
561:
541:
512:
492:
435:
406:
386:
366:
340:
298:
247:
227:
195:
175:
152:
122:
83:
63:
22:differential topology
2213:Covariant derivative
1764:Topological manifold
1549:Spacetime symmetries
1537:Large diffeomorphism
1531:Invariance of domain
1410:
1338:
1307:
1278:
1238:
1218:
1198:
1194:. However, this map
1166:
1146:
1126:
1106:
1086:
1060:
1034:
1014:
994:
980:invariance of domain
945:
906:
876:
850:
804:
772:
749:
729:
725:. This implies that
703:
628:
593:
550:
539:{\displaystyle f(U)}
521:
501:
448:
442:embedded submanifold
433:{\displaystyle f(U)}
415:
411:such that the image
395:
375:
349:
317:
256:
236:
225:{\displaystyle f(U)}
207:
184:
164:
135:
129:local diffeomorphism
99:
72:
52:
26:local diffeomorphism
20:, more specifically
2247:Exterior derivative
1849:Atiyah–Singer index
1798:Riemannian manifold
1543:Local homeomorphism
1393:local homeomorphism
131:if, for each point
2553:Secondary calculus
2507:Singularity theory
2462:Parallel transport
2230:De Rham cohomology
1869:Generalized Stokes
1575:, Proposition 5.22
1474:Flow (mathematics)
1422:{\displaystyle n.}
1419:
1353:
1322:
1293:
1256:
1224:
1204:
1184:
1152:
1132:
1112:
1092:
1072:
1046:
1020:
1000:
961:
931:
892:
862:
828:
790:
755:
735:
715:
697:linear isomorphism
685:
611:
556:
536:
507:
487:
430:
401:
381:
361:
335:
293:
242:
222:
190:
170:
157:, there exists an
147:
117:
78:
58:
2621:Inverse functions
2588:
2587:
2470:
2469:
2235:Differential form
1889:Whitney embedding
1823:Differential form
1699:978-3-031-41026-0
1661:978-1-4419-9981-8
1634:978-0-8218-2003-2
1588:, Proposition 4.8
1510:
1509:
1397:locally injective
1373:Euclidean 2-space
1227:{\displaystyle X}
1207:{\displaystyle f}
1155:{\displaystyle y}
1135:{\displaystyle V}
1115:{\displaystyle x}
1095:{\displaystyle U}
1023:{\displaystyle Y}
1003:{\displaystyle X}
758:{\displaystyle Y}
738:{\displaystyle X}
570:Characterizations
559:{\displaystyle Y}
510:{\displaystyle X}
404:{\displaystyle x}
384:{\displaystyle U}
245:{\displaystyle Y}
193:{\displaystyle x}
173:{\displaystyle U}
81:{\displaystyle Y}
61:{\displaystyle X}
44:Formal definition
28:is intuitively a
2628:
2580:Stratified space
2538:Fréchet manifold
2252:Interior product
2145:
2144:
1842:
1738:
1731:
1724:
1715:
1714:
1710:
1691:
1672:
1645:
1602:
1595:
1589:
1582:
1576:
1569:
1554:
1505:
1502:
1492:You can help by
1485:
1478:
1428:
1426:
1425:
1420:
1395:and therefore a
1362:
1360:
1359:
1354:
1352:
1351:
1346:
1331:
1329:
1328:
1323:
1321:
1320:
1315:
1302:
1300:
1299:
1294:
1292:
1291:
1286:
1265:
1263:
1262:
1257:
1233:
1231:
1230:
1225:
1213:
1211:
1210:
1205:
1193:
1191:
1190:
1185:
1161:
1159:
1158:
1153:
1141:
1139:
1138:
1133:
1121:
1119:
1118:
1113:
1101:
1099:
1098:
1093:
1081:
1079:
1078:
1073:
1055:
1053:
1052:
1047:
1029:
1027:
1026:
1021:
1009:
1007:
1006:
1001:
970:
968:
967:
962:
960:
959:
940:
938:
937:
932:
927:
926:
901:
899:
898:
893:
888:
887:
871:
869:
868:
863:
837:
835:
834:
829:
799:
797:
796:
791:
764:
762:
761:
756:
744:
742:
741:
736:
724:
722:
721:
716:
694:
692:
691:
686:
681:
680:
656:
655:
643:
642:
620:
618:
617:
612:
565:
563:
562:
557:
545:
543:
542:
537:
516:
514:
513:
508:
496:
494:
493:
488:
465:
464:
459:
439:
437:
436:
431:
410:
408:
407:
402:
390:
388:
387:
382:
370:
368:
367:
362:
344:
342:
341:
336:
302:
300:
299:
294:
271:
270:
251:
249:
248:
243:
231:
229:
228:
223:
199:
197:
196:
191:
179:
177:
176:
171:
156:
154:
153:
148:
126:
124:
123:
118:
87:
85:
84:
79:
67:
65:
64:
59:
34:smooth manifolds
2636:
2635:
2631:
2630:
2629:
2627:
2626:
2625:
2606:Diffeomorphisms
2591:
2590:
2589:
2584:
2523:Banach manifold
2516:Generalizations
2511:
2466:
2403:
2300:
2262:Ricci curvature
2218:Cotangent space
2196:
2134:
1976:
1970:
1929:Exponential map
1893:
1838:
1832:
1752:
1742:
1700:
1662:
1635:
1611:
1606:
1605:
1596:
1592:
1583:
1579:
1570:
1566:
1561:
1552:
1515:
1506:
1500:
1497:
1476:
1470:
1434:
1411:
1408:
1407:
1389:
1347:
1342:
1341:
1339:
1336:
1335:
1316:
1311:
1310:
1308:
1305:
1304:
1303:that each make
1287:
1282:
1281:
1279:
1276:
1275:
1239:
1236:
1235:
1219:
1216:
1215:
1199:
1196:
1195:
1167:
1164:
1163:
1147:
1144:
1143:
1127:
1124:
1123:
1107:
1104:
1103:
1087:
1084:
1083:
1061:
1058:
1057:
1035:
1032:
1031:
1015:
1012:
1011:
995:
992:
991:
988:
955:
951:
946:
943:
942:
913:
909:
907:
904:
903:
883:
879:
877:
874:
873:
851:
848:
847:
805:
802:
801:
773:
770:
769:
750:
747:
746:
730:
727:
726:
704:
701:
700:
699:for all points
667:
663:
651:
647:
638:
634:
629:
626:
625:
594:
591:
590:
572:
551:
548:
547:
522:
519:
518:
502:
499:
498:
460:
455:
454:
449:
446:
445:
416:
413:
412:
396:
393:
392:
376:
373:
372:
350:
347:
346:
318:
315:
314:
266:
262:
257:
254:
253:
237:
234:
233:
208:
205:
204:
185:
182:
181:
165:
162:
161:
136:
133:
132:
100:
97:
96:
73:
70:
69:
53:
50:
49:
46:
12:
11:
5:
2634:
2624:
2623:
2618:
2616:Homeomorphisms
2613:
2608:
2603:
2586:
2585:
2583:
2582:
2577:
2572:
2567:
2562:
2561:
2560:
2550:
2545:
2540:
2535:
2530:
2525:
2519:
2517:
2513:
2512:
2510:
2509:
2504:
2499:
2494:
2489:
2484:
2478:
2476:
2472:
2471:
2468:
2467:
2465:
2464:
2459:
2454:
2449:
2444:
2439:
2434:
2429:
2424:
2419:
2413:
2411:
2405:
2404:
2402:
2401:
2396:
2391:
2386:
2381:
2376:
2371:
2361:
2356:
2351:
2341:
2336:
2331:
2326:
2321:
2316:
2310:
2308:
2302:
2301:
2299:
2298:
2293:
2288:
2287:
2286:
2276:
2271:
2270:
2269:
2259:
2254:
2249:
2244:
2243:
2242:
2232:
2227:
2226:
2225:
2215:
2210:
2204:
2202:
2198:
2197:
2195:
2194:
2189:
2184:
2179:
2178:
2177:
2167:
2162:
2157:
2151:
2149:
2142:
2136:
2135:
2133:
2132:
2127:
2117:
2112:
2098:
2093:
2088:
2083:
2078:
2076:Parallelizable
2073:
2068:
2063:
2062:
2061:
2051:
2046:
2041:
2036:
2031:
2026:
2021:
2016:
2011:
2006:
1996:
1986:
1980:
1978:
1972:
1971:
1969:
1968:
1963:
1958:
1956:Lie derivative
1953:
1951:Integral curve
1948:
1943:
1938:
1937:
1936:
1926:
1921:
1920:
1919:
1912:Diffeomorphism
1909:
1903:
1901:
1895:
1894:
1892:
1891:
1886:
1881:
1876:
1871:
1866:
1861:
1856:
1851:
1845:
1843:
1834:
1833:
1831:
1830:
1825:
1820:
1815:
1810:
1805:
1800:
1795:
1790:
1789:
1788:
1783:
1773:
1772:
1771:
1760:
1758:
1757:Basic concepts
1754:
1753:
1741:
1740:
1733:
1726:
1718:
1712:
1711:
1698:
1673:
1660:
1647:
1633:
1610:
1607:
1604:
1603:
1601:, Theorem 3.21
1590:
1577:
1563:
1562:
1560:
1557:
1556:
1555:
1546:
1540:
1534:
1528:
1522:
1519:Diffeomorphism
1514:
1511:
1508:
1507:
1488:
1486:
1469:
1466:
1465:
1464:
1462:
1461:evenly covered
1446:
1439:diffeomorphism
1433:
1430:
1418:
1415:
1388:
1385:
1350:
1345:
1319:
1314:
1290:
1285:
1255:
1252:
1249:
1246:
1243:
1223:
1203:
1183:
1180:
1177:
1174:
1171:
1151:
1131:
1111:
1091:
1071:
1068:
1065:
1045:
1042:
1039:
1019:
999:
987:
984:
958:
954:
950:
930:
925:
922:
919:
916:
912:
891:
886:
882:
861:
858:
855:
827:
824:
821:
818:
815:
812:
809:
789:
786:
783:
780:
777:
754:
734:
714:
711:
708:
684:
679:
676:
673:
670:
666:
662:
659:
654:
650:
646:
641:
637:
633:
610:
607:
604:
601:
598:
571:
568:
555:
535:
532:
529:
526:
506:
486:
483:
480:
477:
474:
471:
468:
463:
458:
453:
429:
426:
423:
420:
400:
380:
360:
357:
354:
334:
331:
328:
325:
322:
305:diffeomorphism
292:
289:
286:
283:
280:
277:
274:
269:
265:
261:
241:
221:
218:
215:
212:
200:such that the
189:
169:
146:
143:
140:
116:
113:
110:
107:
104:
77:
57:
45:
42:
9:
6:
4:
3:
2:
2633:
2622:
2619:
2617:
2614:
2612:
2609:
2607:
2604:
2602:
2599:
2598:
2596:
2581:
2578:
2576:
2575:Supermanifold
2573:
2571:
2568:
2566:
2563:
2559:
2556:
2555:
2554:
2551:
2549:
2546:
2544:
2541:
2539:
2536:
2534:
2531:
2529:
2526:
2524:
2521:
2520:
2518:
2514:
2508:
2505:
2503:
2500:
2498:
2495:
2493:
2490:
2488:
2485:
2483:
2480:
2479:
2477:
2473:
2463:
2460:
2458:
2455:
2453:
2450:
2448:
2445:
2443:
2440:
2438:
2435:
2433:
2430:
2428:
2425:
2423:
2420:
2418:
2415:
2414:
2412:
2410:
2406:
2400:
2397:
2395:
2392:
2390:
2387:
2385:
2382:
2380:
2377:
2375:
2372:
2370:
2366:
2362:
2360:
2357:
2355:
2352:
2350:
2346:
2342:
2340:
2337:
2335:
2332:
2330:
2327:
2325:
2322:
2320:
2317:
2315:
2312:
2311:
2309:
2307:
2303:
2297:
2296:Wedge product
2294:
2292:
2289:
2285:
2282:
2281:
2280:
2277:
2275:
2272:
2268:
2265:
2264:
2263:
2260:
2258:
2255:
2253:
2250:
2248:
2245:
2241:
2240:Vector-valued
2238:
2237:
2236:
2233:
2231:
2228:
2224:
2221:
2220:
2219:
2216:
2214:
2211:
2209:
2206:
2205:
2203:
2199:
2193:
2190:
2188:
2185:
2183:
2180:
2176:
2173:
2172:
2171:
2170:Tangent space
2168:
2166:
2163:
2161:
2158:
2156:
2153:
2152:
2150:
2146:
2143:
2141:
2137:
2131:
2128:
2126:
2122:
2118:
2116:
2113:
2111:
2107:
2103:
2099:
2097:
2094:
2092:
2089:
2087:
2084:
2082:
2079:
2077:
2074:
2072:
2069:
2067:
2064:
2060:
2057:
2056:
2055:
2052:
2050:
2047:
2045:
2042:
2040:
2037:
2035:
2032:
2030:
2027:
2025:
2022:
2020:
2017:
2015:
2012:
2010:
2007:
2005:
2001:
1997:
1995:
1991:
1987:
1985:
1982:
1981:
1979:
1973:
1967:
1964:
1962:
1959:
1957:
1954:
1952:
1949:
1947:
1944:
1942:
1939:
1935:
1934:in Lie theory
1932:
1931:
1930:
1927:
1925:
1922:
1918:
1915:
1914:
1913:
1910:
1908:
1905:
1904:
1902:
1900:
1896:
1890:
1887:
1885:
1882:
1880:
1877:
1875:
1872:
1870:
1867:
1865:
1862:
1860:
1857:
1855:
1852:
1850:
1847:
1846:
1844:
1841:
1837:Main results
1835:
1829:
1826:
1824:
1821:
1819:
1818:Tangent space
1816:
1814:
1811:
1809:
1806:
1804:
1801:
1799:
1796:
1794:
1791:
1787:
1784:
1782:
1779:
1778:
1777:
1774:
1770:
1767:
1766:
1765:
1762:
1761:
1759:
1755:
1750:
1746:
1739:
1734:
1732:
1727:
1725:
1720:
1719:
1716:
1709:
1705:
1701:
1695:
1690:
1685:
1681:
1680:
1674:
1671:
1667:
1663:
1657:
1653:
1648:
1644:
1640:
1636:
1630:
1626:
1622:
1618:
1613:
1612:
1600:
1594:
1587:
1581:
1574:
1568:
1564:
1550:
1547:
1544:
1541:
1538:
1535:
1532:
1529:
1526:
1525:Homeomorphism
1523:
1520:
1517:
1516:
1504:
1495:
1491:
1487:
1484:
1480:
1479:
1475:
1460:
1458:
1454:
1451:
1447:
1444:
1440:
1436:
1435:
1429:
1416:
1413:
1405:
1401:
1398:
1394:
1384:
1382:
1381:compact space
1378:
1374:
1370:
1365:
1363:
1348:
1317:
1288:
1273:
1268:
1253:
1247:
1244:
1241:
1221:
1201:
1181:
1175:
1172:
1169:
1149:
1129:
1109:
1089:
1069:
1066:
1063:
1043:
1040:
1037:
1017:
997:
983:
981:
977:
972:
956:
952:
948:
928:
920:
914:
910:
889:
884:
880:
859:
856:
853:
845:
841:
825:
822:
819:
816:
813:
810:
807:
787:
781:
778:
775:
766:
752:
732:
712:
709:
706:
698:
682:
674:
668:
664:
657:
652:
648:
644:
639:
635:
631:
624:
608:
602:
599:
596:
588:
583:
581:
577:
567:
553:
530:
524:
504:
481:
475:
469:
466:
461:
451:
443:
424:
418:
398:
378:
358:
355:
352:
332:
326:
323:
320:
313:
308:
306:
287:
281:
275:
272:
267:
259:
239:
216:
210:
203:
187:
167:
160:
144:
141:
138:
130:
114:
108:
105:
102:
95:
91:
75:
55:
41:
39:
35:
31:
27:
23:
19:
2502:Moving frame
2497:Morse theory
2487:Gauge theory
2279:Tensor field
2208:Closed/Exact
2187:Vector field
2155:Distribution
2096:Hypercomplex
2091:Quaternionic
1916:
1828:Vector field
1786:Smooth atlas
1678:
1651:
1616:
1598:
1593:
1585:
1580:
1572:
1567:
1498:
1494:adding to it
1489:
1457:neighborhood
1453:covering map
1390:
1366:
1269:
989:
973:
767:
584:
573:
309:
128:
47:
25:
15:
2447:Levi-Civita
2437:Generalized
2409:Connections
2359:Lie algebra
2291:Volume form
2192:Vector flow
2165:Pushforward
2160:Lie bracket
2059:Lie algebra
2024:G-structure
1813:Pushforward
1793:Submanifold
1463:by the map.
391:containing
232:is open in
180:containing
18:mathematics
2595:Categories
2570:Stratifold
2528:Diffeology
2324:Associated
2125:Symplectic
2110:Riemannian
2039:Hyperbolic
1966:Submersion
1874:Hopf–Rinow
1808:Submersion
1803:Smooth map
1609:References
1472:See also:
1387:Properties
1377:continuous
986:Discussion
844:submersion
623:derivative
2452:Principal
2427:Ehresmann
2384:Subbundle
2374:Principal
2349:Fibration
2329:Cotangent
2201:Covectors
2054:Lie group
2034:Hermitian
1977:manifolds
1946:Immersion
1941:Foliation
1879:Noether's
1864:Frobenius
1859:De Rham's
1854:Darboux's
1745:Manifolds
1501:July 2010
1443:bijective
1251:→
1179:→
1067:∈
1041:∈
857:∈
840:immersion
823:
811:
785:→
710:∈
661:→
606:→
576:immersion
473:→
356:∈
330:→
312:immersion
279:→
142:∈
112:→
2548:Orbifold
2543:K-theory
2533:Diffiety
2257:Pullback
2071:Oriented
2049:Kenmotsu
2029:Hadamard
1975:Types of
1924:Geodesic
1749:Glossary
1513:See also
1459:that is
1432:Examples
1400:open map
1369:2-sphere
978:, while
580:open map
159:open set
94:function
32:between
2492:History
2475:Related
2389:Tangent
2367:)
2347:)
2314:Adjoint
2306:Bundles
2284:density
2182:Torsion
2148:Vectors
2140:Tensors
2123:)
2108:)
2104:,
2102:Pseudo−
2081:Poisson
2014:Finsler
2009:Fibered
2004:Contact
2002:)
1994:Complex
1992:)
1961:Section
1708:4696768
1670:2954043
1643:2428390
1597:Axler,
1334:Exotic
872:, both
2457:Vector
2442:Koszul
2422:Cartan
2417:Affine
2399:Vector
2394:Tensor
2379:Spinor
2369:Normal
2365:Stable
2319:Affine
2223:bundle
2175:bundle
2121:Almost
2044:Kähler
2000:Almost
1990:Almost
1984:Closed
1884:Sard's
1840:(list)
1706:
1696:
1668:
1658:
1641:
1631:
1450:smooth
444:, and
440:is an
2565:Sheaf
2339:Fiber
2115:Rizza
2086:Prime
1917:Local
1907:Curve
1769:Atlas
1584:Lee,
1571:Lee,
1559:Notes
1441:is a
695:is a
303:is a
202:image
127:is a
2432:Form
2334:Dual
2267:flow
2130:Tame
2106:Sub−
2019:Flat
1899:Maps
1694:ISBN
1656:ISBN
1629:ISBN
1404:rank
1122:and
1056:and
1010:and
902:and
745:and
585:The
517:and
252:and
92:. A
68:and
48:Let
24:, a
2354:Jet
1684:doi
1496:.
1406:of
1371:to
1364:).
1274:on
1142:of
1102:of
820:dim
808:dim
582:.
307:.
88:be
30:map
16:In
2597::
2345:Co
1704:MR
1702:,
1692:,
1666:MR
1664:,
1639:MR
1637:,
1627:,
1619:,
1448:A
1437:A
566:.
2363:(
2343:(
2119:(
2100:(
1998:(
1988:(
1751:)
1747:(
1737:e
1730:t
1723:v
1686::
1646:.
1503:)
1499:(
1417:.
1414:n
1349:4
1344:R
1318:4
1313:R
1289:4
1284:R
1254:Y
1248:X
1245::
1242:f
1222:X
1202:f
1182:V
1176:U
1173::
1170:f
1150:y
1130:V
1110:x
1090:U
1070:Y
1064:y
1044:X
1038:x
1018:Y
998:X
957:x
953:f
949:D
929:Y
924:)
921:x
918:(
915:f
911:T
890:X
885:x
881:T
860:X
854:x
826:Y
817:=
814:X
788:Y
782:X
779::
776:f
753:Y
733:X
713:X
707:x
683:Y
678:)
675:x
672:(
669:f
665:T
658:X
653:x
649:T
645::
640:x
636:f
632:D
609:Y
603:X
600::
597:f
554:Y
534:)
531:U
528:(
525:f
505:X
485:)
482:U
479:(
476:f
470:U
467::
462:U
457:|
452:f
428:)
425:U
422:(
419:f
399:x
379:U
359:X
353:x
333:Y
327:X
324::
321:f
291:)
288:U
285:(
282:f
276:U
273::
268:U
264:|
260:f
240:Y
220:)
217:U
214:(
211:f
188:x
168:U
145:X
139:x
115:Y
109:X
106::
103:f
76:Y
56:X
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
