1618:
1318:
1495:
813:
1659:
165:
2616:
1738:
1553:
992:
1197:
767:
1038:
318:
925:
2530:
1253:
1919:
1832:
474:
1111:
1071:
1881:
1692:
842:
411:
667:
2026:
697:
594:
1770:
2454:
2229:
2191:
371:
229:
1409:
1223:
1140:
2651:
2314:
2282:
2157:
2125:
1953:
2381:
2474:
2425:
2405:
2358:
2250:
2094:
2074:
2046:
2000:
1980:
1790:
1433:
1382:
1362:
1342:
949:
717:
643:
614:
561:
538:
514:
436:
340:
269:
249:
205:
185:
81:
57:
886:
2284:
is
Hausdorff depends only on the properness of the action (as discussed above); the rest of the claim requires freeness and is a consequence of the
1558:
1258:
2795:
2781:
2738:
2285:
2810:
1438:
2002:-action is automatically proper. An example of proper action by a not necessarily compact Lie group is given by the action a Lie subgroup
1504:, any abstract infinitesimal action of a (finite-dimensional) Lie algebra on a compact manifold can be integrated to a Lie group action.
772:
1623:
102:
1793:
1555:
is injective if and only if the corresponding global Lie group action is free. This follows from the fact that the kernel of
2546:
1700:
1515:
954:
2667:
1145:
732:
564:
36:
2662:
2194:
1412:
855:
848:
2288:. If the "free action" condition (i.e. "having zero stabilizers") is relaxed to "having finite stabilizers",
997:
277:
2672:
892:
485:
1501:
889:, Lie group actions can also be studied from the infinitesimal point of view. Indeed, any Lie group action
2533:
2479:
1228:
272:
1892:
1798:
447:
1076:
1043:
720:
28:
1853:
1664:
818:
383:
1922:
727:
648:
477:
442:
2815:
859:
378:
2005:
680:
573:
2623:
2337:
2333:
1743:
88:
2430:
2791:
2777:
2744:
2734:
2709:
2200:
2162:
994:. Intuitively, this is obtained by differentiating at the identity the Lie group homomorphism
866:
356:
214:
2701:
2384:
1387:
1202:
1119:
873:
2629:
1956:
84:
2291:
2259:
2134:
2102:
1930:
2363:
2459:
2410:
2390:
2343:
2321:
2235:
2079:
2059:
2031:
1985:
1965:
1775:
1418:
1367:
1347:
1327:
1321:
934:
702:
628:
622:
599:
546:
523:
499:
421:
325:
254:
234:
190:
170:
66:
42:
2456:
diagonally. The action is free since it is so on the first factor and is proper since
2804:
1884:
1843:
414:
17:
2128:
2097:
484:
Forgetting the smooth structure, a Lie group action is a particular case of a
2748:
2713:
60:
1613:{\displaystyle \mathrm {d} _{e}\sigma _{x}\colon {\mathfrak {g}}\to T_{x}M}
1313:{\displaystyle \mathrm {d} _{e}\sigma _{x}\colon {\mathfrak {g}}\to T_{x}M}
2728:
2127:
does not admit in general a manifold structure. However, if the action is
2317:
2705:
1792:-bundle: the image of the infinitesimal action is actually equal to the
1199:
is differentiable and one can compute its differential at the identity
1490:{\displaystyle {\mathfrak {g}}\to {\mathfrak {X}}(M),X\mapsto X^{\#}}
320:. A smooth manifold endowed with a Lie group action is also called a
2694:"A global formulation of the Lie theory of transformation groups"
2693:
1842:
An important (and common) class of Lie group actions is that of
808:{\displaystyle G\subseteq \operatorname {GL} (n,\mathbb {R} )}
1654:{\displaystyle {\mathfrak {g}}_{x}\subseteq {\mathfrak {g}}}
616:
by left multiplication, right multiplication or conjugation;
160:{\displaystyle \sigma :G\times M\to M,(g,x)\mapsto g\cdot x}
1073:
as the Lie algebra of the (infinite-dimensional) Lie group
563:
on itself by left multiplication, right multiplication or
1846:
ones. Indeed, such a topological condition implies that
2159:
has a unique smooth structure such that the projection
231:
is differentiable. Equivalently, a Lie group action of
2788:
Foundations of differentiable manifolds and Lie groups
2632:
2611:{\displaystyle H_{G}^{*}(M)=H_{\text{dr}}^{*}(M_{G})}
2549:
2482:
2476:
is compact; thus, one can form the quotient manifold
2462:
2433:
2413:
2393:
2366:
2346:
2294:
2262:
2238:
2203:
2165:
2137:
2105:
2082:
2062:
2034:
2008:
1988:
1968:
1933:
1895:
1856:
1801:
1778:
1746:
1733:{\displaystyle {\mathfrak {g}}\to {\mathfrak {X}}(M)}
1703:
1667:
1626:
1561:
1548:{\displaystyle {\mathfrak {g}}\to {\mathfrak {X}}(M)}
1518:
1441:
1421:
1390:
1370:
1350:
1330:
1261:
1231:
1205:
1148:
1122:
1079:
1046:
1000:
987:{\displaystyle {\mathfrak {g}}\to {\mathfrak {X}}(M)}
957:
937:
895:
821:
775:
735:
705:
683:
651:
631:
602:
576:
549:
526:
502:
450:
424:
386:
359:
328:
280:
257:
237:
217:
193:
173:
105:
69:
45:
1192:{\displaystyle \sigma _{x}:G\to M,g\mapsto g\cdot x}
762:{\displaystyle \operatorname {GL} (n,\mathbb {R} )}
2645:
2610:
2524:
2468:
2448:
2419:
2399:
2375:
2352:
2308:
2276:
2244:
2223:
2185:
2151:
2119:
2088:
2068:
2040:
2020:
1994:
1974:
1947:
1913:
1875:
1826:
1784:
1764:
1732:
1686:
1653:
1612:
1547:
1489:
1427:
1403:
1376:
1356:
1336:
1312:
1247:
1217:
1191:
1134:
1105:
1065:
1032:
986:
943:
919:
836:
807:
761:
711:
691:
661:
637:
608:
588:
555:
532:
508:
468:
430:
405:
373:is smooth has a couple of immediate consequences:
365:
334:
312:
263:
243:
223:
199:
179:
159:
75:
51:
872:more generally, the group action underlying any
2387:, which we can assume to be a manifold since
1740:in general not surjective. For instance, let
1384:. The minus of this vector field, denoted by
673:Other examples of Lie group actions include:
8:
2698:Memoirs of the American Mathematical Society
1040:, and interpreting the set of vector fields
2637:
2631:
2599:
2586:
2581:
2559:
2554:
2548:
2514:
2487:
2481:
2461:
2432:
2412:
2392:
2365:
2345:
2298:
2293:
2266:
2261:
2237:
2213:
2202:
2175:
2164:
2141:
2136:
2109:
2104:
2081:
2061:
2033:
2007:
1987:
1967:
1937:
1932:
1894:
1861:
1855:
1806:
1800:
1777:
1745:
1715:
1714:
1705:
1704:
1702:
1672:
1666:
1645:
1644:
1635:
1629:
1628:
1625:
1601:
1588:
1587:
1578:
1568:
1563:
1560:
1530:
1529:
1520:
1519:
1517:
1481:
1453:
1452:
1443:
1442:
1440:
1420:
1395:
1389:
1369:
1349:
1329:
1301:
1288:
1287:
1278:
1268:
1263:
1260:
1239:
1238:
1230:
1204:
1153:
1147:
1121:
1080:
1078:
1048:
1047:
1045:
1007:
999:
969:
968:
959:
958:
956:
936:
894:
828:
824:
823:
820:
798:
797:
774:
752:
751:
734:
704:
685:
684:
682:
653:
652:
650:
630:
601:
575:
548:
525:
501:
449:
423:
413:of the group action are closed, thus are
391:
385:
358:
327:
287:
279:
256:
236:
216:
192:
172:
104:
68:
44:
2332:An application of this principle is the
167:be a (left) group action of a Lie group
2684:
1033:{\displaystyle G\to \mathrm {Diff} (M)}
516:, the following are Lie group actions:
313:{\displaystyle G\to \mathrm {Diff} (M)}
2622:where the right-hand side denotes the
920:{\displaystyle \sigma :G\times M\to M}
2767:Torus actions on symplectic manifolds
865:the transitive action underlying any
7:
2733:(2nd ed.). New York: Springer.
2525:{\displaystyle M_{G}=(EG\times M)/G}
1512:An infinitesimal Lie algebra action
1248:{\displaystyle X\in {\mathfrak {g}}}
887:Lie group-Lie algebra correspondence
1914:{\displaystyle G\cdot x\subseteq M}
1827:{\displaystyle T^{\pi }P\subset TP}
1716:
1706:
1646:
1630:
1589:
1531:
1521:
1454:
1444:
1289:
1240:
1049:
970:
960:
654:
469:{\displaystyle G\cdot x\subseteq M}
1564:
1482:
1396:
1264:
1106:{\displaystyle \mathrm {Diff} (M)}
1090:
1087:
1084:
1081:
1066:{\displaystyle {\mathfrak {X}}(M)}
1017:
1014:
1011:
1008:
951:, i.e. a Lie algebra homomorphism
297:
294:
291:
288:
25:
2774:Introduction to smooth manifolds
2730:Introduction to smooth manifolds
1876:{\displaystyle G_{x}\subseteq G}
1687:{\displaystyle G_{x}\subseteq G}
1497:is a Lie algebra homomorphism).
929:infinitesimal Lie algebra action
881:Infinitesimal Lie algebra action
837:{\displaystyle \mathbb {R} ^{n}}
406:{\displaystyle G_{x}\subseteq G}
662:{\displaystyle {\mathfrak {g}}}
570:the action of any Lie subgroup
39:adapted to the smooth setting:
2605:
2592:
2571:
2565:
2511:
2496:
2207:
2169:
1756:
1727:
1721:
1711:
1594:
1542:
1536:
1526:
1474:
1465:
1459:
1449:
1364:one obtains a vector field on
1294:
1177:
1165:
1100:
1094:
1060:
1054:
1027:
1021:
1004:
981:
975:
965:
911:
802:
788:
756:
742:
307:
301:
284:
211:(or smooth action) if the map
145:
142:
130:
121:
1:
2668:Equivariant differential form
1435:(the minus sign ensures that
353:The fact that the action map
2056:Given a Lie group action of
2052:Structure of the orbit space
2021:{\displaystyle H\subseteq G}
885:Following the spirit of the
692:{\displaystyle \mathbb {R} }
589:{\displaystyle H\subseteq G}
2811:Group actions (mathematics)
2692:Palais, Richard S. (1957).
1962:In general, if a Lie group
1765:{\displaystyle \pi :P\to M}
1116:More precisely, fixing any
2832:
2449:{\displaystyle EG\times M}
844:by matrix multiplication;
769:and of its Lie subgroups
719:given by the flow of any
2663:Hamiltonian group action
2224:{\displaystyle M\to M/G}
2186:{\displaystyle M\to M/G}
1413:fundamental vector field
856:Hamiltonian group action
849:Lie group representation
476:of the group action are
87:, and the action map is
2673:Isotropy representation
1982:is compact, any smooth
1255:, then its image under
486:continuous group action
366:{\displaystyle \sigma }
224:{\displaystyle \sigma }
2647:
2612:
2534:equivariant cohomology
2526:
2470:
2450:
2421:
2401:
2377:
2354:
2328:Equivariant cohomology
2310:
2278:
2246:
2225:
2187:
2153:
2121:
2090:
2070:
2042:
2022:
1996:
1976:
1949:
1915:
1877:
1828:
1786:
1766:
1734:
1688:
1655:
1614:
1549:
1491:
1429:
1405:
1404:{\displaystyle X^{\#}}
1378:
1358:
1338:
1314:
1249:
1219:
1218:{\displaystyle e\in G}
1193:
1136:
1135:{\displaystyle x\in M}
1107:
1067:
1034:
988:
945:
921:
838:
809:
763:
713:
693:
663:
639:
610:
590:
557:
534:
520:the trivial action of
510:
470:
432:
407:
367:
336:
314:
273:Lie group homomorphism
265:
245:
225:
201:
181:
161:
77:
53:
2727:Lee, John M. (2012).
2648:
2646:{\displaystyle M_{G}}
2613:
2527:
2471:
2451:
2422:
2402:
2378:
2355:
2311:
2279:
2247:
2226:
2188:
2154:
2122:
2091:
2071:
2043:
2023:
1997:
1977:
1950:
1923:embedded submanifolds
1916:
1878:
1829:
1787:
1767:
1735:
1689:
1656:
1615:
1550:
1492:
1430:
1411:, is also called the
1406:
1379:
1359:
1339:
1315:
1250:
1220:
1194:
1137:
1108:
1068:
1035:
989:
946:
922:
839:
810:
764:
721:complete vector field
714:
694:
664:
640:
611:
591:
558:
535:
511:
478:immersed submanifolds
471:
433:
408:
368:
337:
315:
266:
246:
226:
202:
187:on a smooth manifold
182:
162:
78:
54:
29:differential geometry
2630:
2547:
2480:
2460:
2431:
2411:
2407:is compact, and let
2391:
2364:
2344:
2292:
2260:
2236:
2201:
2163:
2135:
2103:
2080:
2060:
2032:
2006:
1986:
1966:
1931:
1893:
1854:
1799:
1776:
1744:
1701:
1665:
1624:
1559:
1516:
1439:
1419:
1388:
1368:
1348:
1328:
1259:
1229:
1203:
1146:
1120:
1077:
1044:
998:
955:
935:
893:
847:more generally, any
819:
773:
733:
728:general linear group
703:
681:
649:
629:
600:
574:
547:
524:
500:
496:For every Lie group
448:
422:
384:
357:
326:
278:
255:
235:
215:
191:
171:
103:
67:
43:
2591:
2564:
2316:becomes instead an
2309:{\displaystyle M/G}
2277:{\displaystyle M/G}
2152:{\displaystyle M/G}
2120:{\displaystyle M/G}
1948:{\displaystyle M/G}
1697:On the other hand,
1620:is the Lie algebra
860:symplectic manifold
726:the actions of the
645:on its Lie algebra
2769:, Birkhauser, 2004
2643:
2624:de Rham cohomology
2608:
2577:
2550:
2522:
2466:
2446:
2417:
2397:
2376:{\displaystyle EG}
2373:
2350:
2338:algebraic topology
2334:Borel construction
2306:
2274:
2242:
2221:
2183:
2149:
2117:
2086:
2066:
2038:
2018:
1992:
1972:
1945:
1911:
1873:
1824:
1794:vertical subbundle
1782:
1762:
1730:
1684:
1661:of the stabilizer
1651:
1610:
1545:
1502:Lie–Palais theorem
1487:
1425:
1401:
1374:
1354:
1334:
1310:
1245:
1215:
1189:
1132:
1103:
1063:
1030:
984:
941:
917:
851:on a vector space;
834:
805:
759:
709:
689:
659:
635:
606:
586:
553:
530:
506:
466:
428:
403:
363:
332:
310:
261:
241:
221:
197:
177:
157:
73:
49:
2796:978-0-387-90894-6
2782:978-1-4419-9981-8
2740:978-1-4419-9982-5
2706:10.1090/memo/0022
2584:
2469:{\displaystyle G}
2420:{\displaystyle G}
2400:{\displaystyle G}
2353:{\displaystyle G}
2245:{\displaystyle G}
2131:and proper, then
2089:{\displaystyle M}
2069:{\displaystyle G}
2041:{\displaystyle G}
1995:{\displaystyle G}
1975:{\displaystyle G}
1785:{\displaystyle G}
1428:{\displaystyle X}
1377:{\displaystyle M}
1357:{\displaystyle x}
1337:{\displaystyle x}
944:{\displaystyle M}
867:homogeneous space
712:{\displaystyle M}
638:{\displaystyle G}
609:{\displaystyle G}
556:{\displaystyle G}
533:{\displaystyle G}
509:{\displaystyle G}
431:{\displaystyle G}
335:{\displaystyle G}
264:{\displaystyle M}
244:{\displaystyle G}
207:; it is called a
200:{\displaystyle M}
180:{\displaystyle G}
76:{\displaystyle M}
52:{\displaystyle G}
18:Quotient manifold
16:(Redirected from
2823:
2753:
2752:
2724:
2718:
2717:
2689:
2652:
2650:
2649:
2644:
2642:
2641:
2626:of the manifold
2617:
2615:
2614:
2609:
2604:
2603:
2590:
2585:
2582:
2563:
2558:
2531:
2529:
2528:
2523:
2518:
2492:
2491:
2475:
2473:
2472:
2467:
2455:
2453:
2452:
2447:
2426:
2424:
2423:
2418:
2406:
2404:
2403:
2398:
2385:universal bundle
2382:
2380:
2379:
2374:
2360:is compact, let
2359:
2357:
2356:
2351:
2340:. Assuming that
2315:
2313:
2312:
2307:
2302:
2283:
2281:
2280:
2275:
2270:
2251:
2249:
2248:
2243:
2230:
2228:
2227:
2222:
2217:
2192:
2190:
2189:
2184:
2179:
2158:
2156:
2155:
2150:
2145:
2126:
2124:
2123:
2118:
2113:
2095:
2093:
2092:
2087:
2075:
2073:
2072:
2067:
2047:
2045:
2044:
2039:
2027:
2025:
2024:
2019:
2001:
1999:
1998:
1993:
1981:
1979:
1978:
1973:
1954:
1952:
1951:
1946:
1941:
1927:the orbit space
1920:
1918:
1917:
1912:
1882:
1880:
1879:
1874:
1866:
1865:
1850:the stabilizers
1833:
1831:
1830:
1825:
1811:
1810:
1791:
1789:
1788:
1783:
1771:
1769:
1768:
1763:
1739:
1737:
1736:
1731:
1720:
1719:
1710:
1709:
1693:
1691:
1690:
1685:
1677:
1676:
1660:
1658:
1657:
1652:
1650:
1649:
1640:
1639:
1634:
1633:
1619:
1617:
1616:
1611:
1606:
1605:
1593:
1592:
1583:
1582:
1573:
1572:
1567:
1554:
1552:
1551:
1546:
1535:
1534:
1525:
1524:
1496:
1494:
1493:
1488:
1486:
1485:
1458:
1457:
1448:
1447:
1434:
1432:
1431:
1426:
1415:associated with
1410:
1408:
1407:
1402:
1400:
1399:
1383:
1381:
1380:
1375:
1363:
1361:
1360:
1355:
1343:
1341:
1340:
1335:
1319:
1317:
1316:
1311:
1306:
1305:
1293:
1292:
1283:
1282:
1273:
1272:
1267:
1254:
1252:
1251:
1246:
1244:
1243:
1224:
1222:
1221:
1216:
1198:
1196:
1195:
1190:
1158:
1157:
1142:, the orbit map
1141:
1139:
1138:
1133:
1112:
1110:
1109:
1104:
1093:
1072:
1070:
1069:
1064:
1053:
1052:
1039:
1037:
1036:
1031:
1020:
993:
991:
990:
985:
974:
973:
964:
963:
950:
948:
947:
942:
926:
924:
923:
918:
874:principal bundle
843:
841:
840:
835:
833:
832:
827:
814:
812:
811:
806:
801:
768:
766:
765:
760:
755:
718:
716:
715:
710:
698:
696:
695:
690:
688:
668:
666:
665:
660:
658:
657:
644:
642:
641:
636:
615:
613:
612:
607:
595:
593:
592:
587:
562:
560:
559:
554:
540:on any manifold;
539:
537:
536:
531:
515:
513:
512:
507:
475:
473:
472:
467:
437:
435:
434:
429:
412:
410:
409:
404:
396:
395:
372:
370:
369:
364:
341:
339:
338:
333:
319:
317:
316:
311:
300:
270:
268:
267:
262:
250:
248:
247:
242:
230:
228:
227:
222:
209:Lie group action
206:
204:
203:
198:
186:
184:
183:
178:
166:
164:
163:
158:
82:
80:
79:
74:
58:
56:
55:
50:
33:Lie group action
21:
2831:
2830:
2826:
2825:
2824:
2822:
2821:
2820:
2801:
2800:
2765:Michele Audin,
2762:
2757:
2756:
2741:
2726:
2725:
2721:
2691:
2690:
2686:
2681:
2659:
2633:
2628:
2627:
2595:
2545:
2544:
2532:and define the
2483:
2478:
2477:
2458:
2457:
2429:
2428:
2409:
2408:
2389:
2388:
2362:
2361:
2342:
2341:
2330:
2290:
2289:
2258:
2257:
2234:
2233:
2231:is a principal
2199:
2198:
2161:
2160:
2133:
2132:
2101:
2100:
2078:
2077:
2058:
2057:
2054:
2030:
2029:
2004:
2003:
1984:
1983:
1964:
1963:
1929:
1928:
1891:
1890:
1857:
1852:
1851:
1840:
1802:
1797:
1796:
1774:
1773:
1772:be a principal
1742:
1741:
1699:
1698:
1668:
1663:
1662:
1627:
1622:
1621:
1597:
1574:
1562:
1557:
1556:
1514:
1513:
1510:
1500:Conversely, by
1477:
1437:
1436:
1417:
1416:
1391:
1386:
1385:
1366:
1365:
1346:
1345:
1326:
1325:
1297:
1274:
1262:
1257:
1256:
1227:
1226:
1201:
1200:
1149:
1144:
1143:
1118:
1117:
1075:
1074:
1042:
1041:
996:
995:
953:
952:
933:
932:
891:
890:
883:
822:
817:
816:
771:
770:
731:
730:
701:
700:
679:
678:
647:
646:
627:
626:
598:
597:
572:
571:
545:
544:
522:
521:
498:
497:
494:
446:
445:
420:
419:
387:
382:
381:
355:
354:
351:
324:
323:
276:
275:
253:
252:
233:
232:
213:
212:
189:
188:
169:
168:
101:
100:
97:
85:smooth manifold
65:
64:
41:
40:
23:
22:
15:
12:
11:
5:
2829:
2827:
2819:
2818:
2813:
2803:
2802:
2799:
2798:
2786:Frank Warner,
2784:
2770:
2761:
2758:
2755:
2754:
2739:
2719:
2683:
2682:
2680:
2677:
2676:
2675:
2670:
2665:
2658:
2655:
2640:
2636:
2620:
2619:
2607:
2602:
2598:
2594:
2589:
2580:
2576:
2573:
2570:
2567:
2562:
2557:
2553:
2521:
2517:
2513:
2510:
2507:
2504:
2501:
2498:
2495:
2490:
2486:
2465:
2445:
2442:
2439:
2436:
2416:
2396:
2372:
2369:
2349:
2329:
2326:
2322:quotient stack
2305:
2301:
2297:
2273:
2269:
2265:
2256:The fact that
2241:
2220:
2216:
2212:
2209:
2206:
2182:
2178:
2174:
2171:
2168:
2148:
2144:
2140:
2116:
2112:
2108:
2085:
2065:
2053:
2050:
2037:
2017:
2014:
2011:
1991:
1971:
1960:
1959:
1944:
1940:
1936:
1925:
1910:
1907:
1904:
1901:
1898:
1887:
1872:
1869:
1864:
1860:
1839:
1838:Proper actions
1836:
1823:
1820:
1817:
1814:
1809:
1805:
1781:
1761:
1758:
1755:
1752:
1749:
1729:
1726:
1723:
1718:
1713:
1708:
1683:
1680:
1675:
1671:
1648:
1643:
1638:
1632:
1609:
1604:
1600:
1596:
1591:
1586:
1581:
1577:
1571:
1566:
1544:
1541:
1538:
1533:
1528:
1523:
1509:
1506:
1484:
1480:
1476:
1473:
1470:
1467:
1464:
1461:
1456:
1451:
1446:
1424:
1398:
1394:
1373:
1353:
1344:, and varying
1333:
1322:tangent vector
1309:
1304:
1300:
1296:
1291:
1286:
1281:
1277:
1271:
1266:
1242:
1237:
1234:
1214:
1211:
1208:
1188:
1185:
1182:
1179:
1176:
1173:
1170:
1167:
1164:
1161:
1156:
1152:
1131:
1128:
1125:
1102:
1099:
1096:
1092:
1089:
1086:
1083:
1062:
1059:
1056:
1051:
1029:
1026:
1023:
1019:
1016:
1013:
1010:
1006:
1003:
983:
980:
977:
972:
967:
962:
940:
916:
913:
910:
907:
904:
901:
898:
882:
879:
878:
877:
870:
863:
852:
845:
831:
826:
804:
800:
796:
793:
790:
787:
784:
781:
778:
758:
754:
750:
747:
744:
741:
738:
724:
708:
687:
677:the action of
671:
670:
656:
634:
623:adjoint action
618:
617:
605:
585:
582:
579:
568:
552:
543:the action of
541:
529:
505:
493:
490:
482:
481:
465:
462:
459:
456:
453:
439:
427:
402:
399:
394:
390:
362:
350:
347:
331:
309:
306:
303:
299:
296:
293:
290:
286:
283:
271:consists of a
260:
240:
220:
196:
176:
156:
153:
150:
147:
144:
141:
138:
135:
132:
129:
126:
123:
120:
117:
114:
111:
108:
96:
93:
89:differentiable
72:
48:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2828:
2817:
2814:
2812:
2809:
2808:
2806:
2797:
2793:
2790:, chapter 3,
2789:
2785:
2783:
2779:
2776:, chapter 9,
2775:
2771:
2768:
2764:
2763:
2759:
2750:
2746:
2742:
2736:
2732:
2731:
2723:
2720:
2715:
2711:
2707:
2703:
2699:
2695:
2688:
2685:
2678:
2674:
2671:
2669:
2666:
2664:
2661:
2660:
2656:
2654:
2638:
2634:
2625:
2600:
2596:
2587:
2578:
2574:
2568:
2560:
2555:
2551:
2543:
2542:
2541:
2539:
2535:
2519:
2515:
2508:
2505:
2502:
2499:
2493:
2488:
2484:
2463:
2443:
2440:
2437:
2434:
2414:
2394:
2386:
2370:
2367:
2347:
2339:
2335:
2327:
2325:
2323:
2319:
2303:
2299:
2295:
2287:
2286:slice theorem
2271:
2267:
2263:
2254:
2252:
2239:
2218:
2214:
2210:
2204:
2196:
2180:
2176:
2172:
2166:
2146:
2142:
2138:
2130:
2114:
2110:
2106:
2099:
2083:
2063:
2051:
2049:
2035:
2015:
2012:
2009:
1989:
1969:
1958:
1942:
1938:
1934:
1926:
1924:
1908:
1905:
1902:
1899:
1896:
1888:
1886:
1870:
1867:
1862:
1858:
1849:
1848:
1847:
1845:
1837:
1835:
1821:
1818:
1815:
1812:
1807:
1803:
1795:
1779:
1759:
1753:
1750:
1747:
1724:
1695:
1681:
1678:
1673:
1669:
1641:
1636:
1607:
1602:
1598:
1584:
1579:
1575:
1569:
1539:
1507:
1505:
1503:
1498:
1478:
1471:
1468:
1462:
1422:
1414:
1392:
1371:
1351:
1331:
1323:
1307:
1302:
1298:
1284:
1279:
1275:
1269:
1235:
1232:
1212:
1209:
1206:
1186:
1183:
1180:
1174:
1171:
1168:
1162:
1159:
1154:
1150:
1129:
1126:
1123:
1114:
1097:
1057:
1024:
1001:
978:
938:
930:
914:
908:
905:
902:
899:
896:
888:
880:
875:
871:
868:
864:
861:
857:
853:
850:
846:
829:
794:
791:
785:
782:
779:
776:
748:
745:
739:
736:
729:
725:
722:
706:
676:
675:
674:
632:
624:
620:
619:
603:
583:
580:
577:
569:
566:
550:
542:
527:
519:
518:
517:
503:
491:
489:
487:
479:
463:
460:
457:
454:
451:
444:
440:
438:
425:
416:
415:Lie subgroups
400:
397:
392:
388:
380:
376:
375:
374:
360:
348:
346:
344:
342:
329:
304:
281:
274:
258:
238:
218:
210:
194:
174:
154:
151:
148:
139:
136:
133:
127:
124:
118:
115:
112:
109:
106:
94:
92:
90:
86:
70:
62:
46:
38:
34:
30:
19:
2787:
2773:
2766:
2729:
2722:
2697:
2687:
2621:
2537:
2331:
2255:
2232:
2055:
1961:
1841:
1696:
1511:
1499:
1115:
928:
884:
672:
495:
483:
418:
352:
322:
321:
208:
98:
37:group action
32:
26:
2383:denote the
2098:orbit space
1889:the orbits
927:induces an
565:conjugation
379:stabilizers
2816:Lie groups
2805:Categories
2772:John Lee,
2760:References
2253:-bundle).
2197:(in fact,
2195:submersion
1508:Properties
349:Properties
95:Definition
2749:808682771
2714:0065-9266
2700:(22): 0.
2588:∗
2561:∗
2506:×
2441:×
2208:→
2170:→
2013:⊆
1957:Hausdorff
1906:⊆
1900:⋅
1868:⊆
1816:⊂
1808:π
1757:→
1748:π
1712:→
1679:⊆
1642:⊆
1595:→
1585::
1576:σ
1527:→
1483:#
1475:↦
1450:→
1397:#
1295:→
1285::
1276:σ
1236:∈
1210:∈
1184:⋅
1178:↦
1166:→
1151:σ
1127:∈
1005:→
966:→
912:→
906:×
897:σ
786:
780:⊆
740:
581:⊆
461:⊆
455:⋅
398:⊆
361:σ
343:-manifold
285:→
219:σ
152:⋅
146:↦
122:→
116:×
107:σ
61:Lie group
2657:See also
2318:orbifold
492:Examples
2427:act on
1885:compact
2794:
2780:
2747:
2737:
2712:
2096:, the
1844:proper
443:orbits
2679:Notes
2336:from
2193:is a
1320:is a
1225:. If
858:on a
83:is a
59:is a
35:is a
2792:ISBN
2778:ISBN
2745:OCLC
2735:ISBN
2710:ISSN
2320:(or
2129:free
1921:are
1883:are
854:any
621:the
441:the
377:the
99:Let
31:, a
2702:doi
2540:as
2536:of
2324:).
2076:on
2028:on
1955:is
1694:.
1324:at
931:on
815:on
699:on
625:of
596:on
417:of
345:.
251:on
27:In
2807::
2743:.
2708:.
2696:.
2653:.
2583:dr
2048:.
1834:.
1113:.
783:GL
737:GL
488:.
91:.
63:,
2751:.
2716:.
2704::
2639:G
2635:M
2618:,
2606:)
2601:G
2597:M
2593:(
2579:H
2575:=
2572:)
2569:M
2566:(
2556:G
2552:H
2538:M
2520:G
2516:/
2512:)
2509:M
2503:G
2500:E
2497:(
2494:=
2489:G
2485:M
2464:G
2444:M
2438:G
2435:E
2415:G
2395:G
2371:G
2368:E
2348:G
2304:G
2300:/
2296:M
2272:G
2268:/
2264:M
2240:G
2219:G
2215:/
2211:M
2205:M
2181:G
2177:/
2173:M
2167:M
2147:G
2143:/
2139:M
2115:G
2111:/
2107:M
2084:M
2064:G
2036:G
2016:G
2010:H
1990:G
1970:G
1943:G
1939:/
1935:M
1909:M
1903:x
1897:G
1871:G
1863:x
1859:G
1822:P
1819:T
1813:P
1804:T
1780:G
1760:M
1754:P
1751::
1728:)
1725:M
1722:(
1717:X
1707:g
1682:G
1674:x
1670:G
1647:g
1637:x
1631:g
1608:M
1603:x
1599:T
1590:g
1580:x
1570:e
1565:d
1543:)
1540:M
1537:(
1532:X
1522:g
1479:X
1472:X
1469:,
1466:)
1463:M
1460:(
1455:X
1445:g
1423:X
1393:X
1372:M
1352:x
1332:x
1308:M
1303:x
1299:T
1290:g
1280:x
1270:e
1265:d
1241:g
1233:X
1213:G
1207:e
1187:x
1181:g
1175:g
1172:,
1169:M
1163:G
1160::
1155:x
1130:M
1124:x
1101:)
1098:M
1095:(
1091:f
1088:f
1085:i
1082:D
1061:)
1058:M
1055:(
1050:X
1028:)
1025:M
1022:(
1018:f
1015:f
1012:i
1009:D
1002:G
982:)
979:M
976:(
971:X
961:g
939:M
915:M
909:M
903:G
900::
876:.
869:;
862:;
830:n
825:R
803:)
799:R
795:,
792:n
789:(
777:G
757:)
753:R
749:,
746:n
743:(
723:;
707:M
686:R
669:.
655:g
633:G
604:G
584:G
578:H
567:;
551:G
528:G
504:G
480:.
464:M
458:x
452:G
426:G
401:G
393:x
389:G
330:G
308:)
305:M
302:(
298:f
295:f
292:i
289:D
282:G
259:M
239:G
195:M
175:G
155:x
149:g
143:)
140:x
137:,
134:g
131:(
128:,
125:M
119:M
113:G
110::
71:M
47:G
20:)
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