17:
240:
1180:
1039:
583:
647:-axis. Thus the necessary null-homotopy follows. Since the Whitehead link is symmetric, that is, a homeomorphism of the 3-sphere switches components, it is also true that the meridian of
1498:
977:
1572:
1282:
1212:
1075:
74:
1540:
1250:
933:
859:
1416:
610:
1449:
1322:
1121:
106:
contractible manifolds are homeomorphic to a ball. For dimensions 1 and 2, the answer is classical and it is "yes". In dimension 2, it follows, for example, from the
815:
702:
515:
332:
302:
150:
2591:
1376:
1349:
886:
785:
758:
731:
672:
637:
542:
481:
447:
416:
389:
362:
272:
234:
184:
1782:
2586:
1873:
1897:
2092:
1502:
Another variation is to pick several subtori at each stage instead of just one. The cones over some of these continua appear as the complements of
1351:
in the iterative process. Each embedding should be an unknotted solid torus in the 3-sphere. The essential properties are that the meridian of
1962:
1658:
2188:
2241:
1769:
1291:
More examples of open, contractible 3-manifolds may be constructed by proceeding in similar fashion and picking different embeddings of
1130:
2525:
1677:
2290:
998:
1882:
2273:
2639:
2485:
2470:
2193:
1967:
1078:
2644:
2515:
547:
2520:
2490:
2198:
2154:
2135:
1902:
1846:
2057:
1922:
2442:
2307:
1999:
1841:
102:
is a contractible manifold. All manifolds homeomorphic to the ball are contractible, too. One can ask whether
1454:
2139:
2109:
2033:
2023:
1979:
1809:
1762:
107:
1907:
2480:
2099:
1994:
1814:
84:
940:
2129:
2124:
1545:
1255:
1185:
1048:
47:
1516:
1226:
2634:
2460:
2398:
2246:
1950:
1940:
1912:
1887:
1797:
1729:
1698:
891:
419:
244:
828:
243:
A thickened
Whitehead link. In the Whitehead manifold construction, the blue (untwisted) torus is a
2598:
2571:
2280:
2158:
2143:
2072:
1831:
1615:
2649:
2540:
2495:
2392:
2263:
2067:
1892:
1755:
1717:
1686:
1582:
206:
202:
76:
2077:
979:
which is a non-compact manifold without boundary. It follows from our previous observation, the
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2455:
2450:
2357:
2268:
2082:
2062:
1917:
1856:
1673:
1654:
1385:
984:
592:
98:
is one that can continuously be shrunk to a point inside the manifold itself. For example, an
1421:
1294:
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2613:
2407:
2362:
2285:
2256:
2114:
2047:
2042:
2037:
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1802:
1737:
1706:
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980:
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307:
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198:
125:
1354:
1327:
864:
763:
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709:
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615:
520:
459:
425:
394:
367:
340:
250:
212:
162:
2556:
2465:
2295:
2251:
2017:
1733:
1702:
2422:
2347:
2317:
2215:
2208:
2148:
2119:
1989:
1984:
1945:
1379:
640:
484:
450:
111:
2628:
2608:
2432:
2427:
2412:
2402:
2352:
2329:
2203:
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1587:
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1503:
2535:
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1646:
992:
41:
37:
16:
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1216:
157:
25:
1741:
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2387:
2300:
1932:
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33:
2417:
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2087:
1974:
1628:
99:
239:
2581:
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2566:
1957:
1778:
187:
153:
95:
91:, theorem 3) where he incorrectly claimed that no such manifold exists.
1747:
2173:
1613:(2011). "The Whitehead manifold is a union of two Euclidean spaces".
195:
991:
is contractible. In fact, a closer analysis involving a result of
186:
inside the sphere. (A solid torus is an ordinary three-dimensional
83:) discovered this puzzling object while he was trying to prove the
1689:(1934), "Certain theorems about three-dimensional manifolds (I)",
1175:{\displaystyle \left(\mathbb {R} ^{3}/W\right)\times \mathbb {R} }
238:
191:
15:
1751:
1653:. Lecture Notes in Mathematics, no. 1374, Springer-Verlag.
1034:{\displaystyle X\times \mathbb {R} \cong \mathbb {R} ^{4}.}
1720:(1935), "A certain open manifold whose group is unity",
1548:
1519:
1457:
1424:
1388:
1357:
1330:
1297:
1258:
1229:
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1133:
1127:
crunched to a point). It is not a manifold. However,
1094:
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1001:
943:
894:
867:
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793:
766:
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310:
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128:
50:
2549:
2508:
2441:
2338:
2234:
2181:
2172:
2008:
1931:
1870:
1790:
20:
First three tori of
Whitehead manifold construction
1566:
1534:
1492:
1443:
1410:
1370:
1343:
1316:
1276:
1244:
1206:
1174:
1115:
1069:
1033:
971:
927:
880:
853:
809:
779:
752:
725:
696:
666:
631:
604:
577:
536:
509:
475:
441:
410:
383:
356:
326:
296:
266:
228:
178:
144:
68:
578:{\displaystyle \mathbb {R} ^{3}\cup \{\infty \}}
861:or more precisely the intersection of all the
1763:
8:
674:is also null-homotopic in the complement of
572:
566:
1252:whose intersection is also homeomorphic to
2178:
1770:
1756:
1748:
87:, correcting an error in an earlier paper
1555:
1551:
1550:
1547:
1526:
1522:
1521:
1518:
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1429:
1423:
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1232:
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1228:
1195:
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1190:
1187:
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1167:
1151:
1145:
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1132:
1105:
1099:
1093:
1058:
1054:
1053:
1050:
1022:
1018:
1017:
1009:
1008:
1000:
954:
942:
893:
872:
866:
842:
830:
798:
792:
771:
765:
744:
738:
717:
711:
685:
679:
658:
652:
623:
617:
594:
557:
553:
552:
549:
528:
522:
498:
492:
467:
461:
433:
427:
402:
396:
375:
369:
348:
342:
315:
309:
285:
279:
258:
252:
220:
214:
170:
164:
133:
127:
88:
80:
57:
53:
52:
49:
1668:Rolfsen, Dale (2003), "Section 3.I.8.",
1605:
1603:
1599:
1513:is not a manifold but its product with
1493:{\displaystyle T_{i}\setminus T_{i+1}.}
1468:
960:
1672:, AMS Chelsea Publishing, p. 82,
937:The Whitehead manifold is defined as
487:in the complement of the meridian of
209:complement of the solid torus inside
7:
304:Everything must be contained within
972:{\displaystyle X=S^{3}\setminus W,}
1084:The one point compactification of
843:
596:
569:
14:
1567:{\displaystyle \mathbb {R} ^{4}.}
1418:and in addition the longitude of
1277:{\displaystyle \mathbb {R} ^{3}.}
1207:{\displaystyle \mathbb {R} ^{4}.}
1070:{\displaystyle \mathbb {R} ^{3}.}
110:. Dimension 3 presents the first
69:{\displaystyle \mathbb {R} ^{3}.}
1722:Quarterly Journal of Mathematics
1691:Quarterly Journal of Mathematics
1535:{\displaystyle \mathbb {R} ^{1}}
1451:should not be null-homotopic in
1245:{\displaystyle \mathbb {R} ^{3}}
517:This can be seen by considering
928:{\displaystyle k=1,2,3,\dots .}
817:and so on; to infinity. Define
156:. Now find a compact unknotted
1810:Differentiable/Smooth manifold
1223:is the union of two copies of
987:on homotopy equivalence, that
854:{\displaystyle W=T_{\infty },}
585:and the meridian curve as the
337:Now take a second solid torus
1:
1077:The reason is that it is not
1079:simply connected at infinity
2516:Classification of manifolds
1651:The topology of 4-manifolds
274:, and the orange torus is
194:, which is topologically a
2666:
114:: the Whitehead manifold.
2592:over commutative algebras
422:of the meridian curve of
247:of the meridian curve of
2308:Riemann curvature tensor
1742:10.1093/qmath/os-6.1.268
1711:10.1093/qmath/os-5.1.308
1411:{\displaystyle T_{i+1},}
605:{\displaystyle \infty .}
236:is another solid torus.
154:three-dimensional sphere
1444:{\displaystyle T_{i+1}}
1317:{\displaystyle T_{i+1}}
1116:{\displaystyle S^{3}/W}
1045:is not homeomorphic to
190:, that is, a filled-in
108:Riemann mapping theorem
2100:Manifold with boundary
1815:Differential structure
1568:
1536:
1494:
1445:
1412:
1372:
1345:
1318:
1278:
1246:
1208:
1176:
1117:
1071:
1035:
973:
929:
882:
855:
811:
810:{\displaystyle T_{1},}
781:
754:
727:
698:
697:{\displaystyle T_{2}.}
668:
633:
606:
579:
538:
511:
510:{\displaystyle T_{1}.}
477:
443:
412:
385:
358:
334:
328:
327:{\displaystyle T_{1}.}
298:
297:{\displaystyle T_{2}.}
268:
230:
180:
146:
145:{\displaystyle S^{3},}
77:J. H. C. Whitehead
70:
21:
2640:Differential geometry
1629:10.1112/jtopol/jtr010
1569:
1537:
1495:
1446:
1413:
1382:in the complement of
1373:
1371:{\displaystyle T_{i}}
1346:
1344:{\displaystyle T_{i}}
1319:
1279:
1247:
1209:
1177:
1118:
1072:
1036:
974:
930:
883:
881:{\displaystyle T_{k}}
856:
812:
782:
780:{\displaystyle T_{2}}
755:
753:{\displaystyle T_{2}}
728:
726:{\displaystyle T_{3}}
699:
669:
667:{\displaystyle T_{1}}
634:
632:{\displaystyle T_{2}}
607:
580:
539:
537:{\displaystyle S^{3}}
512:
478:
476:{\displaystyle T_{2}}
444:
442:{\displaystyle T_{1}}
413:
411:{\displaystyle T_{2}}
386:
384:{\displaystyle T_{1}}
359:
357:{\displaystyle T_{2}}
329:
299:
269:
267:{\displaystyle T_{1}}
242:
231:
229:{\displaystyle S^{3}}
181:
179:{\displaystyle T_{1}}
147:
71:
19:
2247:Covariant derivative
1798:Topological manifold
1546:
1517:
1455:
1422:
1386:
1355:
1328:
1295:
1256:
1227:
1186:
1131:
1092:
1049:
999:
941:
892:
865:
829:
791:
764:
737:
710:
678:
651:
616:
593:
589:-axis together with
548:
521:
491:
460:
426:
420:tubular neighborhood
395:
368:
341:
308:
278:
251:
245:tubular neighborhood
213:
163:
126:
48:
2281:Exterior derivative
1883:AtiyahâSinger index
1832:Riemannian manifold
1734:1935QJMat...6..268W
1718:Whitehead, J. H. C.
1703:1934QJMat...5..308W
1687:Whitehead, J. H. C.
1616:Journal of Topology
1542:is homeomorphic to
1182:is homeomorphic to
985:Whitehead's theorem
823:Whitehead continuum
760:in the same way as
85:Poincaré conjecture
2645:Geometric topology
2587:Secondary calculus
2541:Singularity theory
2496:Parallel transport
2264:De Rham cohomology
1903:Generalized Stokes
1583:List of topologies
1564:
1532:
1490:
1441:
1408:
1368:
1341:
1314:
1274:
1242:
1204:
1172:
1113:
1067:
1031:
969:
925:
878:
851:
807:
777:
750:
723:
694:
664:
629:
602:
575:
534:
507:
473:
439:
408:
381:
354:
335:
324:
294:
264:
226:
176:
142:
66:
30:Whitehead manifold
22:
2622:
2621:
2504:
2503:
2269:Differential form
1923:Whitney embedding
1857:Differential form
1660:978-0-387-51148-1
2657:
2614:Stratified space
2572:Fréchet manifold
2286:Interior product
2179:
1876:
1772:
1765:
1758:
1749:
1744:
1713:
1682:
1664:
1633:
1632:
1607:
1573:
1571:
1570:
1565:
1560:
1559:
1554:
1541:
1539:
1538:
1533:
1531:
1530:
1525:
1499:
1497:
1496:
1491:
1486:
1485:
1467:
1466:
1450:
1448:
1447:
1442:
1440:
1439:
1417:
1415:
1414:
1409:
1404:
1403:
1377:
1375:
1374:
1369:
1367:
1366:
1350:
1348:
1347:
1342:
1340:
1339:
1323:
1321:
1320:
1315:
1313:
1312:
1283:
1281:
1280:
1275:
1270:
1269:
1264:
1251:
1249:
1248:
1243:
1241:
1240:
1235:
1213:
1211:
1210:
1205:
1200:
1199:
1194:
1181:
1179:
1178:
1173:
1171:
1163:
1159:
1155:
1150:
1149:
1144:
1122:
1120:
1119:
1114:
1109:
1104:
1103:
1076:
1074:
1073:
1068:
1063:
1062:
1057:
1040:
1038:
1037:
1032:
1027:
1026:
1021:
1012:
981:Hurewicz theorem
978:
976:
975:
970:
959:
958:
934:
932:
931:
926:
887:
885:
884:
879:
877:
876:
860:
858:
857:
852:
847:
846:
816:
814:
813:
808:
803:
802:
786:
784:
783:
778:
776:
775:
759:
757:
756:
751:
749:
748:
732:
730:
729:
724:
722:
721:
703:
701:
700:
695:
690:
689:
673:
671:
670:
665:
663:
662:
638:
636:
635:
630:
628:
627:
611:
609:
608:
603:
584:
582:
581:
576:
562:
561:
556:
543:
541:
540:
535:
533:
532:
516:
514:
513:
508:
503:
502:
482:
480:
479:
474:
472:
471:
448:
446:
445:
440:
438:
437:
417:
415:
414:
409:
407:
406:
390:
388:
387:
382:
380:
379:
363:
361:
360:
355:
353:
352:
333:
331:
330:
325:
320:
319:
303:
301:
300:
295:
290:
289:
273:
271:
270:
265:
263:
262:
235:
233:
232:
227:
225:
224:
185:
183:
182:
177:
175:
174:
151:
149:
148:
143:
138:
137:
75:
73:
72:
67:
62:
61:
56:
2665:
2664:
2660:
2659:
2658:
2656:
2655:
2654:
2625:
2624:
2623:
2618:
2557:Banach manifold
2550:Generalizations
2545:
2500:
2437:
2334:
2296:Ricci curvature
2252:Cotangent space
2230:
2168:
2010:
2004:
1963:Exponential map
1927:
1872:
1866:
1786:
1776:
1716:
1685:
1680:
1670:Knots and links
1667:
1661:
1645:
1642:
1640:Further reading
1637:
1636:
1609:
1608:
1601:
1596:
1579:
1549:
1544:
1543:
1520:
1515:
1514:
1471:
1458:
1453:
1452:
1425:
1420:
1419:
1389:
1384:
1383:
1358:
1353:
1352:
1331:
1326:
1325:
1298:
1293:
1292:
1289:
1259:
1254:
1253:
1230:
1225:
1224:
1189:
1184:
1183:
1139:
1138:
1134:
1129:
1128:
1095:
1090:
1089:
1052:
1047:
1046:
1016:
997:
996:
950:
939:
938:
890:
889:
868:
863:
862:
838:
827:
826:
794:
789:
788:
767:
762:
761:
740:
735:
734:
713:
708:
707:
681:
676:
675:
654:
649:
648:
619:
614:
613:
591:
590:
551:
546:
545:
524:
519:
518:
494:
489:
488:
463:
458:
457:
449:is a thickened
429:
424:
423:
398:
393:
392:
371:
366:
365:
344:
339:
338:
311:
306:
305:
281:
276:
275:
254:
249:
248:
216:
211:
210:
166:
161:
160:
129:
124:
123:
122:Take a copy of
120:
94:A contractible
89:Whitehead (1934
51:
46:
45:
12:
11:
5:
2663:
2661:
2653:
2652:
2647:
2642:
2637:
2627:
2626:
2620:
2619:
2617:
2616:
2611:
2606:
2601:
2596:
2595:
2594:
2584:
2579:
2574:
2569:
2564:
2559:
2553:
2551:
2547:
2546:
2544:
2543:
2538:
2533:
2528:
2523:
2518:
2512:
2510:
2506:
2505:
2502:
2501:
2499:
2498:
2493:
2488:
2483:
2478:
2473:
2468:
2463:
2458:
2453:
2447:
2445:
2439:
2438:
2436:
2435:
2430:
2425:
2420:
2415:
2410:
2405:
2395:
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2385:
2375:
2370:
2365:
2360:
2355:
2350:
2344:
2342:
2336:
2335:
2333:
2332:
2327:
2322:
2321:
2320:
2310:
2305:
2304:
2303:
2293:
2288:
2283:
2278:
2277:
2276:
2266:
2261:
2260:
2259:
2249:
2244:
2238:
2236:
2232:
2231:
2229:
2228:
2223:
2218:
2213:
2212:
2211:
2201:
2196:
2191:
2185:
2183:
2176:
2170:
2169:
2167:
2166:
2161:
2151:
2146:
2132:
2127:
2122:
2117:
2112:
2110:Parallelizable
2107:
2102:
2097:
2096:
2095:
2085:
2080:
2075:
2070:
2065:
2060:
2055:
2050:
2045:
2040:
2030:
2020:
2014:
2012:
2006:
2005:
2003:
2002:
1997:
1992:
1990:Lie derivative
1987:
1985:Integral curve
1982:
1977:
1972:
1971:
1970:
1960:
1955:
1954:
1953:
1946:Diffeomorphism
1943:
1937:
1935:
1929:
1928:
1926:
1925:
1920:
1915:
1910:
1905:
1900:
1895:
1890:
1885:
1879:
1877:
1868:
1867:
1865:
1864:
1859:
1854:
1849:
1844:
1839:
1834:
1829:
1824:
1823:
1822:
1817:
1807:
1806:
1805:
1794:
1792:
1791:Basic concepts
1788:
1787:
1777:
1775:
1774:
1767:
1760:
1752:
1746:
1745:
1728:(1): 268â279,
1714:
1697:(1): 308â320,
1683:
1679:978-0821834367
1678:
1665:
1659:
1641:
1638:
1635:
1634:
1623:(3): 529â534.
1598:
1597:
1595:
1592:
1591:
1590:
1585:
1578:
1575:
1563:
1558:
1553:
1529:
1524:
1504:Casson handles
1489:
1484:
1481:
1478:
1474:
1470:
1465:
1461:
1438:
1435:
1432:
1428:
1407:
1402:
1399:
1396:
1392:
1380:null-homotopic
1365:
1361:
1338:
1334:
1311:
1308:
1305:
1301:
1288:
1287:Related spaces
1285:
1273:
1268:
1263:
1239:
1234:
1203:
1198:
1193:
1170:
1166:
1162:
1158:
1154:
1148:
1143:
1137:
1112:
1108:
1102:
1098:
1066:
1061:
1056:
1030:
1025:
1020:
1015:
1011:
1007:
1004:
968:
965:
962:
957:
953:
949:
946:
924:
921:
918:
915:
912:
909:
906:
903:
900:
897:
875:
871:
850:
845:
841:
837:
834:
806:
801:
797:
774:
770:
747:
743:
720:
716:
693:
688:
684:
661:
657:
641:winding number
626:
622:
601:
598:
574:
571:
568:
565:
560:
555:
531:
527:
506:
501:
497:
485:null-homotopic
470:
466:
451:Whitehead link
436:
432:
405:
401:
378:
374:
351:
347:
323:
318:
314:
293:
288:
284:
261:
257:
223:
219:
173:
169:
141:
136:
132:
119:
116:
112:counterexample
65:
60:
55:
13:
10:
9:
6:
4:
3:
2:
2662:
2651:
2648:
2646:
2643:
2641:
2638:
2636:
2633:
2632:
2630:
2615:
2612:
2610:
2609:Supermanifold
2607:
2605:
2602:
2600:
2597:
2593:
2590:
2589:
2588:
2585:
2583:
2580:
2578:
2575:
2573:
2570:
2568:
2565:
2563:
2560:
2558:
2555:
2554:
2552:
2548:
2542:
2539:
2537:
2534:
2532:
2529:
2527:
2524:
2522:
2519:
2517:
2514:
2513:
2511:
2507:
2497:
2494:
2492:
2489:
2487:
2484:
2482:
2479:
2477:
2474:
2472:
2469:
2467:
2464:
2462:
2459:
2457:
2454:
2452:
2449:
2448:
2446:
2444:
2440:
2434:
2431:
2429:
2426:
2424:
2421:
2419:
2416:
2414:
2411:
2409:
2406:
2404:
2400:
2396:
2394:
2391:
2389:
2386:
2384:
2380:
2376:
2374:
2371:
2369:
2366:
2364:
2361:
2359:
2356:
2354:
2351:
2349:
2346:
2345:
2343:
2341:
2337:
2331:
2330:Wedge product
2328:
2326:
2323:
2319:
2316:
2315:
2314:
2311:
2309:
2306:
2302:
2299:
2298:
2297:
2294:
2292:
2289:
2287:
2284:
2282:
2279:
2275:
2274:Vector-valued
2272:
2271:
2270:
2267:
2265:
2262:
2258:
2255:
2254:
2253:
2250:
2248:
2245:
2243:
2240:
2239:
2237:
2233:
2227:
2224:
2222:
2219:
2217:
2214:
2210:
2207:
2206:
2205:
2204:Tangent space
2202:
2200:
2197:
2195:
2192:
2190:
2187:
2186:
2184:
2180:
2177:
2175:
2171:
2165:
2162:
2160:
2156:
2152:
2150:
2147:
2145:
2141:
2137:
2133:
2131:
2128:
2126:
2123:
2121:
2118:
2116:
2113:
2111:
2108:
2106:
2103:
2101:
2098:
2094:
2091:
2090:
2089:
2086:
2084:
2081:
2079:
2076:
2074:
2071:
2069:
2066:
2064:
2061:
2059:
2056:
2054:
2051:
2049:
2046:
2044:
2041:
2039:
2035:
2031:
2029:
2025:
2021:
2019:
2016:
2015:
2013:
2007:
2001:
1998:
1996:
1993:
1991:
1988:
1986:
1983:
1981:
1978:
1976:
1973:
1969:
1968:in Lie theory
1966:
1965:
1964:
1961:
1959:
1956:
1952:
1949:
1948:
1947:
1944:
1942:
1939:
1938:
1936:
1934:
1930:
1924:
1921:
1919:
1916:
1914:
1911:
1909:
1906:
1904:
1901:
1899:
1896:
1894:
1891:
1889:
1886:
1884:
1881:
1880:
1878:
1875:
1871:Main results
1869:
1863:
1860:
1858:
1855:
1853:
1852:Tangent space
1850:
1848:
1845:
1843:
1840:
1838:
1835:
1833:
1830:
1828:
1825:
1821:
1818:
1816:
1813:
1812:
1811:
1808:
1804:
1801:
1800:
1799:
1796:
1795:
1793:
1789:
1784:
1780:
1773:
1768:
1766:
1761:
1759:
1754:
1753:
1750:
1743:
1739:
1735:
1731:
1727:
1723:
1719:
1715:
1712:
1708:
1704:
1700:
1696:
1692:
1688:
1684:
1681:
1675:
1671:
1666:
1662:
1656:
1652:
1648:
1647:Kirby, Robion
1644:
1643:
1639:
1630:
1626:
1622:
1618:
1617:
1612:
1606:
1604:
1600:
1593:
1589:
1588:Tame manifold
1586:
1584:
1581:
1580:
1576:
1574:
1561:
1556:
1527:
1512:
1511:dogbone space
1507:
1506:in a 4-ball.
1505:
1500:
1487:
1482:
1479:
1476:
1472:
1463:
1459:
1436:
1433:
1430:
1426:
1405:
1400:
1397:
1394:
1390:
1381:
1363:
1359:
1336:
1332:
1309:
1306:
1303:
1299:
1286:
1284:
1271:
1266:
1237:
1222:
1218:
1214:
1201:
1196:
1164:
1160:
1156:
1152:
1146:
1135:
1126:
1110:
1106:
1100:
1096:
1088:is the space
1087:
1082:
1080:
1064:
1059:
1044:
1028:
1023:
1013:
1005:
1002:
994:
990:
986:
982:
966:
963:
955:
951:
947:
944:
935:
922:
919:
916:
913:
910:
907:
904:
901:
898:
895:
873:
869:
848:
839:
835:
832:
824:
820:
804:
799:
795:
772:
768:
745:
741:
718:
714:
704:
691:
686:
682:
659:
655:
646:
642:
624:
620:
599:
588:
563:
558:
529:
525:
504:
499:
495:
486:
468:
464:
454:
452:
434:
430:
421:
403:
399:
376:
372:
349:
345:
321:
316:
312:
291:
286:
282:
259:
255:
246:
241:
237:
221:
217:
208:
204:
200:
197:
193:
189:
171:
167:
159:
155:
139:
134:
130:
117:
115:
113:
109:
105:
101:
97:
92:
90:
86:
82:
78:
63:
58:
43:
39:
35:
31:
27:
18:
2536:Moving frame
2531:Morse theory
2521:Gauge theory
2313:Tensor field
2242:Closed/Exact
2221:Vector field
2189:Distribution
2130:Hypercomplex
2125:Quaternionic
1862:Vector field
1820:Smooth atlas
1725:
1721:
1694:
1690:
1669:
1650:
1620:
1614:
1611:Gabai, David
1508:
1501:
1290:
1220:
1219:showed that
1215:
1124:
1085:
1083:
1042:
993:Morton Brown
988:
936:
822:
818:
787:lies inside
705:
644:
586:
455:
336:
121:
118:Construction
103:
93:
42:homeomorphic
38:contractible
29:
23:
2635:3-manifolds
2481:Levi-Civita
2471:Generalized
2443:Connections
2393:Lie algebra
2325:Volume form
2226:Vector flow
2199:Pushforward
2194:Lie bracket
2093:Lie algebra
2058:G-structure
1847:Pushforward
1827:Submanifold
1217:David Gabai
995:shows that
643:around the
158:solid torus
32:is an open
26:mathematics
2629:Categories
2604:Stratifold
2562:Diffeology
2358:Associated
2159:Symplectic
2144:Riemannian
2073:Hyperbolic
2000:Submersion
1908:HopfâRinow
1842:Submersion
1837:Smooth map
1594:References
1378:should be
706:Now embed
612:The torus
456:Note that
40:, but not
34:3-manifold
2650:Manifolds
2486:Principal
2461:Ehresmann
2418:Subbundle
2408:Principal
2383:Fibration
2363:Cotangent
2235:Covectors
2088:Lie group
2068:Hermitian
2011:manifolds
1980:Immersion
1975:Foliation
1913:Noether's
1898:Frobenius
1893:De Rham's
1888:Darboux's
1779:Manifolds
1469:∖
1165:×
1041:However,
1014:≅
1006:×
961:∖
920:…
844:∞
639:has zero
597:∞
570:∞
564:∪
100:open ball
2582:Orbifold
2577:K-theory
2567:Diffiety
2291:Pullback
2105:Oriented
2083:Kenmotsu
2063:Hadamard
2009:Types of
1958:Geodesic
1783:Glossary
1649:(1989).
1577:See also
825:, to be
391:so that
188:doughnut
96:manifold
36:that is
2526:History
2509:Related
2423:Tangent
2401:)
2381:)
2348:Adjoint
2340:Bundles
2318:density
2216:Torsion
2182:Vectors
2174:Tensors
2157:)
2142:)
2138:,
2136:Pseudoâ
2115:Poisson
2048:Finsler
2043:Fibered
2038:Contact
2036:)
2028:Complex
2026:)
1995:Section
1730:Bibcode
1699:Bibcode
733:inside
364:inside
205:.) The
79: (
2491:Vector
2476:Koszul
2456:Cartan
2451:Affine
2433:Vector
2428:Tensor
2413:Spinor
2403:Normal
2399:Stable
2353:Affine
2257:bundle
2209:bundle
2155:Almost
2078:KĂ€hler
2034:Almost
2024:Almost
2018:Closed
1918:Sard's
1874:(list)
1676:
1657:
1123:(with
983:, and
821:, the
418:and a
207:closed
196:circle
28:, the
2599:Sheaf
2373:Fiber
2149:Rizza
2120:Prime
1951:Local
1941:Curve
1803:Atlas
199:times
192:torus
2466:Form
2368:Dual
2301:flow
2164:Tame
2140:Subâ
2053:Flat
1933:Maps
1674:ISBN
1655:ISBN
1509:The
888:for
203:disk
152:the
81:1935
2388:Jet
1738:doi
1707:doi
1625:doi
1324:in
544:as
483:is
104:all
44:to
24:In
2631::
2379:Co
1736:,
1724:,
1705:,
1693:,
1619:.
1602:^
1081:.
453:.
201:a
2397:(
2377:(
2153:(
2134:(
2032:(
2022:(
1785:)
1781:(
1771:e
1764:t
1757:v
1740::
1732::
1726:6
1709::
1701::
1695:5
1663:.
1631:.
1627::
1621:4
1562:.
1557:4
1552:R
1528:1
1523:R
1488:.
1483:1
1480:+
1477:i
1473:T
1464:i
1460:T
1437:1
1434:+
1431:i
1427:T
1406:,
1401:1
1398:+
1395:i
1391:T
1364:i
1360:T
1337:i
1333:T
1310:1
1307:+
1304:i
1300:T
1272:.
1267:3
1262:R
1238:3
1233:R
1221:X
1202:.
1197:4
1192:R
1169:R
1161:)
1157:W
1153:/
1147:3
1142:R
1136:(
1125:W
1111:W
1107:/
1101:3
1097:S
1086:X
1065:.
1060:3
1055:R
1043:X
1029:.
1024:4
1019:R
1010:R
1003:X
989:X
967:,
964:W
956:3
952:S
948:=
945:X
923:.
917:,
914:3
911:,
908:2
905:,
902:1
899:=
896:k
874:k
870:T
849:,
840:T
836:=
833:W
819:W
805:,
800:1
796:T
773:2
769:T
746:2
742:T
719:3
715:T
692:.
687:2
683:T
660:1
656:T
645:z
625:2
621:T
600:.
587:z
573:}
567:{
559:3
554:R
530:3
526:S
505:.
500:1
496:T
469:2
465:T
435:1
431:T
404:2
400:T
377:1
373:T
350:2
346:T
322:.
317:1
313:T
292:.
287:2
283:T
260:1
256:T
222:3
218:S
172:1
168:T
140:,
135:3
131:S
64:.
59:3
54:R
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
