235:
122:
457:
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302:
642:
424:
354:
324:
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56:
648:
if, when embedded in a high dimensional euclidean space, its normal bundle is trivial. In particular, every parallelizable manifold is a π-manifold.
1733:
645:
924:
1728:
552:, which allows one to construct a parallelism for each. Proving that other spheres are not parallelizable is more difficult, and requires
1015:
482:(every translation is a diffeomorphism and therefore these translations induce linear isomorphisms between tangent spaces of points in
1039:
1234:
170:
1104:
75:
1330:
1383:
911:
1667:
463:
with opposite edges glued together, to get an idea of the two tangent directions at each point. More generally, every
848:
737:
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1024:
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17:
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540:, in independent work. The parallelizable spheres correspond precisely to elements of unit norm in the
613:, and also for an abstract (that is, non-embedded) manifold with a given stable trivialisation of the
1199:
1064:
1584:
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1141:
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1241:
1136:
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1271:
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1602:
1540:
1388:
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939:
525:
433:
8:
1740:
1422:
1300:
1285:
1214:
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810:
368:
1713:
284:
1786:
1682:
1637:
1534:
1405:
1209:
897:
791:
765:
732:, Annals of Mathematics Studies, vol. 76, Princeton University Press, p. 15,
673:
627:
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505:
409:
402:
to be the unit tangent vector field, say pointing in the anti-clockwise direction. The
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127:
41:
28:
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844:
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609:) is most usually applied to an embedded manifold with a given trivialisation of the
427:
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is the circle, which is parallelizable as has already been explained. The
1520:
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35:
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27:"Parallelizable" redirects here. For the computer science usage, see
1723:
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770:
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549:
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392:
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can be moved around by the action of the translation group of
470:
is parallelizable, since a basis for the tangent space at the
517:
426:
is also parallelizable, as can be seen by expressing it as a
403:
306:
A particular choice of such a basis of vector fields on
630:
436:
412:
371:
342:
312:
287:
251:
173:
150:
130:
78:
44:
843:(First Dover 1980 ed.), The Macmillan Company,
879:
Differentiable manifolds which are homotopy spheres
812:
Differentiable manifolds which are homotopy spheres
836:
636:
489:A classical problem was to determine which of the
451:
418:
383:
348:
318:
296:
257:
229:
156:
136:
116:
50:
1768:
752:Benedetti, Riccardo; Lisca, Paolo (2019-07-23).
713:Bishop, Richard L.; Goldberg, Samuel I. (1968),
856:
831:
751:
727:
712:
496:are parallelizable. The zero-dimensional case
905:
516:is parallelizable, since it is the Lie group
230:{\displaystyle \{V_{1}(p),\ldots ,V_{n}(p)\}}
728:Milnor, John W.; Stasheff, James D. (1974),
224:
174:
111:
79:
912:
898:
520:. The only other parallelizable sphere is
124:on the manifold, such that at every point
769:
919:
459:and construct a torus from a square of
117:{\displaystyle \{V_{1},\ldots ,V_{n}\}}
14:
1769:
872:
808:
544:of the real numbers, complex numbers,
500:is trivially parallelizable. The case
893:
754:"Framing 3-manifolds with bare hands"
620:A related notion is the concept of a
24:
717:, New York: Macmillan, p. 160
25:
1803:
512:is not parallelizable. However
952:Differentiable/Smooth manifold
835:; Goldberg, Samuel I. (1968),
802:
745:
721:
706:
559:The product of parallelizable
524:; this was proved in 1958, by
430:of circles. For example, take
221:
215:
193:
187:
13:
1:
825:
868:, Princeton University Press
839:Tensor Analysis on Manifolds
715:Tensor Analysis on Manifolds
38:, a differentiable manifold
7:
1658:Classification of manifolds
758:L'Enseignement Mathématique
652:
359:
10:
1808:
580:
574:three-dimensional manifold
26:
1734:over commutative algebras
1691:
1650:
1583:
1480:
1376:
1323:
1314:
1150:
1073:
1012:
932:
273:, so that the associated
1450:Riemann curvature tensor
809:Milnor, John W. (1958),
699:
689:Connection (mathematics)
679:Orthonormal frame bundle
542:normed division algebras
281:has a global section on
664:Differentiable manifold
1242:Manifold with boundary
957:Differential structure
866:Characteristic Classes
730:Characteristic Classes
638:
453:
420:
385:
350:
320:
298:
259:
231:
158:
138:
118:
52:
1777:Differential topology
639:
454:
421:
386:
351:
321:
299:
265:. Equivalently, the
260:
232:
159:
139:
119:
53:
1389:Covariant derivative
940:Topological manifold
885:, mimeographed notes
780:10.4171/LEM/64-3/4-9
628:
624:. A smooth manifold
526:Friedrich Hirzebruch
452:{\displaystyle n=2,}
434:
410:
369:
340:
334:absolute parallelism
310:
285:
249:
171:
148:
128:
76:
42:
1423:Exterior derivative
1025:Atiyah–Singer index
974:Riemannian manifold
586:Any parallelizable
384:{\displaystyle n=1}
1729:Secondary calculus
1683:Singularity theory
1638:Parallel transport
1406:De Rham cohomology
1045:Generalized Stokes
862:Stasheff, James D.
833:Bishop, Richard L.
674:Kervaire invariant
634:
576:is parallelizable.
563:is parallelizable.
554:algebraic topology
506:hairy ball theorem
449:
416:
381:
346:
316:
297:{\displaystyle M.}
294:
255:
227:
154:
134:
114:
48:
29:parallel algorithm
1764:
1763:
1646:
1645:
1411:Differential form
1065:Whitney embedding
999:Differential form
637:{\displaystyle M}
428:cartesian product
419:{\displaystyle n}
349:{\displaystyle M}
319:{\displaystyle M}
258:{\displaystyle p}
157:{\displaystyle M}
137:{\displaystyle p}
51:{\displaystyle M}
16:(Redirected from
1799:
1756:Stratified space
1714:Fréchet manifold
1428:Interior product
1321:
1320:
1018:
914:
907:
900:
891:
890:
886:
884:
869:
853:
842:
819:
818:
817:
806:
800:
799:
773:
749:
743:
742:
725:
719:
718:
710:
684:Principal bundle
659:Chart (topology)
643:
641:
640:
635:
472:identity element
458:
456:
455:
450:
425:
423:
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390:
388:
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365:An example with
355:
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301:
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275:principal bundle
264:
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214:
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163:
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91:
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57:
55:
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21:
1807:
1806:
1802:
1801:
1800:
1798:
1797:
1796:
1767:
1766:
1765:
1760:
1699:Banach manifold
1692:Generalizations
1687:
1642:
1579:
1476:
1438:Ricci curvature
1394:Cotangent space
1372:
1310:
1152:
1146:
1105:Exponential map
1069:
1014:
1008:
928:
918:
882:
874:Milnor, John W.
858:Milnor, John W.
851:
828:
823:
822:
815:
807:
803:
750:
746:
740:
726:
722:
711:
707:
702:
655:
629:
626:
625:
606:rigged manifold
600:framed manifold
583:
530:Michel Kervaire
435:
432:
431:
411:
408:
407:
401:
370:
367:
366:
362:
341:
338:
337:
329:parallelization
311:
308:
307:
286:
283:
282:
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246:
209:
205:
181:
177:
172:
169:
168:
166:tangent vectors
149:
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129:
126:
125:
105:
101:
86:
82:
77:
74:
73:
66:if there exist
43:
40:
39:
32:
23:
22:
15:
12:
11:
5:
1805:
1795:
1794:
1792:Vector bundles
1789:
1784:
1779:
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1753:
1748:
1743:
1738:
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1600:
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1401:
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1386:
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1371:
1370:
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1360:
1355:
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1353:
1343:
1338:
1333:
1327:
1325:
1318:
1312:
1311:
1309:
1308:
1303:
1293:
1288:
1274:
1269:
1264:
1259:
1254:
1252:Parallelizable
1249:
1244:
1239:
1238:
1237:
1227:
1222:
1217:
1212:
1207:
1202:
1197:
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1182:
1172:
1162:
1156:
1154:
1148:
1147:
1145:
1144:
1139:
1134:
1132:Lie derivative
1129:
1127:Integral curve
1124:
1119:
1114:
1113:
1112:
1102:
1097:
1096:
1095:
1088:Diffeomorphism
1085:
1079:
1077:
1071:
1070:
1068:
1067:
1062:
1057:
1052:
1047:
1042:
1037:
1032:
1027:
1021:
1019:
1010:
1009:
1007:
1006:
1001:
996:
991:
986:
981:
976:
971:
966:
965:
964:
959:
949:
948:
947:
936:
934:
933:Basic concepts
930:
929:
917:
916:
909:
902:
894:
888:
887:
870:
854:
849:
827:
824:
821:
820:
801:
764:(3): 395–413.
744:
738:
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704:
703:
701:
698:
697:
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686:
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661:
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618:
615:tangent bundle
603:(occasionally
595:
582:
579:
578:
577:
564:
557:
487:
448:
445:
442:
439:
415:
399:
395:: we can take
380:
377:
374:
361:
358:
345:
315:
293:
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271:trivial bundle
267:tangent bundle
254:
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184:
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153:
133:
113:
108:
104:
100:
97:
94:
89:
85:
81:
64:parallelizable
47:
18:Parallelizable
9:
6:
4:
3:
2:
1804:
1793:
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1785:
1783:
1782:Fiber bundles
1780:
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1775:
1774:
1772:
1757:
1754:
1752:
1751:Supermanifold
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1506:
1503:
1501:
1498:
1496:
1493:
1491:
1488:
1487:
1485:
1483:
1479:
1473:
1472:Wedge product
1470:
1468:
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1458:
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1456:
1453:
1451:
1448:
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1441:
1440:
1439:
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1424:
1421:
1417:
1416:Vector-valued
1414:
1413:
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1409:
1407:
1404:
1400:
1397:
1396:
1395:
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1387:
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1379:
1375:
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1359:
1356:
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1347:
1346:Tangent space
1344:
1342:
1339:
1337:
1334:
1332:
1329:
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1326:
1322:
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1158:
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1155:
1149:
1143:
1140:
1138:
1135:
1133:
1130:
1128:
1125:
1123:
1120:
1118:
1115:
1111:
1110:in Lie theory
1108:
1107:
1106:
1103:
1101:
1098:
1094:
1091:
1090:
1089:
1086:
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1080:
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1048:
1046:
1043:
1041:
1038:
1036:
1033:
1031:
1028:
1026:
1023:
1022:
1020:
1017:
1013:Main results
1011:
1005:
1002:
1000:
997:
995:
994:Tangent space
992:
990:
987:
985:
982:
980:
977:
975:
972:
970:
967:
963:
960:
958:
955:
954:
953:
950:
946:
943:
942:
941:
938:
937:
935:
931:
926:
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915:
910:
908:
903:
901:
896:
895:
892:
881:
880:
875:
871:
867:
863:
859:
855:
852:
850:0-486-64039-6
846:
841:
840:
834:
830:
829:
814:
813:
805:
797:
793:
789:
785:
781:
777:
772:
767:
763:
759:
755:
748:
741:
739:0-691-08122-0
735:
731:
724:
716:
709:
705:
695:
692:
690:
687:
685:
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677:
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672:
670:
667:
665:
662:
660:
657:
656:
647:
631:
623:
619:
616:
612:
611:normal bundle
608:
607:
602:
601:
596:
593:
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584:
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569:
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562:
558:
555:
551:
547:
543:
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531:
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523:
519:
515:
511:
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499:
495:
492:
488:
485:
481:
477:
473:
469:
466:
462:
446:
443:
440:
437:
429:
413:
406:of dimension
405:
398:
394:
378:
375:
372:
364:
363:
357:
343:
335:
331:
330:
313:
304:
291:
288:
280:
279:linear frames
276:
272:
268:
252:
244:
243:tangent space
240:
218:
210:
206:
202:
199:
196:
190:
182:
178:
167:
151:
131:
106:
102:
98:
95:
92:
87:
83:
72:
71:vector fields
69:
65:
61:
58:of dimension
45:
37:
30:
19:
1678:Moving frame
1673:Morse theory
1663:Gauge theory
1455:Tensor field
1384:Closed/Exact
1363:Vector field
1331:Distribution
1272:Hypercomplex
1267:Quaternionic
1251:
1004:Vector field
962:Smooth atlas
878:
865:
838:
811:
804:
761:
757:
747:
729:
723:
714:
708:
669:Frame bundle
644:is called a
621:
604:
598:
521:
513:
509:
501:
497:
493:
483:
479:
475:
467:
396:
333:
327:
326:is called a
305:
63:
59:
33:
1623:Levi-Civita
1613:Generalized
1585:Connections
1535:Lie algebra
1467:Volume form
1368:Vector flow
1341:Pushforward
1336:Lie bracket
1235:Lie algebra
1200:G-structure
989:Pushforward
969:Submanifold
694:G-structure
546:quaternions
538:John Milnor
508:shows that
461:graph paper
36:mathematics
1771:Categories
1746:Stratifold
1704:Diffeology
1500:Associated
1301:Symplectic
1286:Riemannian
1215:Hyperbolic
1142:Submersion
1050:Hopf–Rinow
984:Submersion
979:Smooth map
826:References
771:1806.04991
646:π-manifold
622:π-manifold
592:orientable
568:orientable
534:Raoul Bott
237:provide a
62:is called
1787:Manifolds
1628:Principal
1603:Ehresmann
1560:Subbundle
1550:Principal
1525:Fibration
1505:Cotangent
1377:Covectors
1230:Lie group
1210:Hermitian
1153:manifolds
1122:Immersion
1117:Foliation
1055:Noether's
1040:Frobenius
1035:De Rham's
1030:Darboux's
921:Manifolds
796:119711633
788:0013-8584
597:The term
561:manifolds
550:octonions
532:, and by
465:Lie group
200:…
96:…
1724:Orbifold
1719:K-theory
1709:Diffiety
1433:Pullback
1247:Oriented
1225:Kenmotsu
1205:Hadamard
1151:Types of
1100:Geodesic
925:Glossary
876:(1958),
864:(1974),
653:See also
588:manifold
360:Examples
1668:History
1651:Related
1565:Tangent
1543:)
1523:)
1490:Adjoint
1482:Bundles
1460:density
1358:Torsion
1324:Vectors
1316:Tensors
1299:)
1284:)
1280:,
1278:Pseudo−
1257:Poisson
1190:Finsler
1185:Fibered
1180:Contact
1178:)
1170:Complex
1168:)
1137:Section
581:Remarks
491:spheres
391:is the
332:(or an
241:of the
1633:Vector
1618:Koszul
1598:Cartan
1593:Affine
1575:Vector
1570:Tensor
1555:Spinor
1545:Normal
1541:Stable
1495:Affine
1399:bundle
1351:bundle
1297:Almost
1220:Kähler
1176:Almost
1166:Almost
1160:Closed
1060:Sard's
1016:(list)
847:
794:
786:
736:
571:closed
566:Every
548:, and
393:circle
68:smooth
1741:Sheaf
1515:Fiber
1291:Rizza
1262:Prime
1093:Local
1083:Curve
945:Atlas
883:(PDF)
816:(PDF)
792:S2CID
766:arXiv
700:Notes
518:SU(2)
404:torus
336:) of
269:is a
239:basis
1608:Form
1510:Dual
1443:flow
1306:Tame
1282:Sub−
1195:Flat
1075:Maps
845:ISBN
784:ISSN
734:ISBN
536:and
164:the
1530:Jet
776:doi
590:is
478:on
277:of
245:at
144:of
34:In
1773::
1521:Co
860:;
790:.
782:.
774:.
762:64
760:.
756:.
528:,
486:).
356:.
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913:e
906:t
899:v
798:.
778::
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632:M
617:.
594:.
556:.
522:S
514:S
510:S
502:S
498:S
494:S
484:G
480:G
476:G
468:G
447:,
444:2
441:=
438:n
414:n
400:1
397:V
379:1
376:=
373:n
344:M
314:M
292:.
289:M
253:p
225:}
222:)
219:p
216:(
211:n
207:V
203:,
197:,
194:)
191:p
188:(
183:1
179:V
175:{
152:M
132:p
112:}
107:n
103:V
99:,
93:,
88:1
84:V
80:{
60:n
46:M
31:.
20:)
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