1270:
5765:
5340:
40:
2500:
2514:
7229:
Depending on the category of spaces involved, the functions may be assumed to have properties other than continuity. For instance, in the category of differentiable manifolds, the functions are assumed to be smooth. In the category of algebraic varieties, they are regular
4531:
1835:
4637:
6867:
6803:
4407:
1440:
7128:
2436:, but the Möbius strip has an overall "twist". This twist is visible only globally; locally the Möbius strip and the cylinder are identical (making a single vertical cut in either gives the same space).
1320:
5330:
2126:
1598:
1377:
1171:
2773:
5682:
5014:
5275:
652:
in 1933, but his definitions are limited to a very special case. The main difference from the present day conception of a fiber space, however, was that for
Seifert what is now called the
4699:
4923:
4268:
6004:
5759:
5454:
4308:
6251:
4402:
4355:
793:
3351:
1756:
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1525:
5966:
5927:
5847:
5808:
5714:
5633:
5601:
5562:
5396:
5200:
5161:
5888:
4071:
3482:
4847:
4536:
3278:
1766:
4215:
4030:
3894:
2337:
2261:
1210:
1091:
222:
6564:
6447:
2008:
1976:
1656:
878:
85:
3691:
456:
6346:
2430:
1240:
268:
177:
6393:
6303:
6173:
6072:
6024:
3974:
3850:
3772:
3082:
2380:
6476:
6118:
5509:
5480:
4798:
4109:
2287:
1477:
1062:
3598:
3563:
3536:
3509:
3116:
2885:
2687:
1707:
314:
3818:
3642:
3371:
2028:
1903:
1260:
993:
822:
476:
3238:
2976:
2947:
3143:
2908:
2071:
1623:
582:
291:
6496:
6271:
3738:
3718:
3430:
3406:
3298:
3203:
3179:
3046:
3004:
2854:
2822:
2727:
2707:
2664:
2644:
2620:
2404:
2357:
2226:
2203:
2179:
2151:
2091:
2048:
1923:
1680:
1545:
1111:
1029:
965:
937:
909:
842:
614:
555:
427:
394:
370:
346:
242:
151:
2406:(the trivializing neighborhood) to a slice of a cylinder: curved, but not twisted. This pair locally trivializes the strip. The corresponding trivial bundle
8945:
6993:
6969:
6927:
8136:
8940:
3906:. Since bundles do not in general have globally defined sections, one of the purposes of the theory is to account for their existence. The
8227:
6273:
follows by the assumptions already given in this case.) More generally, the assumption of compactness can be relaxed if the submersion
726:
Fiber bundles became their own object of study in the period 1935–1940. The first general definition appeared in the works of
Whitney.
1386:
8251:
5229:
8446:
7824:
5051:). The importance of this is that the transition functions determine the fiber bundle (if one assumes the Čech cocycle condition).
7404:
2448:, which can be viewed as a "twisted" circle bundle over another circle. The corresponding non-twisted (trivial) bundle is the 2-
104:
7973:
6519:
applies to many more categories in mathematics, at the expense of appropriately modifying the local triviality condition; cf.
2339:
in the picture is a (somewhat twisted) slice of the strip four squares wide and one long (i.e. all the points that project to
1280:
8316:
7371:
7349:
7316:
7285:
7264:
6124:. However, this necessary condition is not quite sufficient, and there are a variety of sufficient conditions in common use.
5280:
8542:
1550:
4526:{\displaystyle \varphi _{i}\varphi _{j}^{-1}:\left(U_{i}\cap U_{j}\right)\times F\to \left(U_{i}\cap U_{j}\right)\times F}
8595:
8123:
1325:
1119:
8879:
8008:
7687:
6570:
properties in common with fiber bundles. Specifically, under mild technical assumptions a fiber bundle always has the
4127:
of symmetries that describe the matching conditions between overlapping local trivialization charts. Specifically, let
3301:
2732:
5638:
7467:
6770:
4929:
8644:
7889:
8627:
8236:
7363:
4642:
4853:
4220:
1269:
5971:
5726:
5409:
5087:
4273:
1857:, indicates which space is the fiber, total space and base space, as well as the map from total to base space.
180:
8839:
8246:
7740:
7672:
7391:
6182:
4360:
4313:
8824:
8547:
8321:
7765:
8869:
8003:
7386:
6524:
6520:
5071:
2623:
2206:
748:
729:
Whitney came to the general definition of a fiber bundle from his study of a more particular notion of a
17:
3910:
to the existence of a section can often be measured by a cohomology class, which leads to the theory of
3307:
1714:
8874:
8844:
8552:
8508:
8489:
8256:
8200:
7814:
7634:
4136:
3940:
Often one would like to define sections only locally (especially when global sections do not exist). A
2599:
2455:
1830:{\displaystyle {\begin{matrix}{}\\F\longrightarrow E\ {\xrightarrow {\,\ \pi \ }}\ B\\{}\end{matrix}}}
1485:
8411:
8276:
7486:
7064:
In his early works, Whitney referred to the sphere bundles as the "sphere-spaces". See, for example:
6571:
5932:
5893:
5813:
5774:
5687:
5606:
5567:
5528:
5369:
5166:
5127:
3210:
584:
Fiber bundles can be specialized in a number of ways, the most common of which is requiring that the
7968:
5860:
4043:
3447:
521:
Mappings between total spaces of fiber bundles that "commute" with the projection maps are known as
8796:
8661:
8353:
8195:
8070:
7988:
7942:
7649:
6396:
6140:
6039:
4806:
3243:
405:
323:
318:
4176:
3991:
3855:
2303:
2231:
1176:
1070:
192:
8493:
8463:
8387:
8377:
8333:
8163:
8116:
8040:
7727:
7644:
7614:
7381:
6537:
6413:
6407:
6035:
3437:
2504:
1981:
1949:
1629:
1065:
851:
58:
3647:
432:
8988:
8834:
8453:
8348:
8261:
8168:
7998:
7854:
7809:
6685:
6312:
3783:
2409:
1926:
1869:
1219:
558:
247:
156:
6366:
6276:
6146:
6045:
6009:
5042:
3947:
3823:
3743:
3055:
2365:
51:. This hairbrush is like a fiber bundle in which the base space is a cylinder and the fibers (
8483:
8478:
8080:
8035:
7515:
7460:
7432:
7017:
6585:
is continuously dependent on the input." This property is formally captured in the notion of
6452:
6085:
5485:
3566:
881:
511:
507:
130:
7280:. Princeton Mathematical Series. Vol. 14. Princeton, N.J.: Princeton University Press.
6082:
gives rise to a differentiable fiber bundle. For one thing, the map must be surjective, and
5459:
4773:
4088:
2266:
1456:
1041:
8814:
8752:
8600:
8304:
8294:
8266:
8241:
8151:
8055:
7983:
7869:
7735:
7697:
7629:
7427:
7184:
7084:
6876:
6812:
6610:
6516:
6499:
3911:
3571:
3541:
3514:
3487:
3094:
2863:
2672:
2667:
2563:
1854:
1689:
296:
31:
3803:
3618:
3356:
2013:
1879:
1245:
978:
798:
461:
8:
8952:
8634:
8512:
8497:
8426:
8185:
7932:
7755:
7745:
7594:
7579:
7535:
6582:
5333:
4124:
4112:
4074:
3215:
3007:
2955:
2926:
2912:
2829:
2579:
2562:(to qualify as a vector bundle the structure group of the bundle — see below — must be a
2537:
2433:
2298:
1263:
183:
114:
8925:
7188:
7088:
6880:
6816:
3125:
2890:
2053:
1605:
1262:
agrees with the projection onto the first factor. That is, the following diagram should
564:
273:
8894:
8849:
8746:
8617:
8421:
8109:
8065:
7922:
7775:
7589:
7525:
7436:
7305:
7207:
7107:
7072:
7034:
6899:
6862:
6835:
6798:
6620:
6481:
6256:
4082:
3922:
3915:
3907:
3723:
3703:
3415:
3391:
3283:
3188:
3164:
3146:
3031:
2989:
2839:
2807:
2712:
2692:
2649:
2629:
2626:. The bundle is often specified along with the group by referring to it as a principal
2605:
2389:
2342:
2211:
2188:
2164:
2136:
2106:
2076:
2033:
1908:
1665:
1530:
1096:
1014:
950:
922:
894:
845:
827:
599:
540:
526:
412:
379:
355:
331:
227:
136:
8431:
5048:
4632:{\displaystyle \varphi _{i}\varphi _{j}^{-1}(x,\,\xi )=\left(x,\,t_{ij}(x)\xi \right)}
8829:
8809:
8804:
8711:
8622:
8436:
8416:
8271:
8210:
8060:
7829:
7804:
7619:
7530:
7510:
7367:
7345:
7325:
7312:
7291:
7281:
7260:
7212:
7169:
7112:
7012:
6988:
6964:
6946:
6904:
6840:
6776:
6766:
6660:
4132:
3605:
3182:
3049:
2776:
2596:
1683:
1380:
1213:
720:
716:
589:
478:
is just the projection from the product space to the first factor. This is called a
186:
7148:
8967:
8761:
8716:
8639:
8610:
8468:
8401:
8396:
8391:
8381:
8173:
8156:
8075:
7750:
7717:
7702:
7584:
7453:
7202:
7192:
7143:
7102:
7092:
7026:
6894:
6884:
6830:
6820:
6742:
6723:
6714:
6675:
6655:
6625:
6121:
5716:
5116:
It is useful to have notions of a mapping between two fiber bundles. Suppose that
5055:
4169:
3609:
2587:
2571:
696:
688:
534:
515:
7435:, Fibre bundles, jet manifolds and Lagrangian theory. Lectures for theoreticians,
2125:
533:
with respect to such mappings. A bundle map from the base space itself (with the
8910:
8819:
8649:
8605:
8371:
8045:
7993:
7937:
7917:
7819:
7707:
7574:
7545:
7413:
7337:
7273:
7252:
7165:
7124:
7068:
6922:
6858:
6794:
6705:
6665:
6567:
4148:
3206:
1873:
1032:
712:
708:
700:
669:
649:
530:
7239:
Or is, at least, invertible in the appropriate category; e.g., a diffeomorphism.
2154:
483:
8776:
8701:
8671:
8569:
8562:
8502:
8473:
8343:
8338:
8299:
8085:
8050:
7947:
7780:
7770:
7760:
7682:
7654:
7639:
7624:
7540:
6868:
Proceedings of the
National Academy of Sciences of the United States of America
6804:
Proceedings of the
National Academy of Sciences of the United States of America
6746:
6670:
6650:
6605:
6586:
3930:
3694:
3441:
3011:
2567:
2541:
585:
495:
491:
8030:
7328:(1951). "Les connexions infinitésimales dans un espace fibré différentiable".
3006:, given the Euler class of a bundle, one can calculate its cohomology using a
328:
of the bundle, is regarded as part of the structure of the bundle. The space
8982:
8962:
8786:
8781:
8766:
8756:
8706:
8683:
8557:
8517:
8458:
8406:
8205:
8022:
7927:
7839:
7712:
6680:
6600:
6136:
4156:
3601:
3089:
3085:
3018:
2979:
2825:
2788:
2554:
1114:
730:
684:
503:
122:
7295:
8889:
8884:
8666:
8574:
8215:
8090:
7894:
7879:
7692:
7677:
7216:
7116:
6908:
6844:
6780:
6635:
6615:
3613:
3433:
3385:
2575:
2559:
2532:
2445:
2290:
2182:
487:
7197:
7097:
6889:
6825:
660:
was not part of the structure, but derived from it as a quotient space of
8732:
8721:
8678:
8579:
8180:
7978:
7952:
7874:
7563:
7502:
6765:. W. Threlfall, Joan S. Birman, Julian Eisner. New York: Academic Press.
6640:
5854:
5363:
4745:
3926:
2983:
2920:
2857:
2798:
2592:
2205:, so the Möbius strip is a bundle of the line segment over the circle. A
92:
87:
would take a point on any bristle and map it to its root on the cylinder.
6760:
1113:(which will be called a trivializing neighborhood) such that there is a
8957:
8915:
8741:
8654:
8286:
8190:
8101:
7859:
7038:
6728:
6709:
6306:
5111:
3440:), then the quotient map is a fiber bundle. One example of this is the
3150:
2949:
2109:
1930:
884:
704:
522:
6175:
gives rise to a fiber bundle in the sense that there is a fiber space
8771:
8736:
8441:
8328:
7834:
7785:
7418:
6630:
6531:
4760:
3409:
2986:, which characterizes the topology of the bundle completely. For any
734:
692:
44:
7030:
6578:, 11.7) for details). This is the defining property of a fibration.
5764:
5339:
2574:
of a smooth manifold. From any vector bundle, one can construct the
1797:
8935:
8930:
8920:
8311:
8132:
7864:
7849:
6645:
5355:
3981:
3934:
1659:
1480:
625:
499:
96:
7440:
6042:
of one manifold to another. Not every (differentiable) submersion
7558:
7520:
7015:(1951). "Homologie singulière des espaces fibrés. Applications".
5090:). In this case, it is often a matter of convenience to identify
4741:
3896:
2499:
52:
39:
1435:{\displaystyle \left\{\left(U_{i},\,\varphi _{i}\right)\right\}}
8527:
7884:
7476:
6349:
3119:
2294:
2158:
6991:(1955). "Les prolongements d'un espace fibré différentiable".
733:, that is a fiber bundle whose fiber is a sphere of arbitrary
7332:. Paris: Georges Thone, Liège; Masson et Cie. pp. 29–55.
2449:
2161:
that runs lengthwise along the center of the strip as a base
244:
behaves just like a projection from corresponding regions of
27:
Continuous surjection satisfying a local triviality condition
4767:
is smooth and the transition functions are all smooth maps.
4118:
3740:
a closed subgroup that also happens to be a Lie group, then
7366:, vol. 93, Providence: American Mathematical Society,
3388:
will admit local cross-sections are not known, although if
2887:. When the vector bundle in question is the tangent bundle
7445:
5078:
itself (equivalently, one can specify that the action of
1662:, since projections of products are open maps. Therefore
6581:
A section of a fiber bundle is a "function whose output
3280:
is a fiber bundle, whose fiber is the topological space
2513:
2129:
The Möbius strip is a nontrivial bundle over the circle.
1315:{\displaystyle \operatorname {proj} _{1}:U\times F\to U}
7330:
Colloque de
Topologie (Espaces fibrés), Bruxelles, 1950
5325:{\displaystyle \pi _{F}\circ \varphi =f\circ \pi _{E}.}
2536:
is a fiber bundle such that the bundle projection is a
1011:). We shall assume in what follows that the base space
2775:
as a structure group may be constructed, known as the
1771:
7410:
6540:
6484:
6455:
6416:
6369:
6315:
6279:
6259:
6185:
6149:
6088:
6048:
6012:
5974:
5935:
5896:
5863:
5816:
5777:
5729:
5690:
5641:
5609:
5570:
5531:
5488:
5462:
5412:
5372:
5283:
5232:
5169:
5130:
4932:
4856:
4809:
4776:
4645:
4539:
4410:
4363:
4316:
4276:
4223:
4179:
4091:
4046:
3994:
3950:
3858:
3826:
3806:
3746:
3726:
3706:
3650:
3621:
3574:
3544:
3517:
3490:
3450:
3418:
3394:
3359:
3310:
3286:
3246:
3218:
3191:
3167:
3128:
3097:
3058:
3034:
2992:
2958:
2929:
2893:
2866:
2842:
2810:
2735:
2715:
2695:
2675:
2652:
2632:
2608:
2458:
2412:
2392:
2368:
2345:
2306:
2269:
2234:
2214:
2191:
2167:
2139:
2079:
2056:
2036:
2016:
1984:
1952:
1911:
1882:
1769:
1717:
1692:
1668:
1632:
1608:
1593:{\displaystyle \operatorname {proj} _{1}^{-1}(\{p\})}
1553:
1533:
1488:
1459:
1389:
1328:
1283:
1248:
1222:
1179:
1122:
1099:
1073:
1044:
1017:
981:
953:
925:
897:
854:
830:
801:
751:
602:
567:
543:
464:
435:
415:
382:
358:
334:
299:
276:
250:
230:
195:
159:
139:
61:
7428:
Making John
Robinson's Symbolic Sculpture `Eternity'
7129:"Topological properties of differentiable manifolds"
6925:(1939). "Sur la classification des espaces fibrés".
2952:
class in the total space of the bundle. In the case
2566:). Important examples of vector bundles include the
2133:
Perhaps the simplest example of a nontrivial bundle
482:. Examples of non-trivial fiber bundles include the
3118:has a natural structure of a fiber bundle over the
2666:is also the structure group of the bundle. Given a
656:(topological space) of a fiber (topological) space
588:between the local trivial patches lie in a certain
7304:
6967:(1947). "Sur les espaces fibrés différentiables".
6558:
6490:
6470:
6441:
6387:
6340:
6297:
6265:
6245:
6167:
6112:
6066:
6018:
5998:
5960:
5921:
5882:
5841:
5802:
5753:
5708:
5676:
5627:
5595:
5556:
5503:
5474:
5448:
5390:
5324:
5269:
5194:
5155:
5008:
4917:
4841:
4792:
4693:
4631:
4525:
4396:
4349:
4302:
4262:
4209:
4103:
4065:
4024:
3968:
3888:
3844:
3812:
3766:
3732:
3712:
3685:
3636:
3592:
3557:
3530:
3503:
3476:
3424:
3400:
3365:
3345:
3292:
3272:
3232:
3197:
3173:
3137:
3110:
3076:
3040:
2998:
2970:
2941:
2919:A sphere bundle is partially characterized by its
2902:
2879:
2848:
2816:
2767:
2721:
2701:
2681:
2658:
2638:
2614:
2484:
2424:
2398:
2374:
2351:
2331:
2281:
2255:
2220:
2197:
2173:
2145:
2085:
2065:
2042:
2022:
2002:
1970:
1917:
1897:
1829:
1750:
1701:
1674:
1650:
1617:
1592:
1539:
1519:
1471:
1434:
1372:{\displaystyle \varphi :\pi ^{-1}(U)\to U\times F}
1371:
1314:
1254:
1234:
1204:
1166:{\displaystyle \varphi :\pi ^{-1}(U)\to U\times F}
1165:
1105:
1085:
1056:
1023:
987:
959:
931:
903:
872:
836:
816:
787:
608:
576:
549:
470:
450:
421:
388:
364:
340:
308:
285:
262:
236:
216:
171:
145:
79:
4713:-atlases are equivalent if their union is also a
2768:{\displaystyle \rho (G)\subseteq {\text{Aut}}(V)}
8980:
5677:{\displaystyle \pi _{E}=\pi _{F}\circ \varphi .}
1925:are required to be smooth manifolds and all the
6029:
5019:The third condition applies on triple overlaps
5009:{\displaystyle t_{ik}(x)=t_{ij}(x)t_{jk}(x).\,}
4724:is a fiber bundle with an equivalence class of
2585:Another special class of fiber bundles, called
2547:
133:. Specifically, the similarity between a space
7379:
6743:"Topologie Dreidimensionaler Gefaserter Räume"
6710:"Topologie dreidimensionaler gefaserter Räume"
6179:diffeomorphic to each of the fibers such that
5270:{\displaystyle \varphi :E\to F,\quad f:M\to N}
2010:be the projection onto the first factor. Then
8117:
7461:
6949:(1947). "Sur la théorie des espaces fibrés".
6762:Seifert and Threlfall, A textbook of topology
5354:-spaces (such as a principal bundle), bundle
3353:) to form a fiber bundle is that the mapping
7311:, Reading, Mass: Addison-Wesley publishing,
6436:
6430:
4257:
4224:
3384:The most general conditions under which the
1584:
1578:
1511:
1505:
648:) appeared for the first time in a paper by
5226:consists of a pair of continuous functions
4694:{\displaystyle t_{ij}:U_{i}\cap U_{j}\to G}
4155:so that it may be thought of as a group of
4143:on the left. We lose nothing if we require
2582:, which is a principal bundle (see below).
8124:
8110:
7468:
7454:
7336:
4918:{\displaystyle t_{ij}(x)=t_{ji}(x)^{-1}\,}
4263:{\displaystyle \{(U_{k},\,\varphi _{k})\}}
3484:, which is a fiber bundle over the sphere
2982:and the Euler class is equal to the first
676:, but in 1940 Whitney changed the name to
7324:
7206:
7196:
7147:
7106:
7096:
6994:Comptes rendus de l'Académie des Sciences
6987:
6970:Comptes rendus de l'Académie des Sciences
6963:
6945:
6928:Comptes rendus de l'Académie des Sciences
6898:
6888:
6834:
6824:
6727:
6360:
5999:{\displaystyle f\equiv \mathrm {id} _{M}}
5873:
5754:{\displaystyle f\equiv \mathrm {id} _{M}}
5449:{\displaystyle \varphi (xs)=\varphi (x)s}
5005:
4914:
4838:
4598:
4577:
4380:
4333:
4303:{\displaystyle \varphi _{i},\varphi _{j}}
4243:
4119:Structure groups and transition functions
4056:
3339:
3332:
3317:
2910:, the unit sphere bundle is known as the
2552:A special class of fiber bundles, called
1798:
1741:
1734:
1727:
1412:
1242:is the product space) in such a way that
775:
768:
761:
8131:
7825:Covariance and contravariance of vectors
7302:
7272:
7251:
7052:
6575:
4217:is a set of local trivialization charts
3378:
2512:
2498:
2124:
38:
7307:Gauge Theory and Variational Principles
7164:
7123:
7067:
6921:
6857:
6793:
6758:
6704:
6359:. Another sufficient condition, due to
5346:For fiber bundles with structure group
4755:-bundle is a smooth fiber bundle where
2097:one. Any such fiber bundle is called a
683:The theory of fibered spaces, of which
14:
8981:
7357:
6503:
6246:{\displaystyle (E,B,\pi ,F)=(M,N,f,F)}
5521:coincide, then a bundle morphism over
4397:{\displaystyle (U_{j},\,\varphi _{j})}
4350:{\displaystyle (U_{i},\,\varphi _{i})}
4077:then local sections always exist over
3944:of a fiber bundle is a continuous map
3538:. From the perspective of Lie groups,
2383:
47:showing the intuition behind the term
8105:
7449:
7411:
7011:
6253:is a fiber bundle. (Surjectivity of
5849:are defined over the same base space
4740:of the bundle; the analogous term in
3209:, then under some circumstances, the
2115:
699:are a special case, is attributed to
5761:and the following diagram commutes:
3937:having a nowhere vanishing section.
2797:is a fiber bundle whose fiber is an
1760:
891:condition outlined below. The space
740:
6574:or homotopy covering property (see
6078:to another differentiable manifold
6038:, fiber bundles arise naturally as
5350:and whose total spaces are (right)
3921:The most well-known example is the
2836:, for which the fiber over a point
2832:) one can construct the associated
2444:A similar nontrivial bundle is the
2386:) exists that maps the preimage of
788:{\displaystyle (E,\,B,\,\pi ,\,F),}
24:
7688:Tensors in curvilinear coordinates
6509:
5986:
5983:
5763:
5741:
5738:
5338:
5332:That is, the following diagram is
5098:and so obtain a (right) action of
3346:{\displaystyle G,\,G/H,\,\pi ,\,H}
3302:necessary and sufficient condition
3156:
3145:Mapping tori of homeomorphisms of
2622:is given, so that each fiber is a
2093:is not just locally a product but
1751:{\displaystyle (E,\,B,\,\pi ,\,F)}
1268:
25:
9000:
7398:
7170:"On the theory of sphere bundles"
6863:"On the theory of sphere bundles"
6449:is compact and connected for all
4800:satisfy the following conditions
2828:(such as the tangent bundle to a
2782:
2540:. It follows that the fiber is a
2485:{\displaystyle S^{1}\times S^{1}}
1941:
55:) are line segments. The mapping
4763:and the corresponding action on
4139:continuously on the fiber space
4123:Fiber bundles often come with a
3149:are of particular importance in
2591:, are bundles on whose fibers a
1520:{\displaystyle \pi ^{-1}(\{p\})}
7364:Graduate Studies in Mathematics
7360:Topics in Differential Geometry
7233:
7223:
7158:
7149:10.1090/s0002-9904-1937-06642-0
7058:
7045:
7005:
6074:from a differentiable manifold
5961:{\displaystyle \pi _{F}:F\to M}
5922:{\displaystyle \pi _{E}:E\to M}
5842:{\displaystyle \pi _{F}:F\to M}
5803:{\displaystyle \pi _{E}:E\to M}
5709:{\displaystyle \varphi :E\to F}
5684:This means that the bundle map
5628:{\displaystyle \varphi :E\to F}
5596:{\displaystyle \pi _{F}:F\to M}
5557:{\displaystyle \pi _{E}:E\to M}
5391:{\displaystyle \varphi :E\to F}
5366:on the fibers. This means that
5251:
5195:{\displaystyle \pi _{F}:F\to N}
5156:{\displaystyle \pi _{E}:E\to M}
3240:together with the quotient map
3023:
2525:
2439:
2120:
8164:Differentiable/Smooth manifold
7380:Voitsekhovskii, M.I. (2001) ,
7259:, Princeton University Press,
6981:
6957:
6939:
6915:
6851:
6787:
6752:
6736:
6698:
6550:
6379:
6335:
6329:
6305:is assumed to be a surjective
6289:
6240:
6216:
6210:
6186:
6159:
6107:
6089:
6058:
5952:
5913:
5883:{\displaystyle (\varphi ,\,f)}
5877:
5864:
5833:
5794:
5700:
5619:
5587:
5548:
5440:
5434:
5425:
5416:
5382:
5261:
5242:
5186:
5147:
5105:
4999:
4993:
4977:
4971:
4952:
4946:
4902:
4895:
4876:
4870:
4829:
4823:
4685:
4618:
4612:
4581:
4568:
4481:
4391:
4364:
4344:
4317:
4254:
4227:
4204:
4180:
4095:
4066:{\displaystyle (U,\,\varphi )}
4060:
4047:
4013:
4010:
4004:
3998:
3960:
3877:
3874:
3868:
3862:
3836:
3750:
3680:
3674:
3663:
3657:
3631:
3625:
3587:
3581:
3477:{\displaystyle S^{3}\to S^{2}}
3461:
3432:a closed subgroup (and thus a
3256:
3068:
2978:the sphere bundle is called a
2762:
2756:
2745:
2739:
2326:
2320:
2293:; in the picture, this is the
2244:
2238:
1994:
1783:
1745:
1718:
1642:
1587:
1575:
1514:
1502:
1357:
1354:
1348:
1322:is the natural projection and
1306:
1199:
1193:
1151:
1148:
1142:
864:
779:
752:
745:A fiber bundle is a structure
494:. Fiber bundles, such as the
205:
71:
13:
1:
7741:Exterior covariant derivative
7673:Tensor (intrinsic definition)
7278:The Topology of Fibre Bundles
7257:The Topology of Fibre Bundles
7245:
6348:is compact for every compact
5086:is free and transitive, i.e.
4842:{\displaystyle t_{ii}(x)=1\,}
4701:is a continuous map called a
3720:is any topological group and
3374:
3273:{\displaystyle \pi :G\to G/H}
2558:, are those whose fibers are
7766:Raising and lowering indices
6030:Differentiable fiber bundles
5406:-space to another, that is,
4210:{\displaystyle (E,B,\pi ,F)}
4025:{\displaystyle \pi (f(x))=x}
3889:{\displaystyle \pi (f(x))=x}
2548:Vector and principal bundles
2332:{\displaystyle \pi ^{-1}(U)}
2256:{\displaystyle \pi (x)\in B}
1205:{\displaystyle \pi ^{-1}(U)}
1086:{\displaystyle U\subseteq B}
506:, play an important role in
217:{\displaystyle \pi :E\to B,}
7:
8870:Classification of manifolds
8004:Gluon field strength tensor
7475:
7387:Encyclopedia of Mathematics
6593:
6559:{\displaystyle \pi :E\to B}
6525:torsor (algebraic geometry)
6521:principal homogeneous space
6442:{\displaystyle f^{-1}\{x\}}
5072:principal homogeneous space
4310:for the overlapping charts
3777:
3565:can be identified with the
2624:principal homogeneous space
2507:in three-dimensional space.
2297:of one of the squares. The
2003:{\displaystyle \pi :E\to B}
1971:{\displaystyle E=B\times F}
1936:
1843:
1651:{\displaystyle \pi :E\to B}
873:{\displaystyle \pi :E\to B}
80:{\displaystyle \pi :E\to B}
30:Not to be confused with an
10:
9005:
7815:Cartan formalism (physics)
7635:Penrose graphical notation
5109:
4751:In the smooth category, a
4073:is a local trivialization
3929:is the obstruction to the
3781:
3686:{\displaystyle SU(2)/U(1)}
3016:
2786:
2105:. Any fiber bundle over a
1273:Local triviality condition
1038:We require that for every
664:. The first definition of
619:
451:{\displaystyle B\times F,}
29:
8946:over commutative algebras
8903:
8862:
8795:
8692:
8588:
8535:
8526:
8362:
8285:
8224:
8144:
8021:
7961:
7910:
7903:
7795:
7726:
7663:
7607:
7554:
7501:
7494:
7487:Glossary of tensor theory
7483:
7358:Michor, Peter W. (2008),
6572:homotopy lifting property
6341:{\displaystyle f^{-1}(K)}
6026:is also a homeomorphism.
5102:on the principal bundle.
4770:The transition functions
2425:{\displaystyle B\times F}
1929:above are required to be
1868:is a fiber bundle in the
1235:{\displaystyle U\times F}
529:of fiber bundles forms a
263:{\displaystyle B\times F}
224:that in small regions of
172:{\displaystyle B\times F}
8662:Riemann curvature tensor
8071:Gregorio Ricci-Curbastro
7943:Riemann curvature tensor
7650:Van der Waerden notation
7303:Bleecker, David (1981),
6692:
6502:fiber bundle structure (
6408:differentiable manifolds
6388:{\displaystyle f:M\to N}
6298:{\displaystyle f:M\to N}
6168:{\displaystyle f:M\to N}
6067:{\displaystyle f:M\to N}
6036:differentiable manifolds
6019:{\displaystyle \varphi }
5513:In case the base spaces
5358:are also required to be
5066:-bundle where the fiber
3969:{\displaystyle f:U\to E}
3845:{\displaystyle f:B\to E}
3767:{\displaystyle G\to G/H}
3077:{\displaystyle f:X\to X}
2804:. Given a vector bundle
2384:§ Formal definition
2375:{\displaystyle \varphi }
1853:that, in analogy with a
1379:is a homeomorphism. The
490:, as well as nontrivial
8041:Elwin Bruno Christoffel
7974:Angular momentum tensor
7645:Tetrad (index notation)
7615:Abstract index notation
6471:{\displaystyle x\in N,}
6410:such that the preimage
6113:{\displaystyle (M,N,f)}
5504:{\displaystyle s\in G.}
5202:are fiber bundles over
5074:for the left action of
4081:. Such sections are in
2729:, a vector bundle with
1547:(since this is true of
672:in 1935 under the name
502:and other more general
8454:Manifold with boundary
8169:Differential structure
7855:Levi-Civita connection
7433:Sardanashvily, Gennadi
6560:
6492:
6472:
6443:
6389:
6342:
6299:
6267:
6247:
6169:
6143:, then any submersion
6114:
6068:
6020:
6000:
5962:
5923:
5884:
5843:
5804:
5768:
5755:
5710:
5678:
5629:
5597:
5558:
5525:from the fiber bundle
5505:
5476:
5475:{\displaystyle x\in E}
5450:
5392:
5343:
5326:
5271:
5196:
5157:
5010:
4919:
4843:
4794:
4793:{\displaystyle t_{ij}}
4695:
4633:
4527:
4398:
4351:
4304:
4264:
4211:
4105:
4104:{\displaystyle U\to F}
4067:
4026:
3970:
3912:characteristic classes
3890:
3846:
3814:
3784:Section (fiber bundle)
3768:
3734:
3714:
3687:
3638:
3594:
3559:
3532:
3505:
3478:
3426:
3402:
3367:
3347:
3294:
3274:
3234:
3199:
3175:
3139:
3112:
3078:
3042:
3000:
2972:
2943:
2904:
2881:
2850:
2818:
2769:
2723:
2703:
2683:
2660:
2640:
2616:
2518:
2508:
2486:
2426:
2400:
2376:
2353:
2333:
2283:
2282:{\displaystyle x\in E}
2257:
2222:
2199:
2175:
2147:
2130:
2087:
2067:
2044:
2030:is a fiber bundle (of
2024:
2004:
1972:
1919:
1899:
1831:
1752:
1703:
1686:determined by the map
1676:
1652:
1619:
1594:
1541:
1521:
1473:
1472:{\displaystyle p\in B}
1436:
1373:
1316:
1274:
1256:
1236:
1206:
1167:
1107:
1087:
1058:
1057:{\displaystyle x\in B}
1025:
989:
961:
933:
905:
874:
838:
818:
789:
610:
596:, acting on the fiber
578:
551:
472:
452:
423:
390:
366:
342:
310:
287:
264:
238:
218:
173:
147:
88:
81:
8081:Jan Arnoldus Schouten
8036:Augustin-Louis Cauchy
7516:Differential geometry
7198:10.1073/pnas.26.2.148
7177:Proc. Natl. Acad. Sci
7136:Bull. Amer. Math. Soc
7098:10.1073/pnas.21.7.464
7077:Proc. Natl. Acad. Sci
7018:Annals of Mathematics
6951:Coll. Top. Alg. Paris
6890:10.1073/pnas.26.2.148
6826:10.1073/pnas.21.7.464
6561:
6493:
6473:
6444:
6390:
6343:
6300:
6268:
6248:
6170:
6115:
6069:
6021:
6001:
5963:
5924:
5885:
5844:
5805:
5767:
5756:
5711:
5679:
5630:
5598:
5559:
5506:
5477:
5451:
5393:
5342:
5327:
5272:
5197:
5158:
5124:are base spaces, and
5011:
4920:
4844:
4795:
4696:
4634:
4528:
4399:
4352:
4305:
4265:
4212:
4106:
4085:with continuous maps
4068:
4027:
3971:
3891:
3847:
3815:
3769:
3735:
3715:
3688:
3639:
3595:
3593:{\displaystyle SU(2)}
3567:special unitary group
3560:
3558:{\displaystyle S^{3}}
3533:
3531:{\displaystyle S^{3}}
3511:whose total space is
3506:
3504:{\displaystyle S^{2}}
3479:
3427:
3403:
3368:
3348:
3295:
3275:
3235:
3200:
3176:
3140:
3113:
3111:{\displaystyle M_{f}}
3079:
3043:
3001:
2973:
2944:
2905:
2882:
2880:{\displaystyle E_{x}}
2851:
2819:
2770:
2724:
2704:
2684:
2682:{\displaystyle \rho }
2661:
2641:
2617:
2516:
2502:
2487:
2427:
2401:
2377:
2354:
2334:
2284:
2258:
2223:
2200:
2176:
2148:
2128:
2088:
2068:
2045:
2025:
2005:
1973:
1920:
1900:
1832:
1753:
1704:
1702:{\displaystyle \pi .}
1677:
1653:
1620:
1595:
1542:
1522:
1474:
1437:
1374:
1317:
1272:
1257:
1237:
1207:
1168:
1108:
1088:
1059:
1026:
990:
962:
934:
906:
875:
839:
819:
790:
611:
579:
552:
512:differential topology
508:differential geometry
473:
453:
424:
391:
367:
352:of the fiber bundle,
343:
311:
309:{\displaystyle \pi ,}
288:
265:
239:
219:
174:
148:
131:topological structure
129:may have a different
82:
42:
8601:Covariant derivative
8152:Topological manifold
8056:Carl Friedrich Gauss
7989:stress–energy tensor
7984:Cauchy stress tensor
7736:Covariant derivative
7698:Antisymmetric tensor
7630:Multi-index notation
6759:Seifert, H. (1980).
6611:Characteristic class
6538:
6482:
6453:
6414:
6367:
6313:
6277:
6257:
6183:
6147:
6086:
6046:
6010:
5972:
5933:
5894:
5861:
5814:
5775:
5727:
5688:
5639:
5607:
5568:
5529:
5486:
5460:
5410:
5370:
5281:
5230:
5167:
5128:
4930:
4854:
4807:
4774:
4728:-atlases. The group
4643:
4537:
4408:
4361:
4314:
4274:
4221:
4177:
4089:
4044:
3992:
3948:
3856:
3824:
3820:is a continuous map
3813:{\displaystyle \pi }
3804:
3800:) of a fiber bundle
3744:
3724:
3704:
3648:
3637:{\displaystyle U(1)}
3619:
3572:
3542:
3515:
3488:
3448:
3416:
3392:
3375:local cross-sections
3366:{\displaystyle \pi }
3357:
3308:
3284:
3244:
3216:
3189:
3165:
3126:
3095:
3056:
3032:
2990:
2956:
2927:
2923:, which is a degree
2891:
2864:
2840:
2808:
2733:
2713:
2693:
2673:
2650:
2630:
2606:
2456:
2410:
2390:
2366:
2343:
2304:
2267:
2232:
2212:
2189:
2165:
2137:
2077:
2054:
2034:
2023:{\displaystyle \pi }
2014:
1982:
1950:
1909:
1898:{\displaystyle E,B,}
1880:
1855:short exact sequence
1767:
1715:
1690:
1666:
1630:
1606:
1600:) and is called the
1551:
1531:
1486:
1457:
1446:local trivialization
1387:
1326:
1281:
1255:{\displaystyle \pi }
1246:
1220:
1177:
1120:
1097:
1071:
1042:
1015:
988:{\displaystyle \pi }
979:
951:
923:
895:
852:
828:
817:{\displaystyle E,B,}
799:
749:
600:
565:
541:
471:{\displaystyle \pi }
462:
433:
413:
380:
356:
332:
297:
274:
248:
228:
193:
157:
153:and a product space
137:
106:Commonwealth English
59:
32:optical fiber bundle
8635:Exterior derivative
8237:Atiyah–Singer index
8186:Riemannian manifold
7933:Nonmetricity tensor
7788:(2nd-order tensors)
7756:Hodge star operator
7746:Exterior derivative
7595:Transport phenomena
7580:Continuum mechanics
7536:Multilinear algebra
7344:, Springer Verlag,
7189:1940PNAS...26..148W
7089:1935PNAS...21..464W
6881:1940PNAS...26..148W
6817:1935PNAS...21..464W
6034:In the category of
5402:-morphism from one
5210:, respectively. A
4705:transition function
4567:
4438:
3774:is a fiber bundle.
3700:More generally, if
3644:, and the quotient
3233:{\displaystyle G/H}
3151:3-manifold topology
3008:long exact sequence
2971:{\displaystyle n=1}
2942:{\displaystyle n+1}
2913:unit tangent bundle
2830:Riemannian manifold
2646:-bundle. The group
2538:local homeomorphism
1864:smooth fiber bundle
1808:
1626:Every fiber bundle
1571:
1527:is homeomorphic to
1064:, there is an open
179:is defined using a
95:, and particularly
8941:Secondary calculus
8895:Singularity theory
8850:Parallel transport
8618:De Rham cohomology
8257:Generalized Stokes
8066:Tullio Levi-Civita
8009:Metric tensor (GR)
7923:Levi-Civita symbol
7776:Tensor contraction
7590:General relativity
7526:Euclidean geometry
7326:Ehresmann, Charles
7013:Serre, Jean-Pierre
6989:Ehresmann, Charles
6965:Ehresmann, Charles
6947:Ehresmann, Charles
6729:10.1007/bf02398271
6686:Wu–Yang dictionary
6621:Equivariant bundle
6568:homotopy-theoretic
6556:
6488:
6468:
6439:
6385:
6338:
6295:
6263:
6243:
6165:
6110:
6064:
6016:
5996:
5958:
5919:
5880:
5839:
5800:
5769:
5751:
5706:
5674:
5625:
5593:
5554:
5501:
5472:
5446:
5388:
5344:
5322:
5267:
5192:
5153:
5040:and is called the
5006:
4915:
4839:
4790:
4691:
4629:
4550:
4523:
4421:
4394:
4347:
4300:
4270:such that for any
4260:
4207:
4111:. Sections form a
4101:
4083:1-1 correspondence
4063:
4022:
3966:
3923:hairy ball theorem
3916:algebraic topology
3886:
3842:
3810:
3764:
3730:
3710:
3683:
3634:
3590:
3555:
3528:
3501:
3474:
3422:
3398:
3363:
3343:
3290:
3270:
3230:
3195:
3171:
3138:{\displaystyle X.}
3135:
3108:
3074:
3038:
2996:
2968:
2939:
2903:{\displaystyle TM}
2900:
2877:
2856:is the set of all
2846:
2834:unit sphere bundle
2814:
2765:
2719:
2709:on a vector space
2699:
2679:
2656:
2636:
2612:
2519:
2509:
2482:
2422:
2396:
2372:
2349:
2329:
2279:
2253:
2218:
2195:
2171:
2143:
2131:
2116:Nontrivial bundles
2083:
2066:{\displaystyle B.}
2063:
2040:
2020:
2000:
1968:
1915:
1895:
1827:
1825:
1748:
1699:
1672:
1648:
1618:{\displaystyle p.}
1615:
1590:
1554:
1537:
1517:
1469:
1432:
1369:
1312:
1275:
1252:
1232:
1202:
1163:
1103:
1083:
1054:
1021:
985:
957:
929:
901:
870:
846:topological spaces
834:
814:
785:
606:
577:{\displaystyle E.}
574:
547:
537:as projection) to
468:
448:
419:
386:
362:
338:
306:
286:{\displaystyle B.}
283:
260:
234:
214:
169:
143:
89:
77:
8976:
8975:
8858:
8857:
8623:Differential form
8277:Whitney embedding
8211:Differential form
8099:
8098:
8061:Hermann Grassmann
8017:
8016:
7969:Moment of inertia
7830:Differential form
7805:Affine connection
7620:Einstein notation
7603:
7602:
7531:Exterior calculus
7511:Coordinate system
7373:978-0-8218-2003-2
7351:978-0-387-94087-8
7318:978-0-201-10096-9
7287:978-0-691-00548-5
7276:(April 5, 1999).
7266:978-0-691-08055-0
6953:. C.N.R.S.: 3–15.
6661:Projective bundle
6566:that has certain
6491:{\displaystyle f}
6266:{\displaystyle f}
5771:Assume that both
4133:topological group
3733:{\displaystyle H}
3713:{\displaystyle G}
3606:diagonal matrices
3425:{\displaystyle H}
3401:{\displaystyle G}
3293:{\displaystyle H}
3198:{\displaystyle H}
3183:topological group
3174:{\displaystyle G}
3050:topological space
3041:{\displaystyle X}
2999:{\displaystyle n}
2849:{\displaystyle x}
2817:{\displaystyle E}
2777:associated bundle
2754:
2722:{\displaystyle V}
2702:{\displaystyle G}
2659:{\displaystyle G}
2639:{\displaystyle G}
2615:{\displaystyle G}
2588:principal bundles
2523:
2522:
2503:The Klein bottle
2399:{\displaystyle U}
2362:A homeomorphism (
2352:{\displaystyle U}
2221:{\displaystyle U}
2198:{\displaystyle F}
2174:{\displaystyle B}
2146:{\displaystyle E}
2086:{\displaystyle E}
2043:{\displaystyle F}
1918:{\displaystyle F}
1851:
1850:
1813:
1809:
1807:
1801:
1791:
1758:is often denoted
1684:quotient topology
1675:{\displaystyle B}
1540:{\displaystyle F}
1214:subspace topology
1106:{\displaystyle x}
1024:{\displaystyle B}
1007:bundle projection
960:{\displaystyle F}
932:{\displaystyle E}
904:{\displaystyle B}
837:{\displaystyle F}
741:Formal definition
721:Jean-Pierre Serre
717:Charles Ehresmann
697:fibered manifolds
689:principal bundles
609:{\displaystyle F}
590:topological group
550:{\displaystyle E}
516:principal bundles
422:{\displaystyle E}
389:{\displaystyle F}
365:{\displaystyle B}
341:{\displaystyle E}
237:{\displaystyle E}
146:{\displaystyle E}
16:(Redirected from
8996:
8968:Stratified space
8926:Fréchet manifold
8640:Interior product
8533:
8532:
8230:
8126:
8119:
8112:
8103:
8102:
8076:Bernhard Riemann
7908:
7907:
7751:Exterior product
7718:Two-point tensor
7703:Symmetric tensor
7585:Electromagnetism
7499:
7498:
7470:
7463:
7456:
7447:
7446:
7424:
7423:
7394:
7376:
7354:
7338:Husemoller, Dale
7333:
7321:
7310:
7299:
7274:Steenrod, Norman
7269:
7253:Steenrod, Norman
7240:
7237:
7231:
7227:
7221:
7220:
7210:
7200:
7174:
7166:Whitney, Hassler
7162:
7156:
7153:
7151:
7133:
7125:Whitney, Hassler
7120:
7110:
7100:
7069:Whitney, Hassler
7062:
7056:
7049:
7043:
7042:
7009:
7003:
7002:
6985:
6979:
6978:
6961:
6955:
6954:
6943:
6937:
6936:
6923:Feldbau, Jacques
6919:
6913:
6912:
6902:
6892:
6859:Whitney, Hassler
6855:
6849:
6848:
6838:
6828:
6795:Whitney, Hassler
6791:
6785:
6784:
6756:
6750:
6740:
6734:
6733:
6731:
6715:Acta Mathematica
6706:Seifert, Herbert
6702:
6676:Universal bundle
6656:Principal bundle
6626:Fibered manifold
6565:
6563:
6562:
6557:
6515:The notion of a
6497:
6495:
6494:
6489:
6477:
6475:
6474:
6469:
6448:
6446:
6445:
6440:
6429:
6428:
6395:is a surjective
6394:
6392:
6391:
6386:
6361:Ehresmann (1951)
6347:
6345:
6344:
6339:
6328:
6327:
6304:
6302:
6301:
6296:
6272:
6270:
6269:
6264:
6252:
6250:
6249:
6244:
6174:
6172:
6171:
6166:
6122:fibered manifold
6119:
6117:
6116:
6111:
6073:
6071:
6070:
6065:
6025:
6023:
6022:
6017:
6005:
6003:
6002:
5997:
5995:
5994:
5989:
5967:
5965:
5964:
5959:
5945:
5944:
5928:
5926:
5925:
5920:
5906:
5905:
5889:
5887:
5886:
5881:
5857:is a bundle map
5848:
5846:
5845:
5840:
5826:
5825:
5809:
5807:
5806:
5801:
5787:
5786:
5760:
5758:
5757:
5752:
5750:
5749:
5744:
5719:the identity of
5715:
5713:
5712:
5707:
5683:
5681:
5680:
5675:
5664:
5663:
5651:
5650:
5634:
5632:
5631:
5626:
5602:
5600:
5599:
5594:
5580:
5579:
5563:
5561:
5560:
5555:
5541:
5540:
5510:
5508:
5507:
5502:
5481:
5479:
5478:
5473:
5455:
5453:
5452:
5447:
5397:
5395:
5394:
5389:
5331:
5329:
5328:
5323:
5318:
5317:
5293:
5292:
5276:
5274:
5273:
5268:
5224:
5223:
5216:
5215:
5201:
5199:
5198:
5193:
5179:
5178:
5162:
5160:
5159:
5154:
5140:
5139:
5015:
5013:
5012:
5007:
4992:
4991:
4970:
4969:
4945:
4944:
4924:
4922:
4921:
4916:
4913:
4912:
4894:
4893:
4869:
4868:
4848:
4846:
4845:
4840:
4822:
4821:
4799:
4797:
4796:
4791:
4789:
4788:
4738:
4737:
4707:
4706:
4700:
4698:
4697:
4692:
4684:
4683:
4671:
4670:
4658:
4657:
4638:
4636:
4635:
4630:
4628:
4624:
4611:
4610:
4566:
4558:
4549:
4548:
4532:
4530:
4529:
4524:
4516:
4512:
4511:
4510:
4498:
4497:
4474:
4470:
4469:
4468:
4456:
4455:
4437:
4429:
4420:
4419:
4403:
4401:
4400:
4395:
4390:
4389:
4376:
4375:
4356:
4354:
4353:
4348:
4343:
4342:
4329:
4328:
4309:
4307:
4306:
4301:
4299:
4298:
4286:
4285:
4269:
4267:
4266:
4261:
4253:
4252:
4239:
4238:
4216:
4214:
4213:
4208:
4110:
4108:
4107:
4102:
4072:
4070:
4069:
4064:
4031:
4029:
4028:
4023:
3975:
3973:
3972:
3967:
3895:
3893:
3892:
3887:
3851:
3849:
3848:
3843:
3819:
3817:
3816:
3811:
3794:
3793:
3773:
3771:
3770:
3765:
3760:
3739:
3737:
3736:
3731:
3719:
3717:
3716:
3711:
3692:
3690:
3689:
3684:
3670:
3643:
3641:
3640:
3635:
3599:
3597:
3596:
3591:
3564:
3562:
3561:
3556:
3554:
3553:
3537:
3535:
3534:
3529:
3527:
3526:
3510:
3508:
3507:
3502:
3500:
3499:
3483:
3481:
3480:
3475:
3473:
3472:
3460:
3459:
3438:Cartan's theorem
3431:
3429:
3428:
3423:
3407:
3405:
3404:
3399:
3372:
3370:
3369:
3364:
3352:
3350:
3349:
3344:
3325:
3299:
3297:
3296:
3291:
3279:
3277:
3276:
3271:
3266:
3239:
3237:
3236:
3231:
3226:
3204:
3202:
3201:
3196:
3180:
3178:
3177:
3172:
3144:
3142:
3141:
3136:
3117:
3115:
3114:
3109:
3107:
3106:
3083:
3081:
3080:
3075:
3047:
3045:
3044:
3039:
3005:
3003:
3002:
2997:
2977:
2975:
2974:
2969:
2948:
2946:
2945:
2940:
2909:
2907:
2906:
2901:
2886:
2884:
2883:
2878:
2876:
2875:
2855:
2853:
2852:
2847:
2823:
2821:
2820:
2815:
2774:
2772:
2771:
2766:
2755:
2752:
2728:
2726:
2725:
2720:
2708:
2706:
2705:
2700:
2688:
2686:
2685:
2680:
2665:
2663:
2662:
2657:
2645:
2643:
2642:
2637:
2621:
2619:
2618:
2613:
2572:cotangent bundle
2495:
2494:
2491:
2489:
2488:
2483:
2481:
2480:
2468:
2467:
2431:
2429:
2428:
2423:
2405:
2403:
2402:
2397:
2381:
2379:
2378:
2373:
2358:
2356:
2355:
2350:
2338:
2336:
2335:
2330:
2319:
2318:
2288:
2286:
2285:
2280:
2262:
2260:
2259:
2254:
2227:
2225:
2224:
2219:
2204:
2202:
2201:
2196:
2180:
2178:
2177:
2172:
2152:
2150:
2149:
2144:
2103:
2102:
2092:
2090:
2089:
2084:
2072:
2070:
2069:
2064:
2049:
2047:
2046:
2041:
2029:
2027:
2026:
2021:
2009:
2007:
2006:
2001:
1977:
1975:
1974:
1969:
1924:
1922:
1921:
1916:
1904:
1902:
1901:
1896:
1874:smooth manifolds
1866:
1865:
1845:
1836:
1834:
1833:
1828:
1826:
1822:
1811:
1810:
1805:
1799:
1793:
1789:
1775:
1761:
1757:
1755:
1754:
1749:
1708:
1706:
1705:
1700:
1681:
1679:
1678:
1673:
1657:
1655:
1654:
1649:
1624:
1622:
1621:
1616:
1599:
1597:
1596:
1591:
1570:
1562:
1546:
1544:
1543:
1538:
1526:
1524:
1523:
1518:
1501:
1500:
1478:
1476:
1475:
1470:
1448:
1447:
1441:
1439:
1438:
1433:
1431:
1427:
1423:
1422:
1421:
1408:
1407:
1378:
1376:
1375:
1370:
1347:
1346:
1321:
1319:
1318:
1313:
1293:
1292:
1261:
1259:
1258:
1253:
1241:
1239:
1238:
1233:
1211:
1209:
1208:
1203:
1192:
1191:
1172:
1170:
1169:
1164:
1141:
1140:
1112:
1110:
1109:
1104:
1092:
1090:
1089:
1084:
1063:
1061:
1060:
1055:
1030:
1028:
1027:
1022:
1009:
1008:
1001:
1000:
994:
992:
991:
986:
973:
972:
966:
964:
963:
958:
945:
944:
938:
936:
935:
930:
917:
916:
910:
908:
907:
902:
889:local triviality
879:
877:
876:
871:
843:
841:
840:
835:
823:
821:
820:
815:
794:
792:
791:
786:
615:
613:
612:
607:
583:
581:
580:
575:
556:
554:
553:
548:
535:identity mapping
477:
475:
474:
469:
457:
455:
454:
449:
428:
426:
425:
420:
395:
393:
392:
387:
371:
369:
368:
363:
348:is known as the
347:
345:
344:
339:
315:
313:
312:
307:
292:
290:
289:
284:
269:
267:
266:
261:
243:
241:
240:
235:
223:
221:
220:
215:
178:
176:
175:
170:
152:
150:
149:
144:
86:
84:
83:
78:
21:
9004:
9003:
8999:
8998:
8997:
8995:
8994:
8993:
8979:
8978:
8977:
8972:
8911:Banach manifold
8904:Generalizations
8899:
8854:
8791:
8688:
8650:Ricci curvature
8606:Cotangent space
8584:
8522:
8364:
8358:
8317:Exponential map
8281:
8226:
8220:
8140:
8130:
8100:
8095:
8046:Albert Einstein
8013:
7994:Einstein tensor
7957:
7938:Ricci curvature
7918:Kronecker delta
7904:Notable tensors
7899:
7820:Connection form
7797:
7791:
7722:
7708:Tensor operator
7665:
7659:
7599:
7575:Computer vision
7568:
7550:
7546:Tensor calculus
7490:
7479:
7474:
7412:Rowland, Todd.
7401:
7374:
7352:
7319:
7288:
7267:
7248:
7243:
7238:
7234:
7228:
7224:
7172:
7163:
7159:
7142:(12): 785–805.
7131:
7073:"Sphere spaces"
7063:
7059:
7050:
7046:
7031:10.2307/1969485
7010:
7006:
6986:
6982:
6962:
6958:
6944:
6940:
6920:
6916:
6856:
6852:
6799:"Sphere spaces"
6792:
6788:
6773:
6757:
6753:
6741:
6737:
6703:
6699:
6695:
6690:
6666:Pullback bundle
6596:
6539:
6536:
6535:
6530:In topology, a
6512:
6510:Generalizations
6483:
6480:
6479:
6454:
6451:
6450:
6421:
6417:
6415:
6412:
6411:
6368:
6365:
6364:
6320:
6316:
6314:
6311:
6310:
6309:, meaning that
6278:
6275:
6274:
6258:
6255:
6254:
6184:
6181:
6180:
6148:
6145:
6144:
6087:
6084:
6083:
6047:
6044:
6043:
6032:
6011:
6008:
6007:
5990:
5982:
5981:
5973:
5970:
5969:
5940:
5936:
5934:
5931:
5930:
5901:
5897:
5895:
5892:
5891:
5862:
5859:
5858:
5821:
5817:
5815:
5812:
5811:
5782:
5778:
5776:
5773:
5772:
5745:
5737:
5736:
5728:
5725:
5724:
5689:
5686:
5685:
5659:
5655:
5646:
5642:
5640:
5637:
5636:
5608:
5605:
5604:
5575:
5571:
5569:
5566:
5565:
5536:
5532:
5530:
5527:
5526:
5487:
5484:
5483:
5461:
5458:
5457:
5411:
5408:
5407:
5371:
5368:
5367:
5313:
5309:
5288:
5284:
5282:
5279:
5278:
5231:
5228:
5227:
5222:bundle morphism
5221:
5220:
5213:
5212:
5174:
5170:
5168:
5165:
5164:
5135:
5131:
5129:
5126:
5125:
5114:
5108:
5049:Čech cohomology
5038:
5031:
5024:
4984:
4980:
4962:
4958:
4937:
4933:
4931:
4928:
4927:
4905:
4901:
4886:
4882:
4861:
4857:
4855:
4852:
4851:
4814:
4810:
4808:
4805:
4804:
4781:
4777:
4775:
4772:
4771:
4736:structure group
4735:
4734:
4704:
4703:
4679:
4675:
4666:
4662:
4650:
4646:
4644:
4641:
4640:
4603:
4599:
4591:
4587:
4559:
4554:
4544:
4540:
4538:
4535:
4534:
4506:
4502:
4493:
4489:
4488:
4484:
4464:
4460:
4451:
4447:
4446:
4442:
4430:
4425:
4415:
4411:
4409:
4406:
4405:
4385:
4381:
4371:
4367:
4362:
4359:
4358:
4338:
4334:
4324:
4320:
4315:
4312:
4311:
4294:
4290:
4281:
4277:
4275:
4272:
4271:
4248:
4244:
4234:
4230:
4222:
4219:
4218:
4178:
4175:
4174:
4173:for the bundle
4121:
4090:
4087:
4086:
4045:
4042:
4041:
3993:
3990:
3989:
3949:
3946:
3945:
3857:
3854:
3853:
3825:
3822:
3821:
3805:
3802:
3801:
3791:
3790:
3786:
3780:
3756:
3745:
3742:
3741:
3725:
3722:
3721:
3705:
3702:
3701:
3697:to the sphere.
3666:
3649:
3646:
3645:
3620:
3617:
3616:
3573:
3570:
3569:
3549:
3545:
3543:
3540:
3539:
3522:
3518:
3516:
3513:
3512:
3495:
3491:
3489:
3486:
3485:
3468:
3464:
3455:
3451:
3449:
3446:
3445:
3417:
3414:
3413:
3393:
3390:
3389:
3358:
3355:
3354:
3321:
3309:
3306:
3305:
3285:
3282:
3281:
3262:
3245:
3242:
3241:
3222:
3217:
3214:
3213:
3207:closed subgroup
3190:
3187:
3186:
3166:
3163:
3162:
3159:
3157:Quotient spaces
3127:
3124:
3123:
3102:
3098:
3096:
3093:
3092:
3057:
3054:
3053:
3033:
3030:
3029:
3026:
3021:
2991:
2988:
2987:
2957:
2954:
2953:
2928:
2925:
2924:
2892:
2889:
2888:
2871:
2867:
2865:
2862:
2861:
2841:
2838:
2837:
2809:
2806:
2805:
2791:
2785:
2751:
2734:
2731:
2730:
2714:
2711:
2710:
2694:
2691:
2690:
2674:
2671:
2670:
2651:
2648:
2647:
2631:
2628:
2627:
2607:
2604:
2603:
2550:
2528:
2476:
2472:
2463:
2459:
2457:
2454:
2453:
2442:
2411:
2408:
2407:
2391:
2388:
2387:
2367:
2364:
2363:
2344:
2341:
2340:
2311:
2307:
2305:
2302:
2301:
2268:
2265:
2264:
2233:
2230:
2229:
2213:
2210:
2209:
2190:
2187:
2186:
2166:
2163:
2162:
2138:
2135:
2134:
2123:
2118:
2100:
2099:
2078:
2075:
2074:
2055:
2052:
2051:
2035:
2032:
2031:
2015:
2012:
2011:
1983:
1980:
1979:
1951:
1948:
1947:
1944:
1939:
1910:
1907:
1906:
1881:
1878:
1877:
1863:
1862:
1824:
1823:
1821:
1818:
1817:
1792:
1777:
1776:
1774:
1770:
1768:
1765:
1764:
1716:
1713:
1712:
1711:A fiber bundle
1691:
1688:
1687:
1667:
1664:
1663:
1631:
1628:
1627:
1607:
1604:
1603:
1563:
1558:
1552:
1549:
1548:
1532:
1529:
1528:
1493:
1489:
1487:
1484:
1483:
1458:
1455:
1454:
1450:of the bundle.
1445:
1444:
1417:
1413:
1403:
1399:
1398:
1394:
1390:
1388:
1385:
1384:
1339:
1335:
1327:
1324:
1323:
1288:
1284:
1282:
1279:
1278:
1247:
1244:
1243:
1221:
1218:
1217:
1184:
1180:
1178:
1175:
1174:
1133:
1129:
1121:
1118:
1117:
1098:
1095:
1094:
1072:
1069:
1068:
1043:
1040:
1039:
1016:
1013:
1012:
1006:
1005:
998:
997:
980:
977:
976:
970:
969:
952:
949:
948:
942:
941:
924:
921:
920:
919:of the bundle,
914:
913:
896:
893:
892:
853:
850:
849:
829:
826:
825:
800:
797:
796:
750:
747:
746:
743:
713:Norman Steenrod
709:Jacques Feldbau
701:Herbert Seifert
670:Hassler Whitney
650:Herbert Seifert
646:gefaserter Raum
622:
601:
598:
597:
594:structure group
592:, known as the
586:transition maps
566:
563:
562:
542:
539:
538:
492:covering spaces
463:
460:
459:
434:
431:
430:
414:
411:
410:
381:
378:
377:
357:
354:
353:
333:
330:
329:
298:
295:
294:
275:
272:
271:
249:
246:
245:
229:
226:
225:
194:
191:
190:
158:
155:
154:
138:
135:
134:
60:
57:
56:
35:
28:
23:
22:
15:
12:
11:
5:
9002:
8992:
8991:
8974:
8973:
8971:
8970:
8965:
8960:
8955:
8950:
8949:
8948:
8938:
8933:
8928:
8923:
8918:
8913:
8907:
8905:
8901:
8900:
8898:
8897:
8892:
8887:
8882:
8877:
8872:
8866:
8864:
8860:
8859:
8856:
8855:
8853:
8852:
8847:
8842:
8837:
8832:
8827:
8822:
8817:
8812:
8807:
8801:
8799:
8793:
8792:
8790:
8789:
8784:
8779:
8774:
8769:
8764:
8759:
8749:
8744:
8739:
8729:
8724:
8719:
8714:
8709:
8704:
8698:
8696:
8690:
8689:
8687:
8686:
8681:
8676:
8675:
8674:
8664:
8659:
8658:
8657:
8647:
8642:
8637:
8632:
8631:
8630:
8620:
8615:
8614:
8613:
8603:
8598:
8592:
8590:
8586:
8585:
8583:
8582:
8577:
8572:
8567:
8566:
8565:
8555:
8550:
8545:
8539:
8537:
8530:
8524:
8523:
8521:
8520:
8515:
8505:
8500:
8486:
8481:
8476:
8471:
8466:
8464:Parallelizable
8461:
8456:
8451:
8450:
8449:
8439:
8434:
8429:
8424:
8419:
8414:
8409:
8404:
8399:
8394:
8384:
8374:
8368:
8366:
8360:
8359:
8357:
8356:
8351:
8346:
8344:Lie derivative
8341:
8339:Integral curve
8336:
8331:
8326:
8325:
8324:
8314:
8309:
8308:
8307:
8300:Diffeomorphism
8297:
8291:
8289:
8283:
8282:
8280:
8279:
8274:
8269:
8264:
8259:
8254:
8249:
8244:
8239:
8233:
8231:
8222:
8221:
8219:
8218:
8213:
8208:
8203:
8198:
8193:
8188:
8183:
8178:
8177:
8176:
8171:
8161:
8160:
8159:
8148:
8146:
8145:Basic concepts
8142:
8141:
8129:
8128:
8121:
8114:
8106:
8097:
8096:
8094:
8093:
8088:
8086:Woldemar Voigt
8083:
8078:
8073:
8068:
8063:
8058:
8053:
8051:Leonhard Euler
8048:
8043:
8038:
8033:
8027:
8025:
8023:Mathematicians
8019:
8018:
8015:
8014:
8012:
8011:
8006:
8001:
7996:
7991:
7986:
7981:
7976:
7971:
7965:
7963:
7959:
7958:
7956:
7955:
7950:
7948:Torsion tensor
7945:
7940:
7935:
7930:
7925:
7920:
7914:
7912:
7905:
7901:
7900:
7898:
7897:
7892:
7887:
7882:
7877:
7872:
7867:
7862:
7857:
7852:
7847:
7842:
7837:
7832:
7827:
7822:
7817:
7812:
7807:
7801:
7799:
7793:
7792:
7790:
7789:
7783:
7781:Tensor product
7778:
7773:
7771:Symmetrization
7768:
7763:
7761:Lie derivative
7758:
7753:
7748:
7743:
7738:
7732:
7730:
7724:
7723:
7721:
7720:
7715:
7710:
7705:
7700:
7695:
7690:
7685:
7683:Tensor density
7680:
7675:
7669:
7667:
7661:
7660:
7658:
7657:
7655:Voigt notation
7652:
7647:
7642:
7640:Ricci calculus
7637:
7632:
7627:
7625:Index notation
7622:
7617:
7611:
7609:
7605:
7604:
7601:
7600:
7598:
7597:
7592:
7587:
7582:
7577:
7571:
7569:
7567:
7566:
7561:
7555:
7552:
7551:
7549:
7548:
7543:
7541:Tensor algebra
7538:
7533:
7528:
7523:
7521:Dyadic algebra
7518:
7513:
7507:
7505:
7496:
7492:
7491:
7484:
7481:
7480:
7473:
7472:
7465:
7458:
7450:
7444:
7443:
7430:
7425:
7414:"Fiber Bundle"
7408:
7400:
7399:External links
7397:
7396:
7395:
7377:
7372:
7355:
7350:
7334:
7322:
7317:
7300:
7286:
7270:
7265:
7247:
7244:
7242:
7241:
7232:
7222:
7183:(2): 148–153.
7157:
7155:
7154:
7121:
7083:(7): 462–468.
7057:
7053:Steenrod (1951
7044:
7025:(3): 425–505.
7004:
6980:
6956:
6938:
6914:
6875:(2): 148–153.
6850:
6811:(7): 464–468.
6786:
6771:
6751:
6747:Project Euclid
6735:
6696:
6694:
6691:
6689:
6688:
6683:
6678:
6673:
6671:Quasifibration
6668:
6663:
6658:
6653:
6651:Natural bundle
6648:
6643:
6638:
6633:
6628:
6623:
6618:
6613:
6608:
6606:Algebra bundle
6603:
6597:
6595:
6592:
6591:
6590:
6587:dependent type
6579:
6576:Steenrod (1951
6555:
6552:
6549:
6546:
6543:
6528:
6511:
6508:
6487:
6467:
6464:
6461:
6458:
6438:
6435:
6432:
6427:
6424:
6420:
6384:
6381:
6378:
6375:
6372:
6337:
6334:
6331:
6326:
6323:
6319:
6294:
6291:
6288:
6285:
6282:
6262:
6242:
6239:
6236:
6233:
6230:
6227:
6224:
6221:
6218:
6215:
6212:
6209:
6206:
6203:
6200:
6197:
6194:
6191:
6188:
6164:
6161:
6158:
6155:
6152:
6109:
6106:
6103:
6100:
6097:
6094:
6091:
6063:
6060:
6057:
6054:
6051:
6031:
6028:
6015:
6006:and such that
5993:
5988:
5985:
5980:
5977:
5957:
5954:
5951:
5948:
5943:
5939:
5918:
5915:
5912:
5909:
5904:
5900:
5879:
5876:
5872:
5869:
5866:
5838:
5835:
5832:
5829:
5824:
5820:
5799:
5796:
5793:
5790:
5785:
5781:
5748:
5743:
5740:
5735:
5732:
5705:
5702:
5699:
5696:
5693:
5673:
5670:
5667:
5662:
5658:
5654:
5649:
5645:
5624:
5621:
5618:
5615:
5612:
5592:
5589:
5586:
5583:
5578:
5574:
5553:
5550:
5547:
5544:
5539:
5535:
5500:
5497:
5494:
5491:
5471:
5468:
5465:
5445:
5442:
5439:
5436:
5433:
5430:
5427:
5424:
5421:
5418:
5415:
5387:
5384:
5381:
5378:
5375:
5321:
5316:
5312:
5308:
5305:
5302:
5299:
5296:
5291:
5287:
5266:
5263:
5260:
5257:
5254:
5250:
5247:
5244:
5241:
5238:
5235:
5191:
5188:
5185:
5182:
5177:
5173:
5152:
5149:
5146:
5143:
5138:
5134:
5110:Main article:
5107:
5104:
5036:
5029:
5022:
5017:
5016:
5004:
5001:
4998:
4995:
4990:
4987:
4983:
4979:
4976:
4973:
4968:
4965:
4961:
4957:
4954:
4951:
4948:
4943:
4940:
4936:
4925:
4911:
4908:
4904:
4900:
4897:
4892:
4889:
4885:
4881:
4878:
4875:
4872:
4867:
4864:
4860:
4849:
4837:
4834:
4831:
4828:
4825:
4820:
4817:
4813:
4787:
4784:
4780:
4732:is called the
4690:
4687:
4682:
4678:
4674:
4669:
4665:
4661:
4656:
4653:
4649:
4627:
4623:
4620:
4617:
4614:
4609:
4606:
4602:
4597:
4594:
4590:
4586:
4583:
4580:
4576:
4573:
4570:
4565:
4562:
4557:
4553:
4547:
4543:
4522:
4519:
4515:
4509:
4505:
4501:
4496:
4492:
4487:
4483:
4480:
4477:
4473:
4467:
4463:
4459:
4454:
4450:
4445:
4441:
4436:
4433:
4428:
4424:
4418:
4414:
4393:
4388:
4384:
4379:
4374:
4370:
4366:
4346:
4341:
4337:
4332:
4327:
4323:
4319:
4297:
4293:
4289:
4284:
4280:
4259:
4256:
4251:
4247:
4242:
4237:
4233:
4229:
4226:
4206:
4203:
4200:
4197:
4194:
4191:
4188:
4185:
4182:
4157:homeomorphisms
4120:
4117:
4100:
4097:
4094:
4062:
4059:
4055:
4052:
4049:
4021:
4018:
4015:
4012:
4009:
4006:
4003:
4000:
3997:
3965:
3962:
3959:
3956:
3953:
3931:tangent bundle
3885:
3882:
3879:
3876:
3873:
3870:
3867:
3864:
3861:
3841:
3838:
3835:
3832:
3829:
3809:
3782:Main article:
3779:
3776:
3763:
3759:
3755:
3752:
3749:
3729:
3709:
3682:
3679:
3676:
3673:
3669:
3665:
3662:
3659:
3656:
3653:
3633:
3630:
3627:
3624:
3589:
3586:
3583:
3580:
3577:
3552:
3548:
3525:
3521:
3498:
3494:
3471:
3467:
3463:
3458:
3454:
3442:Hopf fibration
3421:
3397:
3362:
3342:
3338:
3335:
3331:
3328:
3324:
3320:
3316:
3313:
3289:
3269:
3265:
3261:
3258:
3255:
3252:
3249:
3229:
3225:
3221:
3211:quotient space
3194:
3170:
3158:
3155:
3134:
3131:
3105:
3101:
3073:
3070:
3067:
3064:
3061:
3037:
3025:
3022:
3012:Gysin sequence
2995:
2967:
2964:
2961:
2938:
2935:
2932:
2899:
2896:
2874:
2870:
2845:
2813:
2787:Main article:
2784:
2783:Sphere bundles
2781:
2764:
2761:
2758:
2750:
2747:
2744:
2741:
2738:
2718:
2698:
2678:
2668:representation
2655:
2635:
2611:
2568:tangent bundle
2555:vector bundles
2549:
2546:
2542:discrete space
2533:covering space
2527:
2524:
2521:
2520:
2510:
2479:
2475:
2471:
2466:
2462:
2441:
2438:
2421:
2418:
2415:
2395:
2371:
2348:
2328:
2325:
2322:
2317:
2314:
2310:
2278:
2275:
2272:
2252:
2249:
2246:
2243:
2240:
2237:
2217:
2194:
2185:for the fiber
2170:
2142:
2122:
2119:
2117:
2114:
2101:trivial bundle
2082:
2062:
2059:
2039:
2019:
1999:
1996:
1993:
1990:
1987:
1967:
1964:
1961:
1958:
1955:
1943:
1942:Trivial bundle
1940:
1938:
1935:
1914:
1894:
1891:
1888:
1885:
1849:
1848:
1839:
1837:
1820:
1819:
1816:
1804:
1796:
1788:
1785:
1782:
1779:
1778:
1773:
1772:
1747:
1744:
1740:
1737:
1733:
1730:
1726:
1723:
1720:
1698:
1695:
1671:
1647:
1644:
1641:
1638:
1635:
1614:
1611:
1589:
1586:
1583:
1580:
1577:
1574:
1569:
1566:
1561:
1557:
1536:
1516:
1513:
1510:
1507:
1504:
1499:
1496:
1492:
1468:
1465:
1462:
1430:
1426:
1420:
1416:
1411:
1406:
1402:
1397:
1393:
1368:
1365:
1362:
1359:
1356:
1353:
1350:
1345:
1342:
1338:
1334:
1331:
1311:
1308:
1305:
1302:
1299:
1296:
1291:
1287:
1251:
1231:
1228:
1225:
1201:
1198:
1195:
1190:
1187:
1183:
1162:
1159:
1156:
1153:
1150:
1147:
1144:
1139:
1136:
1132:
1128:
1125:
1102:
1082:
1079:
1076:
1053:
1050:
1047:
1020:
999:projection map
995:is called the
984:
956:
928:
911:is called the
900:
869:
866:
863:
860:
857:
833:
813:
810:
807:
804:
784:
781:
778:
774:
771:
767:
764:
760:
757:
754:
742:
739:
723:, and others.
691:, topological
685:vector bundles
621:
618:
605:
573:
570:
546:
504:vector bundles
496:tangent bundle
480:trivial bundle
467:
447:
444:
441:
438:
418:
385:
361:
337:
305:
302:
282:
279:
259:
256:
253:
233:
213:
210:
207:
204:
201:
198:
168:
165:
162:
142:
128:
120:
76:
73:
70:
67:
64:
43:A cylindrical
26:
9:
6:
4:
3:
2:
9001:
8990:
8989:Fiber bundles
8987:
8986:
8984:
8969:
8966:
8964:
8963:Supermanifold
8961:
8959:
8956:
8954:
8951:
8947:
8944:
8943:
8942:
8939:
8937:
8934:
8932:
8929:
8927:
8924:
8922:
8919:
8917:
8914:
8912:
8909:
8908:
8906:
8902:
8896:
8893:
8891:
8888:
8886:
8883:
8881:
8878:
8876:
8873:
8871:
8868:
8867:
8865:
8861:
8851:
8848:
8846:
8843:
8841:
8838:
8836:
8833:
8831:
8828:
8826:
8823:
8821:
8818:
8816:
8813:
8811:
8808:
8806:
8803:
8802:
8800:
8798:
8794:
8788:
8785:
8783:
8780:
8778:
8775:
8773:
8770:
8768:
8765:
8763:
8760:
8758:
8754:
8750:
8748:
8745:
8743:
8740:
8738:
8734:
8730:
8728:
8725:
8723:
8720:
8718:
8715:
8713:
8710:
8708:
8705:
8703:
8700:
8699:
8697:
8695:
8691:
8685:
8684:Wedge product
8682:
8680:
8677:
8673:
8670:
8669:
8668:
8665:
8663:
8660:
8656:
8653:
8652:
8651:
8648:
8646:
8643:
8641:
8638:
8636:
8633:
8629:
8628:Vector-valued
8626:
8625:
8624:
8621:
8619:
8616:
8612:
8609:
8608:
8607:
8604:
8602:
8599:
8597:
8594:
8593:
8591:
8587:
8581:
8578:
8576:
8573:
8571:
8568:
8564:
8561:
8560:
8559:
8558:Tangent space
8556:
8554:
8551:
8549:
8546:
8544:
8541:
8540:
8538:
8534:
8531:
8529:
8525:
8519:
8516:
8514:
8510:
8506:
8504:
8501:
8499:
8495:
8491:
8487:
8485:
8482:
8480:
8477:
8475:
8472:
8470:
8467:
8465:
8462:
8460:
8457:
8455:
8452:
8448:
8445:
8444:
8443:
8440:
8438:
8435:
8433:
8430:
8428:
8425:
8423:
8420:
8418:
8415:
8413:
8410:
8408:
8405:
8403:
8400:
8398:
8395:
8393:
8389:
8385:
8383:
8379:
8375:
8373:
8370:
8369:
8367:
8361:
8355:
8352:
8350:
8347:
8345:
8342:
8340:
8337:
8335:
8332:
8330:
8327:
8323:
8322:in Lie theory
8320:
8319:
8318:
8315:
8313:
8310:
8306:
8303:
8302:
8301:
8298:
8296:
8293:
8292:
8290:
8288:
8284:
8278:
8275:
8273:
8270:
8268:
8265:
8263:
8260:
8258:
8255:
8253:
8250:
8248:
8245:
8243:
8240:
8238:
8235:
8234:
8232:
8229:
8225:Main results
8223:
8217:
8214:
8212:
8209:
8207:
8206:Tangent space
8204:
8202:
8199:
8197:
8194:
8192:
8189:
8187:
8184:
8182:
8179:
8175:
8172:
8170:
8167:
8166:
8165:
8162:
8158:
8155:
8154:
8153:
8150:
8149:
8147:
8143:
8138:
8134:
8127:
8122:
8120:
8115:
8113:
8108:
8107:
8104:
8092:
8089:
8087:
8084:
8082:
8079:
8077:
8074:
8072:
8069:
8067:
8064:
8062:
8059:
8057:
8054:
8052:
8049:
8047:
8044:
8042:
8039:
8037:
8034:
8032:
8029:
8028:
8026:
8024:
8020:
8010:
8007:
8005:
8002:
8000:
7997:
7995:
7992:
7990:
7987:
7985:
7982:
7980:
7977:
7975:
7972:
7970:
7967:
7966:
7964:
7960:
7954:
7951:
7949:
7946:
7944:
7941:
7939:
7936:
7934:
7931:
7929:
7928:Metric tensor
7926:
7924:
7921:
7919:
7916:
7915:
7913:
7909:
7906:
7902:
7896:
7893:
7891:
7888:
7886:
7883:
7881:
7878:
7876:
7873:
7871:
7868:
7866:
7863:
7861:
7858:
7856:
7853:
7851:
7848:
7846:
7843:
7841:
7840:Exterior form
7838:
7836:
7833:
7831:
7828:
7826:
7823:
7821:
7818:
7816:
7813:
7811:
7808:
7806:
7803:
7802:
7800:
7794:
7787:
7784:
7782:
7779:
7777:
7774:
7772:
7769:
7767:
7764:
7762:
7759:
7757:
7754:
7752:
7749:
7747:
7744:
7742:
7739:
7737:
7734:
7733:
7731:
7729:
7725:
7719:
7716:
7714:
7713:Tensor bundle
7711:
7709:
7706:
7704:
7701:
7699:
7696:
7694:
7691:
7689:
7686:
7684:
7681:
7679:
7676:
7674:
7671:
7670:
7668:
7662:
7656:
7653:
7651:
7648:
7646:
7643:
7641:
7638:
7636:
7633:
7631:
7628:
7626:
7623:
7621:
7618:
7616:
7613:
7612:
7610:
7606:
7596:
7593:
7591:
7588:
7586:
7583:
7581:
7578:
7576:
7573:
7572:
7570:
7565:
7562:
7560:
7557:
7556:
7553:
7547:
7544:
7542:
7539:
7537:
7534:
7532:
7529:
7527:
7524:
7522:
7519:
7517:
7514:
7512:
7509:
7508:
7506:
7504:
7500:
7497:
7493:
7489:
7488:
7482:
7478:
7471:
7466:
7464:
7459:
7457:
7452:
7451:
7448:
7442:
7438:
7434:
7431:
7429:
7426:
7421:
7420:
7415:
7409:
7406:
7403:
7402:
7393:
7389:
7388:
7383:
7382:"Fibre space"
7378:
7375:
7369:
7365:
7361:
7356:
7353:
7347:
7343:
7342:Fibre Bundles
7339:
7335:
7331:
7327:
7323:
7320:
7314:
7309:
7308:
7301:
7297:
7293:
7289:
7283:
7279:
7275:
7271:
7268:
7262:
7258:
7254:
7250:
7249:
7236:
7226:
7218:
7214:
7209:
7204:
7199:
7194:
7190:
7186:
7182:
7178:
7171:
7167:
7161:
7150:
7145:
7141:
7137:
7130:
7126:
7122:
7118:
7114:
7109:
7104:
7099:
7094:
7090:
7086:
7082:
7078:
7074:
7070:
7066:
7065:
7061:
7054:
7048:
7040:
7036:
7032:
7028:
7024:
7020:
7019:
7014:
7008:
7000:
6996:
6995:
6990:
6984:
6976:
6972:
6971:
6966:
6960:
6952:
6948:
6942:
6934:
6930:
6929:
6924:
6918:
6910:
6906:
6901:
6896:
6891:
6886:
6882:
6878:
6874:
6870:
6869:
6864:
6860:
6854:
6846:
6842:
6837:
6832:
6827:
6822:
6818:
6814:
6810:
6806:
6805:
6800:
6796:
6790:
6782:
6778:
6774:
6772:0-12-634850-2
6768:
6764:
6763:
6755:
6748:
6744:
6739:
6730:
6725:
6721:
6717:
6716:
6711:
6707:
6701:
6697:
6687:
6684:
6682:
6681:Vector bundle
6679:
6677:
6674:
6672:
6669:
6667:
6664:
6662:
6659:
6657:
6654:
6652:
6649:
6647:
6644:
6642:
6639:
6637:
6634:
6632:
6629:
6627:
6624:
6622:
6619:
6617:
6614:
6612:
6609:
6607:
6604:
6602:
6601:Affine bundle
6599:
6598:
6588:
6584:
6580:
6577:
6573:
6569:
6553:
6547:
6544:
6541:
6534:is a mapping
6533:
6529:
6526:
6522:
6518:
6514:
6513:
6507:
6505:
6501:
6485:
6465:
6462:
6459:
6456:
6433:
6425:
6422:
6418:
6409:
6406:
6402:
6398:
6382:
6376:
6373:
6370:
6363:, is that if
6362:
6358:
6354:
6351:
6332:
6324:
6321:
6317:
6308:
6292:
6286:
6283:
6280:
6260:
6237:
6234:
6231:
6228:
6225:
6222:
6219:
6213:
6207:
6204:
6201:
6198:
6195:
6192:
6189:
6178:
6162:
6156:
6153:
6150:
6142:
6138:
6134:
6130:
6125:
6123:
6104:
6101:
6098:
6095:
6092:
6081:
6077:
6061:
6055:
6052:
6049:
6041:
6037:
6027:
6013:
5991:
5978:
5975:
5955:
5949:
5946:
5941:
5937:
5916:
5910:
5907:
5902:
5898:
5874:
5870:
5867:
5856:
5852:
5836:
5830:
5827:
5822:
5818:
5797:
5791:
5788:
5783:
5779:
5766:
5762:
5746:
5733:
5730:
5722:
5718:
5703:
5697:
5694:
5691:
5671:
5668:
5665:
5660:
5656:
5652:
5647:
5643:
5622:
5616:
5613:
5610:
5590:
5584:
5581:
5576:
5572:
5551:
5545:
5542:
5537:
5533:
5524:
5520:
5516:
5511:
5498:
5495:
5492:
5489:
5469:
5466:
5463:
5443:
5437:
5431:
5428:
5422:
5419:
5413:
5405:
5401:
5385:
5379:
5376:
5373:
5365:
5361:
5357:
5353:
5349:
5341:
5337:
5335:
5319:
5314:
5310:
5306:
5303:
5300:
5297:
5294:
5289:
5285:
5264:
5258:
5255:
5252:
5248:
5245:
5239:
5236:
5233:
5225:
5217:
5209:
5205:
5189:
5183:
5180:
5175:
5171:
5150:
5144:
5141:
5136:
5132:
5123:
5119:
5113:
5103:
5101:
5097:
5093:
5089:
5085:
5082:on the fiber
5081:
5077:
5073:
5069:
5065:
5061:
5059:
5052:
5050:
5046:
5044:
5039:
5032:
5025:
5002:
4996:
4988:
4985:
4981:
4974:
4966:
4963:
4959:
4955:
4949:
4941:
4938:
4934:
4926:
4909:
4906:
4898:
4890:
4887:
4883:
4879:
4873:
4865:
4862:
4858:
4850:
4835:
4832:
4826:
4818:
4815:
4811:
4803:
4802:
4801:
4785:
4782:
4778:
4768:
4766:
4762:
4758:
4754:
4749:
4747:
4743:
4739:
4731:
4727:
4723:
4721:
4716:
4712:
4708:
4688:
4680:
4676:
4672:
4667:
4663:
4659:
4654:
4651:
4647:
4625:
4621:
4615:
4607:
4604:
4600:
4595:
4592:
4588:
4584:
4578:
4574:
4571:
4563:
4560:
4555:
4551:
4545:
4541:
4520:
4517:
4513:
4507:
4503:
4499:
4494:
4490:
4485:
4478:
4475:
4471:
4465:
4461:
4457:
4452:
4448:
4443:
4439:
4434:
4431:
4426:
4422:
4416:
4412:
4404:the function
4386:
4382:
4377:
4372:
4368:
4339:
4335:
4330:
4325:
4321:
4295:
4291:
4287:
4282:
4278:
4249:
4245:
4240:
4235:
4231:
4201:
4198:
4195:
4192:
4189:
4186:
4183:
4172:
4171:
4167:
4162:
4158:
4154:
4150:
4146:
4142:
4138:
4134:
4130:
4126:
4116:
4114:
4098:
4092:
4084:
4080:
4076:
4057:
4053:
4050:
4039:
4035:
4019:
4016:
4007:
4001:
3995:
3987:
3983:
3979:
3963:
3957:
3954:
3951:
3943:
3942:local section
3938:
3936:
3932:
3928:
3924:
3919:
3917:
3913:
3909:
3905:
3901:
3898:
3883:
3880:
3871:
3865:
3859:
3839:
3833:
3830:
3827:
3807:
3799:
3798:cross section
3795:
3785:
3775:
3761:
3757:
3753:
3747:
3727:
3707:
3698:
3696:
3695:diffeomorphic
3677:
3671:
3667:
3660:
3654:
3651:
3628:
3622:
3615:
3611:
3607:
3603:
3584:
3578:
3575:
3568:
3550:
3546:
3523:
3519:
3496:
3492:
3469:
3465:
3456:
3452:
3443:
3439:
3435:
3419:
3411:
3395:
3387:
3382:
3380:
3379:Steenrod 1951
3376:
3360:
3340:
3336:
3333:
3329:
3326:
3322:
3318:
3314:
3311:
3303:
3287:
3267:
3263:
3259:
3253:
3250:
3247:
3227:
3223:
3219:
3212:
3208:
3192:
3184:
3168:
3154:
3152:
3148:
3132:
3129:
3121:
3103:
3099:
3091:
3090:mapping torus
3087:
3086:homeomorphism
3071:
3065:
3062:
3059:
3051:
3035:
3020:
3019:Wang sequence
3015:
3013:
3009:
2993:
2985:
2981:
2980:circle bundle
2965:
2962:
2959:
2951:
2936:
2933:
2930:
2922:
2917:
2915:
2914:
2897:
2894:
2872:
2868:
2859:
2843:
2835:
2831:
2827:
2811:
2803:
2801:
2796:
2795:sphere bundle
2790:
2789:Sphere bundle
2780:
2778:
2759:
2748:
2742:
2736:
2716:
2696:
2676:
2669:
2653:
2633:
2625:
2609:
2601:
2598:
2594:
2590:
2589:
2583:
2581:
2577:
2573:
2569:
2565:
2561:
2560:vector spaces
2557:
2556:
2545:
2543:
2539:
2535:
2534:
2515:
2511:
2506:
2501:
2497:
2496:
2493:
2477:
2473:
2469:
2464:
2460:
2451:
2447:
2437:
2435:
2419:
2416:
2413:
2393:
2385:
2369:
2360:
2346:
2323:
2315:
2312:
2308:
2300:
2296:
2292:
2276:
2273:
2270:
2250:
2247:
2241:
2235:
2215:
2208:
2192:
2184:
2168:
2160:
2157:. It has the
2156:
2140:
2127:
2113:
2111:
2108:
2104:
2096:
2080:
2060:
2057:
2037:
2017:
1997:
1991:
1988:
1985:
1965:
1962:
1959:
1956:
1953:
1934:
1932:
1928:
1912:
1892:
1889:
1886:
1883:
1875:
1871:
1867:
1858:
1856:
1847:
1840:
1838:
1814:
1802:
1794:
1786:
1780:
1763:
1762:
1759:
1742:
1738:
1735:
1731:
1728:
1724:
1721:
1709:
1696:
1693:
1685:
1669:
1661:
1645:
1639:
1636:
1633:
1625:
1612:
1609:
1581:
1572:
1567:
1564:
1559:
1555:
1534:
1508:
1497:
1494:
1490:
1482:
1466:
1463:
1460:
1453:Thus for any
1451:
1449:
1428:
1424:
1418:
1414:
1409:
1404:
1400:
1395:
1391:
1382:
1366:
1363:
1360:
1351:
1343:
1340:
1336:
1332:
1329:
1309:
1303:
1300:
1297:
1294:
1289:
1285:
1271:
1267:
1265:
1249:
1229:
1226:
1223:
1215:
1212:is given the
1196:
1188:
1185:
1181:
1160:
1157:
1154:
1145:
1137:
1134:
1130:
1126:
1123:
1116:
1115:homeomorphism
1100:
1080:
1077:
1074:
1067:
1051:
1048:
1045:
1036:
1034:
1018:
1010:
1002:
982:
974:
954:
946:
926:
918:
898:
890:
887:satisfying a
886:
883:
867:
861:
858:
855:
847:
831:
811:
808:
805:
802:
782:
776:
772:
769:
765:
762:
758:
755:
738:
736:
732:
731:sphere bundle
727:
724:
722:
718:
714:
710:
706:
702:
698:
694:
690:
686:
681:
679:
678:sphere bundle
675:
671:
668:was given by
667:
663:
659:
655:
651:
647:
643:
642:
637:
633:
632:
627:
617:
603:
595:
591:
587:
571:
568:
560:
544:
536:
532:
528:
524:
519:
517:
513:
509:
505:
501:
497:
493:
489:
485:
481:
465:
445:
442:
439:
436:
416:
408:
407:
401:
399:
383:
375:
359:
351:
335:
327:
326:
321:
320:
303:
300:
280:
277:
257:
254:
251:
231:
211:
208:
202:
199:
196:
188:
185:
182:
166:
163:
160:
140:
132:
126:
124:
123:product space
118:
116:
112:
108:
107:
102:
98:
94:
74:
68:
65:
62:
54:
50:
46:
41:
37:
33:
19:
8890:Moving frame
8885:Morse theory
8875:Gauge theory
8726:
8693:
8667:Tensor field
8596:Closed/Exact
8575:Vector field
8543:Distribution
8484:Hypercomplex
8479:Quaternionic
8216:Vector field
8174:Smooth atlas
8091:Hermann Weyl
7895:Vector space
7880:Pseudotensor
7845:Fiber bundle
7844:
7798:abstractions
7693:Mixed tensor
7678:Tensor field
7485:
7417:
7407:, PlanetMath
7405:Fiber Bundle
7385:
7359:
7341:
7329:
7306:
7277:
7256:
7235:
7225:
7180:
7176:
7160:
7139:
7135:
7080:
7076:
7060:
7047:
7022:
7016:
7007:
7001:: 1755–1757.
6998:
6992:
6983:
6977:: 1611–1612.
6974:
6968:
6959:
6950:
6941:
6935:: 1621–1623.
6932:
6926:
6917:
6872:
6866:
6853:
6808:
6802:
6789:
6761:
6754:
6738:
6719:
6713:
6700:
6636:Gauge theory
6616:Covering map
6404:
6400:
6356:
6352:
6176:
6132:
6128:
6126:
6120:is called a
6079:
6075:
6033:
5850:
5770:
5720:
5522:
5518:
5514:
5512:
5403:
5399:
5359:
5351:
5347:
5345:
5219:
5211:
5207:
5203:
5121:
5117:
5115:
5099:
5095:
5091:
5083:
5079:
5075:
5067:
5063:
5057:
5053:
5041:
5034:
5027:
5020:
5018:
4769:
4764:
4756:
4752:
4750:
4733:
4729:
4725:
4719:
4718:
4714:
4710:
4702:
4533:is given by
4165:
4164:
4160:
4152:
4144:
4140:
4128:
4122:
4078:
4037:
4033:
3985:
3977:
3941:
3939:
3925:, where the
3920:
3903:
3899:
3797:
3789:
3787:
3699:
3614:circle group
3604:subgroup of
3434:Lie subgroup
3386:quotient map
3383:
3160:
3027:
3024:Mapping tori
2918:
2911:
2858:unit vectors
2833:
2799:
2794:
2792:
2586:
2584:
2576:frame bundle
2564:linear group
2553:
2551:
2531:
2529:
2526:Covering map
2446:Klein bottle
2443:
2440:Klein bottle
2361:
2207:neighborhood
2183:line segment
2155:Möbius strip
2132:
2121:Möbius strip
2112:is trivial.
2107:contractible
2098:
2094:
1945:
1861:
1859:
1852:
1841:
1710:
1682:carries the
1601:
1452:
1443:
1442:is called a
1276:
1066:neighborhood
1037:
1004:
996:
968:
940:
912:
888:
744:
728:
725:
682:
677:
674:sphere space
673:
665:
661:
657:
653:
645:
640:
639:
635:
630:
629:
628:, the terms
623:
593:
557:is called a
520:
488:Klein bottle
484:Möbius strip
479:
458:and the map
404:
402:
397:
373:
349:
324:
317:
111:fibre bundle
110:
105:
101:fiber bundle
100:
90:
49:fiber bundle
48:
36:
8835:Levi-Civita
8825:Generalized
8797:Connections
8747:Lie algebra
8679:Volume form
8580:Vector flow
8553:Pushforward
8548:Lie bracket
8447:Lie algebra
8412:G-structure
8201:Pushforward
8181:Submanifold
8031:Élie Cartan
7979:Spin tensor
7953:Weyl tensor
7911:Mathematics
7875:Multivector
7666:definitions
7564:Engineering
7503:Mathematics
6722:: 147–238.
6641:Hopf bundle
6504:Michor 2008
6040:submersions
5855:isomorphism
5853:. A bundle
5723:. That is,
5364:equivariant
5334:commutative
5106:Bundle maps
4746:gauge group
3927:Euler class
3908:obstruction
3122:with fiber
3010:called the
2984:Chern class
2921:Euler class
2602:by a group
2432:would be a
1931:smooth maps
1876:. That is,
1602:fiber over
943:total space
711:, Whitney,
666:fiber space
641:fiber space
523:bundle maps
350:total space
316:called the
93:mathematics
18:Total space
8958:Stratifold
8916:Diffeology
8712:Associated
8513:Symplectic
8498:Riemannian
8427:Hyperbolic
8354:Submersion
8262:Hopf–Rinow
8196:Submersion
8191:Smooth map
7860:Linear map
7728:Operations
7246:References
7230:morphisms.
7055:, Preface)
6500:compatible
6397:submersion
6307:proper map
5968:such that
5635:such that
5277:such that
5214:bundle map
5112:Bundle map
5056:principal
4717:-atlas. A
4149:faithfully
3852:such that
3610:isomorphic
3017:See also:
2950:cohomology
2597:transitive
2110:CW-complex
975:. The map
915:base space
885:surjection
882:continuous
705:Heinz Hopf
693:fibrations
654:base space
525:, and the
374:base space
325:submersion
319:projection
184:surjective
181:continuous
8840:Principal
8815:Ehresmann
8772:Subbundle
8762:Principal
8737:Fibration
8717:Cotangent
8589:Covectors
8442:Lie group
8422:Hermitian
8365:manifolds
8334:Immersion
8329:Foliation
8267:Noether's
8252:Frobenius
8247:De Rham's
8242:Darboux's
8133:Manifolds
7999:EM tensor
7835:Dimension
7786:Transpose
7441:0908.1886
7419:MathWorld
7392:EMS Press
6631:Fibration
6551:→
6542:π
6532:fibration
6498:admits a
6460:∈
6423:−
6380:→
6322:−
6290:→
6202:π
6160:→
6141:connected
6059:→
6014:φ
5979:≡
5953:→
5938:π
5914:→
5899:π
5868:φ
5834:→
5819:π
5795:→
5780:π
5734:≡
5701:→
5692:φ
5669:φ
5666:∘
5657:π
5644:π
5620:→
5611:φ
5603:is a map
5588:→
5573:π
5549:→
5534:π
5493:∈
5467:∈
5432:φ
5414:φ
5383:→
5374:φ
5356:morphisms
5311:π
5307:∘
5298:φ
5295:∘
5286:π
5262:→
5243:→
5234:φ
5187:→
5172:π
5148:→
5133:π
5045:condition
4907:−
4761:Lie group
4686:→
4673:∩
4622:ξ
4579:ξ
4561:−
4552:φ
4542:φ
4518:×
4500:∩
4482:→
4476:×
4458:∩
4432:−
4423:φ
4413:φ
4383:φ
4336:φ
4292:φ
4279:φ
4246:φ
4196:π
4096:→
4058:φ
3996:π
3961:→
3860:π
3837:→
3808:π
3751:→
3462:→
3410:Lie group
3361:π
3334:π
3257:→
3248:π
3088:then the
3069:→
2749:⊆
2737:ρ
2677:ρ
2470:×
2417:×
2370:φ
2313:−
2309:π
2274:∈
2248:∈
2236:π
2018:π
1995:→
1986:π
1963:×
1927:functions
1803:π
1784:⟶
1736:π
1694:π
1643:→
1634:π
1573:
1565:−
1495:−
1491:π
1464:∈
1415:φ
1364:×
1358:→
1341:−
1337:π
1330:φ
1307:→
1301:×
1250:π
1227:×
1186:−
1182:π
1158:×
1152:→
1135:−
1131:π
1124:φ
1078:⊆
1049:∈
1033:connected
983:π
865:→
856:π
770:π
735:dimension
634:(German:
466:π
440:×
301:π
255:×
206:→
197:π
164:×
72:→
63:π
45:hairbrush
8983:Category
8936:Orbifold
8931:K-theory
8921:Diffiety
8645:Pullback
8459:Oriented
8437:Kenmotsu
8417:Hadamard
8363:Types of
8312:Geodesic
8137:Glossary
7865:Manifold
7850:Geodesic
7608:Notation
7340:(1994),
7296:40734875
7255:(1951),
7217:16588328
7168:(1940).
7127:(1937).
7117:16588001
7071:(1935).
6909:16588328
6861:(1940).
6845:16588001
6797:(1935).
6708:(1933).
6646:I-bundle
6594:See also
6506:, §17).
5890:between
5456:for all
5398:is also
4032:for all
3982:open set
3935:2-sphere
3778:Sections
3147:surfaces
2517:A torus.
2505:immersed
2434:cylinder
2299:preimage
2289:) is an
2095:globally
1978:and let
1937:Examples
1870:category
1795:→
1660:open map
1481:preimage
626:topology
531:category
514:, as do
500:manifold
429:is just
293:The map
127:globally
117:that is
97:topology
53:bristles
8880:History
8863:Related
8777:Tangent
8755:)
8735:)
8702:Adjoint
8694:Bundles
8672:density
8570:Torsion
8536:Vectors
8528:Tensors
8511:)
8496:)
8492:,
8490:Pseudo−
8469:Poisson
8402:Finsler
8397:Fibered
8392:Contact
8390:)
8382:Complex
8380:)
8349:Section
7962:Physics
7796:Related
7559:Physics
7477:Tensors
7208:1078023
7185:Bibcode
7108:1076627
7085:Bibcode
7039:1969485
6900:1078023
6877:Bibcode
6836:1076627
6813:Bibcode
6781:5831391
6137:compact
5088:regular
5060:-bundle
5043:cocycle
4742:physics
4722:-bundle
4147:to act
3933:of the
3897:for all
3792:section
3612:to the
3602:abelian
3381:, §7).
3373:admits
2824:with a
2802:-sphere
2263:(where
2153:is the
2050:) over
1383:of all
1264:commute
1173:(where
620:History
559:section
406:trivial
403:In the
372:as the
119:locally
113:) is a
8845:Vector
8830:Koszul
8810:Cartan
8805:Affine
8787:Vector
8782:Tensor
8767:Spinor
8757:Normal
8753:Stable
8707:Affine
8611:bundle
8563:bundle
8509:Almost
8432:Kähler
8388:Almost
8378:Almost
8372:Closed
8272:Sard's
8228:(list)
7890:Vector
7885:Spinor
7870:Matrix
7664:Tensor
7370:
7348:
7315:
7294:
7284:
7263:
7215:
7205:
7115:
7105:
7037:
6907:
6897:
6843:
6833:
6779:
6769:
6517:bundle
6350:subset
5717:covers
4709:. Two
4639:where
3980:is an
3976:where
3600:. The
3120:circle
2826:metric
2600:action
2295:length
2181:and a
2159:circle
1812:
1806:
1800:
1790:
1658:is an
1479:, the
1277:where
1216:, and
947:, and
795:where
638:) and
409:case,
376:, and
125:, but
8953:Sheaf
8727:Fiber
8503:Rizza
8474:Prime
8305:Local
8295:Curve
8157:Atlas
7810:Basis
7495:Scope
7437:arXiv
7173:(PDF)
7132:(PDF)
7035:JSTOR
6693:Notes
6583:range
6478:then
6399:with
5094:with
5070:is a
5062:is a
5047:(see
4759:is a
4170:atlas
4135:that
4131:be a
4125:group
4113:sheaf
4075:chart
4040:. If
3408:is a
3304:for (
3205:is a
3181:is a
3084:is a
3048:is a
2580:bases
2450:torus
2073:Here
971:fiber
880:is a
636:Faser
631:fiber
527:class
498:of a
398:fiber
115:space
8820:Form
8722:Dual
8655:flow
8518:Tame
8494:Sub−
8407:Flat
8287:Maps
7368:ISBN
7346:ISBN
7313:ISBN
7292:OCLC
7282:ISBN
7261:ISBN
7213:PMID
7113:PMID
7051:See
6905:PMID
6841:PMID
6777:OCLC
6767:ISBN
6523:and
6403:and
6139:and
6135:are
6131:and
5929:and
5810:and
5517:and
5482:and
5206:and
5163:and
5120:and
4357:and
4163:. A
4137:acts
3988:and
3796:(or
3412:and
3300:. A
3185:and
3052:and
2595:and
2593:free
2570:and
1946:Let
1905:and
1556:proj
1286:proj
1003:(or
967:the
939:the
848:and
844:are
824:and
695:and
510:and
486:and
396:the
99:, a
8742:Jet
7203:PMC
7193:doi
7144:doi
7103:PMC
7093:doi
7027:doi
6999:240
6975:224
6933:208
6895:PMC
6885:doi
6831:PMC
6821:doi
6745:on
6724:doi
6355:of
6127:If
5564:to
5218:or
4744:is
4159:of
4151:on
4036:in
3984:in
3914:in
3902:in
3693:is
3608:is
3436:by
3161:If
3028:If
2860:in
2753:Aut
2689:of
2578:of
2382:in
2359:).
2291:arc
2228:of
1872:of
1381:set
1093:of
1031:is
624:In
561:of
322:or
270:to
187:map
91:In
8985::
8733:Co
7416:.
7390:,
7384:,
7362:,
7290:.
7211:.
7201:.
7191:.
7181:26
7179:.
7175:.
7140:43
7138:.
7134:.
7111:.
7101:.
7091:.
7081:21
7079:.
7075:.
7033:.
7023:54
7021:.
6997:.
6973:.
6931:.
6903:.
6893:.
6883:.
6873:26
6871:.
6865:.
6839:.
6829:.
6819:.
6809:21
6807:.
6801:.
6775:.
6720:60
6718:.
6712:.
5336::
5054:A
5033:∩
5026:∩
4748:.
4115:.
3918:.
3788:A
3444:,
3153:.
3014:.
2916:.
2793:A
2779:.
2544:.
2530:A
2492:.
2452:,
1933:.
1860:A
1266::
1035:.
737:.
719:,
715:,
707:,
703:,
687:,
680:.
616:.
518:.
400:.
189:,
121:a
109::
8751:(
8731:(
8507:(
8488:(
8386:(
8376:(
8139:)
8135:(
8125:e
8118:t
8111:v
7469:e
7462:t
7455:v
7439::
7422:.
7298:.
7219:.
7195::
7187::
7152:.
7146::
7119:.
7095::
7087::
7041:.
7029::
6911:.
6887::
6879::
6847:.
6823::
6815::
6783:.
6749:.
6732:.
6726::
6589:.
6554:B
6548:E
6545::
6527:.
6486:f
6466:,
6463:N
6457:x
6437:}
6434:x
6431:{
6426:1
6419:f
6405:N
6401:M
6383:N
6377:M
6374::
6371:f
6357:N
6353:K
6336:)
6333:K
6330:(
6325:1
6318:f
6293:N
6287:M
6284::
6281:f
6261:f
6241:)
6238:F
6235:,
6232:f
6229:,
6226:N
6223:,
6220:M
6217:(
6214:=
6211:)
6208:F
6205:,
6199:,
6196:B
6193:,
6190:E
6187:(
6177:F
6163:N
6157:M
6154::
6151:f
6133:N
6129:M
6108:)
6105:f
6102:,
6099:N
6096:,
6093:M
6090:(
6080:N
6076:M
6062:N
6056:M
6053::
6050:f
5992:M
5987:d
5984:i
5976:f
5956:M
5950:F
5947::
5942:F
5917:M
5911:E
5908::
5903:E
5878:)
5875:f
5871:,
5865:(
5851:M
5837:M
5831:F
5828::
5823:F
5798:M
5792:E
5789::
5784:E
5747:M
5742:d
5739:i
5731:f
5721:M
5704:F
5698:E
5695::
5672:.
5661:F
5653:=
5648:E
5623:F
5617:E
5614::
5591:M
5585:F
5582::
5577:F
5552:M
5546:E
5543::
5538:E
5523:M
5519:N
5515:M
5499:.
5496:G
5490:s
5470:E
5464:x
5444:s
5441:)
5438:x
5435:(
5429:=
5426:)
5423:s
5420:x
5417:(
5404:G
5400:G
5386:F
5380:E
5377::
5362:-
5360:G
5352:G
5348:G
5320:.
5315:E
5304:f
5301:=
5290:F
5265:N
5259:M
5256::
5253:f
5249:,
5246:F
5240:E
5237::
5208:N
5204:M
5190:N
5184:F
5181::
5176:F
5151:M
5145:E
5142::
5137:E
5122:N
5118:M
5100:G
5096:G
5092:F
5084:F
5080:G
5076:G
5068:F
5064:G
5058:G
5037:k
5035:U
5030:j
5028:U
5023:i
5021:U
5003:.
5000:)
4997:x
4994:(
4989:k
4986:j
4982:t
4978:)
4975:x
4972:(
4967:j
4964:i
4960:t
4956:=
4953:)
4950:x
4947:(
4942:k
4939:i
4935:t
4910:1
4903:)
4899:x
4896:(
4891:i
4888:j
4884:t
4880:=
4877:)
4874:x
4871:(
4866:j
4863:i
4859:t
4836:1
4833:=
4830:)
4827:x
4824:(
4819:i
4816:i
4812:t
4786:j
4783:i
4779:t
4765:F
4757:G
4753:G
4730:G
4726:G
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4711:G
4689:G
4681:j
4677:U
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4660::
4655:j
4652:i
4648:t
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4616:x
4613:(
4608:j
4605:i
4601:t
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4593:x
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4585:=
4582:)
4575:,
4572:x
4569:(
4564:1
4556:j
4546:i
4521:F
4514:)
4508:j
4504:U
4495:i
4491:U
4486:(
4479:F
4472:)
4466:j
4462:U
4453:i
4449:U
4444:(
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