5336:
2160:
4127:
1993:
31:
4195:
2478:
543:
3329:
2313:
2601:
5883:
3872:. Restricting to vector bundles for which the spaces are manifolds (and the bundle projections are smooth maps) and smooth bundle morphisms we obtain the category of smooth vector bundles. Vector bundle morphisms are a special case of the notion of a
2473:{\displaystyle {\begin{aligned}\varphi _{U}\colon U\times \mathbb {R} ^{k}&\mathrel {\xrightarrow {\cong } } \pi ^{-1}(U),\\\varphi _{V}\colon V\times \mathbb {R} ^{k}&\mathrel {\xrightarrow {\cong } } \pi ^{-1}(V)\end{aligned}}}
2714:
6234:
1144:
5564:
2962:
2814:
5307:
5245:
is obtained. Specifically, one must require that the local trivializations are Banach space isomorphisms (rather than just linear isomorphisms) on each of the fibers and that, furthermore, the transitions
3649:
A subbundle of a trivial bundle need not be trivial, and indeed every real vector bundle over a compact space can be viewed as a subbundle of a trivial bundle of sufficiently high rank. For example, the
5323:, those whose fibers are vector spaces and whose cocycle respects the vector space structure. More general fiber bundles can be constructed in which the fiber may have other structures; for example
2492:
2318:
1938:
3578:
3035:
3244:
3088:
1243:
5748:
1897:
1529:
6629:
1339:
3492:
6079:
3618:
2153:
2115:
2085:
3304:
1597:
1420:
1375:
1295:
957:
3275:
3188:
3153:
2748:
4681:
There are many functorial operations which can be performed on pairs of vector spaces (over the same field), and these extend straightforwardly to pairs of vector bundles
1029:
3426:
2024:
852:
751:
3455:
2051:
1663:
669:
459:
1464:
3644:
3122:
2245:
331:
7922:
2997:
1852:
1718:
1627:
716:
611:
3512:
1494:
230:
7113:
6679:
4188:
4168:
4148:
3532:
3390:
3370:
3350:
2305:
2285:
2265:
1982:
1962:
1824:
1802:
1778:
1758:
1738:
1691:
1549:
1444:
1266:
1187:
1167:
1072:
1049:
1000:
980:
899:
879:
807:
783:
689:
631:
584:
564:
479:
433:
413:
393:
373:
353:
296:
272:
250:
201:
181:
149:
129:
7917:
5221:
4307:
a vector from the attached vector space, in a continuous manner. As an example, sections of the tangent bundle of a differential manifold are nothing but
7204:
499:
to the manifold at that point. Tangent bundles are not, in general, trivial bundles. For example, the tangent bundle of the sphere is non-trivial by the
2612:
7228:
7423:
5339:
The regularity of transition functions describing a vector bundle determines the type of the vector bundle. If the continuous transition functions
6116:
4964:. Though this construction is natural, unless care is taken, the resulting object will not have local trivializations. Consider the case of
7293:
7029:
6989:
6964:
6938:
6900:
1083:
7519:
5501:
5220:
in the fibers. More generally, one can typically understand the additional structure imposed on a vector bundle in terms of the resulting
4010:
3779:
7572:
7100:
7856:
2846:
2756:
7010:
6449:
5252:
5216:, which may also be obtained by replacing real vector spaces in the definition with complex ones and requiring that all mappings be
5030:
Each of these operations is a particular example of a general feature of bundles: that many operations that can be performed on the
3311:
2155:). Different choices of transition functions may result in different vector bundles which are non-trivial after gluing is complete.
7621:
4972:
having isolated zeroes. The fiber over these zeroes in the resulting "eigenbundle" will be isomorphic to the fiber over them in
7213:
7604:
6981:
2596:{\displaystyle \varphi _{U}^{-1}\circ \varphi _{V}\colon (U\cap V)\times \mathbb {R} ^{k}\to (U\cap V)\times \mathbb {R} ^{k}}
531:
6587:
7816:
7057:
6884:
5471:
A smooth vector bundle can be characterized by the fact that it admits transition functions as described above which are
530:
can be viewed as real vector bundles with additional structure. In the following, we focus on real vector bundles in the
7801:
7524:
7298:
3396:
One simple method of constructing vector bundles is by taking subbundles of other vector bundles. Given a vector bundle
7846:
7052:
6085:
characterizes completely the smooth vector bundle structure in the following manner. As a preparation, note that when
5410:
7851:
7821:
7529:
7485:
7466:
7233:
7177:
6812:
6806:
6475:
1902:
278:
The simplest example is the case that the family of vector spaces is constant, i.e., there is a fixed vector space
5878:{\displaystyle \operatorname {vl} _{v}w:=\left.{\frac {d}{dt}}\right|_{t=0}f(v+tw),\quad f\in C^{\infty }(E_{x}).}
3537:
3002:
7388:
7253:
6835:
5619:
5416:
5031:
4525:
4521:
3196:
3040:
1195:
7773:
7638:
7330:
7172:
5217:
5209:
5181:
4024:
719:
1867:
1499:
3992:. As morphisms in this category we take those morphisms of vector bundles whose map on the base space is the
7965:
7470:
7440:
7364:
7354:
7310:
7140:
7093:
6825:
5635:
4586:
1670:
1303:
504:
7238:
3463:
7811:
7430:
7325:
7145:
6815:: the general study of connections on vector bundles and principal bundles and their relations to physics.
6578:
5197:
5193:
4243:
3869:
1630:
6036:
7460:
7455:
6634:
6574:
5213:
4769:
4734:
4366:
3583:
2118:
527:
5335:
2128:
2090:
2060:
3280:
1557:
1380:
1351:
1271:
917:
7791:
7729:
7577:
7281:
7271:
7243:
7218:
7128:
6784:
6746:
6737:
6682:
6240:
4574:
4437:
4429:
3458:
1468:
634:
3249:
3162:
3127:
2722:
7929:
7902:
7611:
7489:
7474:
7403:
7162:
7047:
6789:
5192:
Vector bundles are often given more structure. For instance, vector bundles may be equipped with a
4378:
4001:
817:
109:
1008:
7871:
7826:
7723:
7594:
7398:
7223:
7086:
6888:
6729:
4474:
4433:
4203:
3399:
2484:
2002:
825:
724:
500:
7408:
6490:. The vector bundle operations in this secondary vector bundle structure are the push-forwards +
3434:
2029:
1635:
526:, in which case the vector bundle is said to be a real or complex vector bundle (respectively).
640:
438:
7806:
7786:
7781:
7688:
7599:
7413:
7393:
7248:
7187:
7025:
7006:
6985:
6960:
6934:
6896:
6771:
6570:
5624:
5603:
5225:
4627:
3993:
3307:
2833:
2159:
2054:
1862:
1449:
1003:
232:
in such a way that these vector spaces fit together to form another space of the same kind as
160:
152:
81:
7944:
7738:
7693:
7616:
7587:
7445:
7378:
7373:
7368:
7358:
7150:
7133:
6775:
6697:
6031:
5487:
is smooth if it admits a covering by trivializing open sets such that for any two such sets
4589:
on vector spaces can be extended to vector bundles by performing the vector space operation
4552:
4542:
4126:
4020:
3623:
3097:
2224:
2214:
1666:
1246:
301:
2970:
1830:
1827:. Often the definition of a vector bundle includes that the rank is well defined, so that
1696:
1605:
694:
589:
7887:
7796:
7626:
7582:
7348:
6956:
6930:
6750:
6690:
6562:
5640:
5313:
5205:
5044:
4702:
4105:
3902:
3658:
over the circle, can be seen as a subbundle of the trivial rank 2 bundle over the circle.
3497:
3429:
1479:
902:
206:
2837:
2164:
35:
7000:
6914:
7753:
7678:
7648:
7546:
7539:
7479:
7450:
7320:
7315:
7276:
6733:
6582:
5440:
5402:
5039:
4781:
4567:
4173:
4153:
4133:
4027:
of a morphism of vector bundles is in general not a vector bundle in any natural way.)
3651:
3517:
3375:
3355:
3335:
2821:
2290:
2270:
2250:
1967:
1947:
1809:
1787:
1763:
1743:
1723:
1676:
1534:
1429:
1251:
1172:
1152:
1057:
1052:
1034:
985:
965:
884:
864:
792:
768:
674:
616:
569:
549:
523:
488:
464:
418:
398:
378:
358:
338:
281:
257:
235:
186:
166:
134:
114:
101:
69:
3314:
for vector bundles, and can be taken as an alternative definition of a vector bundle.
7959:
7939:
7758:
7743:
7733:
7683:
7660:
7534:
7494:
7435:
7383:
7182:
6910:
6794:
6689:
is a version of this construction which considers real vector bundles. K-theory with
6566:
5423:
5324:
5241:
5147:
4199:
2122:
1075:
496:
54:
6975:
1992:
7866:
7861:
7703:
7670:
7643:
7551:
7192:
7068:
6830:
6767:
5570:
5320:
5236:
4891:
4308:
3877:
3156:
3091:
912:
908:
515:
482:
252:(e.g. a topological space, manifold, or algebraic variety), which is then called a
105:
2709:{\displaystyle \varphi _{U}^{-1}\circ \varphi _{V}(x,v)=\left(x,g_{UV}(x)v\right)}
30:
5316:. In the corresponding theory for C bundles, all mappings are required to be C.
4377:) becomes itself a real vector space. The collection of these vector spaces is a
7709:
7698:
7655:
7556:
7157:
6779:
6745:, as well as the K-theory groups of vector bundles on the scheme with the above
6374:
6366:
4980:
4935:
4614:
4351:
4009:
3988:
We can also consider the category of all vector bundles over a fixed base space
3824:
3778:
3655:
1855:
1345:
905:
519:
97:
89:
39:
2213:, the Möbius strip. This can be visualised as a "twisting" of one of the local
7934:
7892:
7718:
7631:
7263:
7167:
6948:
6701:
5430:
5406:
4675:
4674:: the key point here is that the operation of taking the dual vector space is
4194:
3873:
3832:
1342:
820:
17:
7748:
7713:
7418:
7305:
6686:
6229:{\displaystyle C_{x}(X):T_{x}M\to T_{x}M;\quad C_{x}(X)Y=(\nabla _{Y}X)_{x}}
5578:
4976:, while everywhere else the fiber is the trivial 0-dimensional vector space.
4856:
4663:
4362:
3323:
1600:
542:
4477:
ones do. (The reason: locally we are looking for sections of a projection
2167:
can be constructed by a non-trivial gluing of two trivial bundles on open
962:
where the following compatibility condition is satisfied: for every point
7912:
7907:
7897:
7288:
7109:
6749:. The two constructs are the same provided that the underlying scheme is
6358:
6081:
given by the fibrewise scalar multiplication. The canonical vector field
5390:
4556:
3668:
2179:
1997:
492:
156:
43:
7078:
5035:
3868:
The class of all vector bundles together with bundle morphisms forms a
2117:
which serve to stick the shaded grey regions together after applying a
1139:{\displaystyle \varphi \colon U\times \mathbb {R} ^{k}\to \pi ^{-1}(U)}
333:
5350:
is only continuous but not smooth. If the smooth transition functions
3328:
7504:
5559:{\displaystyle g_{UV}:U\cap V\to \operatorname {GL} (k,\mathbb {R} )}
5184:
over the infinite real projective space does not have this property.
5158:
is a direct summand of a trivial bundle; i.e., there exists a bundle
3981:). The definition of a vector bundle shows that any vector bundle is
2193:) one obtains the trivial bundle, but with the non-trivial gluing of
2168:
6030:), and it can also be defined as the infinitesimal generator of the
2437:
2361:
7073:
5611:(this requires the matrix group to have a real analytic structure),
4551:; this latter category is abelian, so this is where we can compute
4450:
denotes the structure sheaf of continuous real-valued functions on
7064:
6404:
satisfying 1–4, then there is a unique vector bundle structure on
4502:
4193:
487:. A more complicated (and prototypical) class of examples are the
29:
7074:
Why is it useful to classify the vector bundles of a space ?
6809:: the notion needed to differentiate sections of vector bundles.
4855:)). The Hom-bundle is so-called (and useful) because there is a
1858:, while those of rank 2 are less commonly called plane bundles.
7082:
5108:
can be defined as the pullback bundle of the diagonal map from
4473:-modules arises in this fashion from a vector bundle: only the
2957:{\displaystyle g_{UU}(x)=I,\quad g_{UV}(x)g_{VW}(x)g_{WU}(x)=I}
2809:{\displaystyle g_{UV}\colon U\cap V\to \operatorname {GL} (k).}
5302:{\displaystyle g_{UV}\colon U\cap V\to \operatorname {GL} (F)}
5231:
If instead of a finite-dimensional vector space, if the fiber
3580:
the structure of a vector bundle also. In this case the fibre
518:. Also, the vector spaces are usually required to be over the
27:
Mathematical parametrization of vector spaces by another space
5034:
can also be performed on the category of vector bundles in a
72:
including the point), but the total bundle is different from
6984:, vol. 107, Providence: American Mathematical Society,
5663:) have a very important property not shared by more general
4000:. That is, bundle morphisms for which the following diagram
5987:
of any smooth vector bundle carries a natural vector field
5782:
3543:
3469:
4537:
So we can think of the category of real vector bundles on
2209:
on the second overlap, one obtains the non-trivial bundle
1599:
is a finite-dimensional real vector space and hence has a
1472:
of the vector bundle. The local trivialization shows that
4206:
can be thought of as a section. The surface is the space
6693:
can also be defined, as well as higher K-theory groups.
435:
and these copies fit together to form the vector bundle
5010:. There is a canonical vector bundle isomorphism Hom(
6637:
6590:
6119:
6039:
5751:
5504:
5255:
4176:
4156:
4136:
3626:
3586:
3540:
3520:
3500:
3466:
3437:
3402:
3378:
3358:
3338:
3283:
3252:
3199:
3165:
3130:
3100:
3043:
3005:
2973:
2849:
2759:
2725:
2615:
2495:
2316:
2293:
2273:
2253:
2227:
2131:
2093:
2063:
2032:
2005:
1970:
1950:
1905:
1870:
1833:
1812:
1790:
1766:
1746:
1726:
1699:
1679:
1638:
1608:
1560:
1537:
1502:
1482:
1452:
1432:
1383:
1354:
1306:
1274:
1254:
1198:
1175:
1155:
1086:
1060:
1037:
1011:
988:
968:
920:
887:
867:
828:
795:
771:
727:
697:
677:
643:
619:
592:
572:
552:
467:
441:
421:
401:
381:
361:
341:
304:
284:
260:
238:
209:
189:
169:
137:
117:
5959:, a natural vector subbundle of the tangent bundle (
5705:
itself. This identification is obtained through the
4090:
can also be viewed as a vector bundle morphism over
7880:
7839:
7772:
7669:
7565:
7512:
7503:
7339:
7262:
7201:
7121:
5401:is a smooth map, and the local trivializations are
4662:)*. The dual bundle is locally trivial because the
4334:) always contains at least one element, namely the
4303:. Essentially, a section assigns to every point of
2121:to the fibres (note the transformation of the blue
6732:, one considers the K-theory groups consisting of
6673:
6623:
6228:
6073:
5877:
5558:
5301:
4886:Building on the previous example, given a section
4516:Even more: the category of real vector bundles on
4214:there is a vector in the vector space attached at
4182:
4162:
4142:
3638:
3612:
3572:
3526:
3506:
3486:
3449:
3428:over a topological space, a subbundle is simply a
3420:
3384:
3364:
3344:
3298:
3269:
3238:
3182:
3147:
3116:
3082:
3029:
2991:
2956:
2808:
2742:
2708:
2595:
2472:
2299:
2279:
2259:
2239:
2147:
2109:
2079:
2045:
2018:
1976:
1956:
1932:
1891:
1846:
1818:
1796:
1772:
1752:
1732:
1712:
1685:
1657:
1621:
1591:
1543:
1523:
1488:
1458:
1438:
1414:
1369:
1333:
1289:
1260:
1237:
1181:
1161:
1138:
1066:
1043:
1023:
994:
974:
951:
893:
873:
846:
801:
777:
745:
710:
683:
663:
625:
605:
578:
558:
495:: to every point of such a manifold we attach the
473:
453:
427:
407:
387:
367:
347:
325:
290:
266:
244:
224:
195:
175:
143:
123:
5429:-bundles. In this section we will concentrate on
5038:manner. This is made precise in the language of
1854:is constant. Vector bundles of rank 1 are called
5888:The vertical lift can also be seen as a natural
4030:A vector bundle morphism between vector bundles
3277:cocycle acting in the standard way on the fiber
510:Vector bundles are almost always required to be
507:if, and only if, its tangent bundle is trivial.
7065:Why is it useful to study vector bundles ?
5409:, there are different corresponding notions of
4693:(over the given field). A few examples follow.
4489:; these are precisely the continuous functions
6895:, London: Benjamin-Cummings, see section 1.5,
6089:is a smooth vector field on a smooth manifold
2606:is well-defined on the overlap, and satisfies
100:construction that makes precise the idea of a
7094:
5475:functions on overlaps of trivializing charts
5042:. An operation of a different nature is the
4780:is defined in a similar way, using fiberwise
2999:over which the bundle trivializes satisfying
1933:{\displaystyle X\times \mathbb {R} ^{k}\to X}
691:, and this fibre is mapped down to the point
8:
6239:does not depend on the choice of the linear
5584:Similarly, if the transition functions are:
5222:reduction of the structure group of a bundle
4666:of the inverse of a local trivialization of
3573:{\displaystyle \left.\pi \right|_{F}:F\to X}
3310:a vector bundle. This is an example of the
3030:{\displaystyle U\cap V\cap W\neq \emptyset }
1583:
1577:
1406:
1400:
943:
937:
5696:can be naturally identified with the fibre
5357:are used, then the resulting vector bundle
3239:{\displaystyle (E,X,\pi ,\mathbb {R} ^{k})}
3083:{\displaystyle (E,X,\pi ,\mathbb {R} ^{k})}
1238:{\displaystyle (\pi \circ \varphi )(x,v)=x}
7509:
7101:
7087:
7079:
6927:Riemannian Geometry and Geometric Analysis
5667:-fibre bundles. Namely, the tangent space
5435:-bundles. The most important example of a
3941:vector bundles. An isomorphism of a (rank
3905:which is also a bundle homomorphism (from
1629:. The local trivializations show that the
203:we associate (or "attach") a vector space
6636:
6589:
6220:
6207:
6179:
6162:
6146:
6124:
6118:
6062:
6038:
5863:
5850:
5803:
5784:
5756:
5750:
5639:(this requires the matrix group to be an
5549:
5548:
5509:
5503:
5260:
5254:
5188:Additional structures and generalizations
4859:between vector bundle homomorphisms from
4175:
4155:
4135:
3625:
3604:
3591:
3585:
3552:
3539:
3519:
3499:
3478:
3465:
3436:
3401:
3377:
3357:
3337:
3290:
3286:
3285:
3282:
3253:
3251:
3227:
3223:
3222:
3198:
3166:
3164:
3131:
3129:
3105:
3099:
3071:
3067:
3066:
3042:
3004:
2972:
2930:
2908:
2886:
2854:
2848:
2764:
2758:
2726:
2724:
2680:
2641:
2625:
2620:
2614:
2587:
2583:
2582:
2554:
2550:
2549:
2521:
2505:
2500:
2494:
2448:
2432:
2422:
2418:
2417:
2401:
2372:
2356:
2346:
2342:
2341:
2325:
2317:
2315:
2292:
2272:
2252:
2226:
2136:
2130:
2098:
2092:
2068:
2062:
2037:
2031:
2010:
2004:
1969:
1949:
1918:
1914:
1913:
1904:
1883:
1879:
1878:
1869:
1838:
1832:
1811:
1789:
1765:
1745:
1725:
1704:
1698:
1678:
1649:
1637:
1613:
1607:
1565:
1559:
1536:
1515:
1511:
1510:
1501:
1481:
1451:
1431:
1388:
1382:
1361:
1357:
1356:
1353:
1305:
1281:
1277:
1276:
1273:
1253:
1197:
1174:
1154:
1118:
1105:
1101:
1100:
1085:
1059:
1036:
1010:
987:
967:
925:
919:
886:
866:
827:
794:
770:
726:
702:
696:
676:
653:
648:
642:
618:
597:
591:
571:
551:
466:
440:
420:
400:
380:
360:
340:
303:
283:
259:
237:
208:
188:
168:
136:
116:
7022:Algebraic Geometry, a concise dictionary
5623:(this requires the matrix group to be a
5334:
5196:. Usually this metric is required to be
4412:(pointwise scalar multiplication) is in
4125:
3327:
2158:
1991:
1892:{\displaystyle X\times \mathbb {R} ^{k}}
1524:{\displaystyle U\times \mathbb {R} ^{k}}
541:
6854:
6847:
6704:asserts that the K-theory of any space
3159:on the fiber is the standard action of
503:. In general, a manifold is said to be
6624:{\displaystyle 0\to A\to B\to C\to 0,}
5405:. Depending on the required degree of
5346:are used, the resulting vector bundle
3714:is given by a pair of continuous maps
2307:over which the bundle trivializes via
1334:{\displaystyle v\mapsto \varphi (x,v)}
6953:Differential and Riemannian manifolds
3487:{\displaystyle \left.\pi \right|_{F}}
481:. Such vector bundles are said to be
7:
7024:, Berlin/Boston: Walter De Gruyter,
6866:
6565:topological space is defined as the
5048:construction. Given a vector bundle
4799:) is a vector bundle whose fiber at
4641:can be defined as the set of pairs (
4524:to the category of locally free and
1669:, and is therefore constant on each
493:smooth (or differentiable) manifolds
6977:Manifolds and Differential Geometry
6074:{\displaystyle (t,v)\mapsto e^{tv}}
5224:. Vector bundles over more general
5088:is essentially just the fiber over
514:, which means they are examples of
6929:(3rd ed.), Berlin, New York:
6801:Topology and differential geometry
6204:
5851:
4541:as sitting inside the category of
3613:{\displaystyle F_{x}\subset E_{x}}
3352:of a trivial rank 2 vector bundle
3024:
375:: in this case there is a copy of
25:
6450:secondary vector bundle structure
5180:is not compact: for example, the
4918:by taking the fiber over a point
4803:is the space of linear maps from
4122:Sections and locally free sheaves
3953:with the trivial bundle (of rank
3312:fibre bundle construction theorem
3193:Conversely, given a fiber bundle
2148:{\displaystyle g_{\alpha \beta }}
2110:{\displaystyle g_{\alpha \beta }}
2080:{\displaystyle U_{\alpha \beta }}
538:Definition and first consequences
7002:Introduction to Smooth Manifolds
6408:whose canonical vector field is
4559:of morphisms of vector bundles.
4352:zero element of the vector space
4322:) be the set of all sections on
4008:
3777:
3372:over a one-dimensional manifold
3299:{\displaystyle \mathbb {R} ^{k}}
2836:of transition functions forms a
1996:Two trivial vector bundles over
1592:{\displaystyle \pi ^{-1}(\{x\})}
1446:together with the homeomorphism
1415:{\displaystyle \pi ^{-1}(\{x\})}
1370:{\displaystyle \mathbb {R} ^{k}}
1290:{\displaystyle \mathbb {R} ^{k}}
952:{\displaystyle \pi ^{-1}(\{x\})}
49:. Locally around every point in
6982:Graduate Studies in Mathematics
6820:Algebraic and analytic geometry
6774:for vector bundle, among which
6174:
5839:
3620:is a vector subspace for every
2881:
1899:, equipped with the projection
1496:"looks like" the projection of
7141:Differentiable/Smooth manifold
6668:
6662:
6656:
6650:
6644:
6638:
6612:
6606:
6600:
6594:
6415:For any smooth vector bundle (
6217:
6200:
6191:
6185:
6155:
6136:
6130:
6055:
6052:
6040:
5912:) is the pull-back bundle of (
5869:
5856:
5833:
5818:
5774:
5768:
5569:is a smooth function into the
5553:
5539:
5530:
5296:
5290:
5281:
5200:, in which case each fibre of
3564:
3412:
3270:{\displaystyle {\text{GL}}(k)}
3264:
3258:
3233:
3200:
3183:{\displaystyle {\text{GL}}(k)}
3177:
3171:
3148:{\displaystyle {\text{GL}}(k)}
3142:
3136:
3077:
3044:
2945:
2939:
2923:
2917:
2901:
2895:
2869:
2863:
2800:
2794:
2785:
2743:{\displaystyle {\text{GL}}(k)}
2737:
2731:
2695:
2689:
2659:
2647:
2575:
2563:
2560:
2542:
2530:
2463:
2457:
2387:
2381:
2267:, and a pair of neighborhoods
2231:
1924:
1642:
1586:
1574:
1409:
1397:
1328:
1316:
1310:
1226:
1214:
1211:
1199:
1133:
1127:
1111:
946:
934:
838:
737:
532:category of topological spaces
314:
308:
219:
213:
1:
6916:Vector Bundles & K-Theory
6710:is isomorphic to that of the
6534:and scalar multiplication λ:
6247:. The canonical vector field
5633:then the vector bundle is an
4998:) of bundle homomorphisms of
4670:is a local trivialization of
3882:(vector) bundle homomorphisms
3155:structure group in which the
3094:; the additional data of the
2185:. When glued trivially (with
6522:of the original addition +:
6478:of the canonical projection
6400:is a smooth vector field on
5617:then the vector bundle is a
5607:then the vector bundle is a
5591:then the vector bundle is a
4817:(which is often denoted Hom(
4581:Operations on vector bundles
4256:, i.e. continuous functions
4016:(Note that this category is
1024:{\displaystyle U\subseteq X}
7847:Classification of manifolds
7053:Encyclopedia of Mathematics
6719:, the double suspension of
6396:is any smooth manifold and
5892:-vector bundle isomorphism
5609:real analytic vector bundle
5483:. That is, a vector bundle
5319:Vector bundles are special
5312:are continuous mappings of
4968:being the zero section and
4497:, and such a function is a
4408:is a continuous map, then α
3921:(vector) bundle isomorphism
3887:A bundle homomorphism from
3880:, and are sometimes called
3421:{\displaystyle \pi :E\to X}
2019:{\displaystyle U_{\alpha }}
847:{\displaystyle \pi :E\to X}
746:{\displaystyle \pi :E\to M}
7982:
6581:that, whenever we have an
5495:, the transition function
5361:is a smooth vector bundle.
5176:is trivial. This fails if
3450:{\displaystyle F\subset E}
3321:
2827:coordinate transformations
2046:{\displaystyle U_{\beta }}
1784:of the vector bundle, and
1658:{\displaystyle x\to k_{x}}
1348:between the vector spaces
34:The (infinitely extended)
7923:over commutative algebras
6836:Holomorphic vector bundle
5620:holomorphic vector bundle
5417:infinitely differentiable
5208:. A vector bundle with a
5100:. Hence, Whitney summing
5080:. The fiber over a point
5032:category of vector spaces
4604:, then there is a bundle
4566:vector bundle is trivial
664:{\displaystyle E_{m_{1}}}
454:{\displaystyle X\times V}
108:parameterized by another
7639:Riemann curvature tensor
6974:Lee, Jeffrey M. (2009),
6893:Foundations of mechanics
6377:is equal to the rank of
6110:= 0, the linear mapping
6013:is a smooth section of (
5327:are fibered by spheres.
5182:tautological line bundle
4600:is a vector bundle over
4505:of continuous functions
4342:that maps every element
2829:) of the vector bundle.
2087:by transition functions
1459:{\displaystyle \varphi }
6826:Algebraic vector bundle
6431:of its tangent bundle (
5957:vertical tangent bundle
5636:algebraic vector bundle
5002:and the trivial bundle
4914:, one can construct an
3672:from the vector bundle
3662:Vector bundle morphisms
1720:is equal to a constant
7431:Manifold with boundary
7146:Differential structure
7005:, New York: Springer,
6675:
6625:
6575:complex vector bundles
6230:
6075:
6007:canonical vector field
5879:
5560:
5439:-vector bundle is the
5362:
5303:
5120:where the bundle over
4986:is the Hom bundle Hom(
4222:Given a vector bundle
4219:
4198:The map associating a
4191:
4184:
4164:
4144:
3835:between vector spaces.
3640:
3639:{\displaystyle x\in X}
3614:
3574:
3528:
3508:
3488:
3451:
3422:
3393:
3386:
3366:
3346:
3300:
3271:
3240:
3184:
3149:
3118:
3117:{\displaystyle g_{UV}}
3084:
3031:
2993:
2958:
2810:
2744:
2710:
2597:
2474:
2301:
2281:
2261:
2241:
2240:{\displaystyle E\to X}
2221:Given a vector bundle
2218:
2156:
2149:
2111:
2081:
2057:over the intersection
2047:
2020:
1978:
1958:
1934:
1893:
1848:
1820:
1806:vector bundle of rank
1798:
1774:
1754:
1734:
1714:
1687:
1659:
1623:
1593:
1545:
1525:
1490:
1460:
1440:
1426:The open neighborhood
1416:
1371:
1335:
1291:
1262:
1239:
1183:
1163:
1140:
1068:
1045:
1025:
996:
976:
953:
895:
875:
848:
803:
779:
754:
747:
712:
685:
665:
627:
607:
580:
560:
528:Complex vector bundles
475:
455:
429:
409:
389:
369:
349:
327:
326:{\displaystyle V(x)=V}
292:
268:
246:
226:
197:
177:
145:
125:
85:
7020:Rubei, Elena (2014),
6999:Lee, John M. (2003),
6925:Jost, Jürgen (2002),
6676:
6626:
6292:there is a unique lim
6255:satisfies the axioms
6231:
6076:
5976:) of the total space
5880:
5561:
5338:
5331:Smooth vector bundles
5304:
5214:complex vector bundle
5056:and a continuous map
4770:tensor product bundle
4750:of the vector spaces
4438:real-valued functions
4367:scalar multiplication
4197:
4185:
4165:
4145:
4129:
3693:to the vector bundle
3641:
3615:
3575:
3529:
3509:
3489:
3452:
3423:
3387:
3367:
3347:
3331:
3301:
3272:
3241:
3185:
3150:
3119:
3085:
3032:
2994:
2992:{\displaystyle U,V,W}
2959:
2819:These are called the
2811:
2745:
2711:
2598:
2475:
2302:
2282:
2262:
2242:
2162:
2150:
2119:linear transformation
2112:
2082:
2048:
2021:
1995:
1979:
1959:
1935:
1894:
1849:
1847:{\displaystyle k_{x}}
1821:
1799:
1775:
1755:
1735:
1715:
1713:{\displaystyle k_{x}}
1688:
1660:
1624:
1622:{\displaystyle k_{x}}
1594:
1546:
1526:
1491:
1461:
1441:
1417:
1372:
1336:
1292:
1263:
1240:
1184:
1164:
1141:
1069:
1046:
1026:
997:
977:
954:
901:, the structure of a
896:
876:
849:
804:
780:
748:
713:
711:{\displaystyle m_{1}}
686:
671:of the vector bundle
666:
628:
608:
606:{\displaystyle m_{1}}
581:
561:
545:
476:
456:
430:
410:
390:
370:
350:
328:
293:
269:
247:
227:
198:
178:
146:
126:
33:
7578:Covariant derivative
7129:Topological manifold
6955:, Berlin, New York:
6785:Characteristic class
6747:equivalence relation
6683:topological K-theory
6635:
6588:
6550:The K-theory group,
6281:is globally defined.
6241:covariant derivative
6117:
6037:
5749:
5502:
5253:
5194:vector bundle metric
5150:. Any vector bundle
5068:one can "pull back"
4871:and sections of Hom(
4575:linearly independent
4467:Not every sheaf of O
4458:becomes a sheaf of O
4381:of vector spaces on
4268:where the composite
4210:, and at each point
4174:
4154:
4134:
3854:is surjective), and
3624:
3584:
3538:
3518:
3507:{\displaystyle \pi }
3498:
3464:
3435:
3400:
3376:
3356:
3336:
3281:
3250:
3197:
3163:
3128:
3098:
3041:
3003:
2971:
2847:
2822:transition functions
2757:
2723:
2613:
2493:
2314:
2291:
2271:
2251:
2225:
2129:
2125:under the effect of
2091:
2061:
2030:
2003:
1988:Transition functions
1968:
1948:
1903:
1868:
1831:
1810:
1788:
1764:
1744:
1724:
1697:
1677:
1636:
1606:
1558:
1535:
1500:
1489:{\displaystyle \pi }
1480:
1469:local trivialization
1450:
1430:
1381:
1352:
1304:
1272:
1252:
1196:
1173:
1153:
1084:
1058:
1035:
1009:
986:
966:
918:
885:
865:
826:
793:
769:
725:
695:
675:
641:
617:
590:
570:
550:
465:
439:
419:
399:
379:
359:
339:
302:
282:
258:
236:
225:{\displaystyle V(x)}
207:
187:
167:
135:
115:
7612:Exterior derivative
7214:Atiyah–Singer index
7163:Riemannian manifold
6889:Marsden, Jerrold E.
6790:Splitting principle
6698:periodicity theorem
6571:isomorphism classes
5631:algebraic functions
5072:to a vector bundle
4717:is a vector bundle
4444:. Furthermore, if O
4234:and an open subset
4202:to each point on a
3973:is then said to be
2633:
2513:
2441:
2365:
2201:on one overlap and
1671:connected component
765:topological spaces
633:corresponds to the
254:vector bundle over
7918:Secondary calculus
7872:Singularity theory
7827:Parallel transport
7595:De Rham cohomology
7234:Generalized Stokes
6944:, see section 1.5.
6772:classifying spaces
6730:algebraic geometry
6674:{\displaystyle =+}
6671:
6621:
6427:) the total space
6226:
6071:
5875:
5556:
5363:
5299:
5228:may also be used.
5226:topological fields
4981:dual vector bundle
4938:of the linear map
4526:finitely generated
4311:on that manifold.
4242:, we can consider
4220:
4192:
4180:
4160:
4140:
3636:
3610:
3570:
3524:
3504:
3484:
3447:
3418:
3394:
3382:
3362:
3342:
3296:
3267:
3236:
3180:
3145:
3114:
3080:
3027:
2989:
2954:
2840:in the sense that
2806:
2740:
2706:
2616:
2593:
2496:
2485:composite function
2470:
2468:
2297:
2277:
2257:
2237:
2219:
2157:
2145:
2107:
2077:
2043:
2016:
1974:
1954:
1930:
1889:
1844:
1816:
1794:
1770:
1750:
1730:
1710:
1683:
1655:
1619:
1589:
1541:
1521:
1486:
1456:
1436:
1412:
1367:
1331:
1287:
1258:
1235:
1179:
1159:
1149:such that for all
1136:
1064:
1041:
1021:
992:
972:
949:
903:finite-dimensional
891:
871:
844:
799:
775:
759:real vector bundle
755:
743:
708:
681:
661:
623:
603:
576:
556:
501:hairy ball theorem
471:
451:
425:
405:
385:
365:
345:
323:
288:
264:
242:
222:
193:
173:
163:): to every point
141:
121:
86:
7953:
7952:
7835:
7834:
7600:Differential form
7254:Whitney embedding
7188:Differential form
7031:978-3-11-031622-3
6991:978-0-8218-4815-9
6966:978-0-387-94338-1
6940:978-3-540-42627-1
6902:978-0-8053-0102-1
6885:Abraham, Ralph H.
6776:projective spaces
6009:. More formally,
5797:
5651:-vector bundles (
5625:complex Lie group
5365:A vector bundle (
5235:is taken to be a
5212:corresponds to a
5210:complex structure
5198:positive definite
4902:) and a function
4784:of vector spaces.
4729:whose fiber over
4707:direct sum bundle
4628:dual vector space
4618:, whose fiber at
4577:global sections.
4392:is an element of
4183:{\displaystyle s}
4163:{\displaystyle M}
4143:{\displaystyle E}
3843:is determined by
3527:{\displaystyle F}
3385:{\displaystyle M}
3365:{\displaystyle E}
3345:{\displaystyle L}
3332:A line subbundle
3256:
3169:
3134:
3037:. Thus the data
2750:-valued function
2729:
2442:
2366:
2300:{\displaystyle V}
2280:{\displaystyle U}
2260:{\displaystyle k}
1977:{\displaystyle X}
1957:{\displaystyle k}
1863:Cartesian product
1819:{\displaystyle k}
1797:{\displaystyle E}
1773:{\displaystyle k}
1753:{\displaystyle X}
1733:{\displaystyle k}
1686:{\displaystyle X}
1544:{\displaystyle U}
1439:{\displaystyle U}
1261:{\displaystyle v}
1182:{\displaystyle U}
1162:{\displaystyle x}
1067:{\displaystyle k}
1044:{\displaystyle p}
1004:open neighborhood
995:{\displaystyle X}
975:{\displaystyle p}
894:{\displaystyle X}
874:{\displaystyle x}
856:bundle projection
802:{\displaystyle E}
778:{\displaystyle X}
684:{\displaystyle E}
626:{\displaystyle M}
579:{\displaystyle M}
559:{\displaystyle E}
474:{\displaystyle X}
428:{\displaystyle X}
408:{\displaystyle x}
388:{\displaystyle V}
368:{\displaystyle X}
348:{\displaystyle x}
291:{\displaystyle V}
267:{\displaystyle X}
245:{\displaystyle X}
196:{\displaystyle X}
176:{\displaystyle x}
161:algebraic variety
153:topological space
144:{\displaystyle X}
124:{\displaystyle X}
16:(Redirected from
7973:
7945:Stratified space
7903:Fréchet manifold
7617:Interior product
7510:
7207:
7103:
7096:
7089:
7080:
7061:
7034:
7015:
6994:
6969:
6943:
6920:
6905:
6870:
6864:
6858:
6852:
6744:
6734:coherent sheaves
6724:
6718:
6709:
6691:compact supports
6680:
6678:
6677:
6672:
6630:
6628:
6627:
6622:
6560:
6448:) has a natural
6438:
6235:
6233:
6232:
6227:
6225:
6224:
6212:
6211:
6184:
6183:
6167:
6166:
6151:
6150:
6129:
6128:
6080:
6078:
6077:
6072:
6070:
6069:
6032:Lie-group action
6020:
5983:The total space
5966:
5884:
5882:
5881:
5876:
5868:
5867:
5855:
5854:
5814:
5813:
5802:
5798:
5796:
5785:
5761:
5760:
5565:
5563:
5562:
5557:
5552:
5517:
5516:
5450:
5391:smooth manifolds
5314:Banach manifolds
5308:
5306:
5305:
5300:
5268:
5267:
5175:
5164:
4596:For example, if
4356:
4294:
4281:
4271:
4251:
4225:
4189:
4187:
4186:
4181:
4169:
4167:
4166:
4161:
4149:
4147:
4146:
4141:
4130:A vector bundle
4054:
4033:
4012:
3945:) vector bundle
3858:is then said to
3850:
3811:
3800:
3781:
3767:
3760:
3696:
3675:
3654:, a non-trivial
3645:
3643:
3642:
3637:
3619:
3617:
3616:
3611:
3609:
3608:
3596:
3595:
3579:
3577:
3576:
3571:
3557:
3556:
3551:
3533:
3531:
3530:
3525:
3513:
3511:
3510:
3505:
3493:
3491:
3490:
3485:
3483:
3482:
3477:
3456:
3454:
3453:
3448:
3427:
3425:
3424:
3419:
3391:
3389:
3388:
3383:
3371:
3369:
3368:
3363:
3351:
3349:
3348:
3343:
3305:
3303:
3302:
3297:
3295:
3294:
3289:
3276:
3274:
3273:
3268:
3257:
3254:
3245:
3243:
3242:
3237:
3232:
3231:
3226:
3189:
3187:
3186:
3181:
3170:
3167:
3154:
3152:
3151:
3146:
3135:
3132:
3123:
3121:
3120:
3115:
3113:
3112:
3089:
3087:
3086:
3081:
3076:
3075:
3070:
3036:
3034:
3033:
3028:
2998:
2996:
2995:
2990:
2963:
2961:
2960:
2955:
2938:
2937:
2916:
2915:
2894:
2893:
2862:
2861:
2815:
2813:
2812:
2807:
2772:
2771:
2749:
2747:
2746:
2741:
2730:
2727:
2715:
2713:
2712:
2707:
2705:
2701:
2688:
2687:
2646:
2645:
2632:
2624:
2602:
2600:
2599:
2594:
2592:
2591:
2586:
2559:
2558:
2553:
2526:
2525:
2512:
2504:
2479:
2477:
2476:
2471:
2469:
2456:
2455:
2443:
2433:
2427:
2426:
2421:
2406:
2405:
2380:
2379:
2367:
2357:
2351:
2350:
2345:
2330:
2329:
2306:
2304:
2303:
2298:
2286:
2284:
2283:
2278:
2266:
2264:
2263:
2258:
2246:
2244:
2243:
2238:
2154:
2152:
2151:
2146:
2144:
2143:
2116:
2114:
2113:
2108:
2106:
2105:
2086:
2084:
2083:
2078:
2076:
2075:
2052:
2050:
2049:
2044:
2042:
2041:
2025:
2023:
2022:
2017:
2015:
2014:
1983:
1981:
1980:
1975:
1963:
1961:
1960:
1955:
1940:, is called the
1939:
1937:
1936:
1931:
1923:
1922:
1917:
1898:
1896:
1895:
1890:
1888:
1887:
1882:
1853:
1851:
1850:
1845:
1843:
1842:
1825:
1823:
1822:
1817:
1804:is said to be a
1803:
1801:
1800:
1795:
1779:
1777:
1776:
1771:
1759:
1757:
1756:
1751:
1739:
1737:
1736:
1731:
1719:
1717:
1716:
1711:
1709:
1708:
1692:
1690:
1689:
1684:
1667:locally constant
1664:
1662:
1661:
1656:
1654:
1653:
1628:
1626:
1625:
1620:
1618:
1617:
1598:
1596:
1595:
1590:
1573:
1572:
1550:
1548:
1547:
1542:
1530:
1528:
1527:
1522:
1520:
1519:
1514:
1495:
1493:
1492:
1487:
1465:
1463:
1462:
1457:
1445:
1443:
1442:
1437:
1421:
1419:
1418:
1413:
1396:
1395:
1376:
1374:
1373:
1368:
1366:
1365:
1360:
1340:
1338:
1337:
1332:
1296:
1294:
1293:
1288:
1286:
1285:
1280:
1267:
1265:
1264:
1259:
1244:
1242:
1241:
1236:
1188:
1186:
1185:
1180:
1168:
1166:
1165:
1160:
1145:
1143:
1142:
1137:
1126:
1125:
1110:
1109:
1104:
1073:
1071:
1070:
1065:
1050:
1048:
1047:
1042:
1030:
1028:
1027:
1022:
1001:
999:
998:
993:
981:
979:
978:
973:
958:
956:
955:
950:
933:
932:
900:
898:
897:
892:
880:
878:
877:
872:
853:
851:
850:
845:
808:
806:
805:
800:
784:
782:
781:
776:
752:
750:
749:
744:
717:
715:
714:
709:
707:
706:
690:
688:
687:
682:
670:
668:
667:
662:
660:
659:
658:
657:
632:
630:
629:
624:
612:
610:
609:
604:
602:
601:
585:
583:
582:
577:
565:
563:
562:
557:
546:A vector bundle
480:
478:
477:
472:
460:
458:
457:
452:
434:
432:
431:
426:
414:
412:
411:
406:
394:
392:
391:
386:
374:
372:
371:
366:
354:
352:
351:
346:
332:
330:
329:
324:
297:
295:
294:
289:
273:
271:
270:
265:
251:
249:
248:
243:
231:
229:
228:
223:
202:
200:
199:
194:
182:
180:
179:
174:
150:
148:
147:
142:
130:
128:
127:
122:
21:
7981:
7980:
7976:
7975:
7974:
7972:
7971:
7970:
7956:
7955:
7954:
7949:
7888:Banach manifold
7881:Generalizations
7876:
7831:
7768:
7665:
7627:Ricci curvature
7583:Cotangent space
7561:
7499:
7341:
7335:
7294:Exponential map
7258:
7203:
7197:
7117:
7107:
7048:"Vector bundle"
7046:
7043:
7038:
7032:
7019:
7013:
6998:
6992:
6973:
6967:
6957:Springer-Verlag
6947:
6941:
6931:Springer-Verlag
6924:
6909:
6903:
6883:
6879:
6874:
6873:
6865:
6861:
6853:
6849:
6844:
6822:
6803:
6764:
6762:General notions
6759:
6740:
6720:
6711:
6705:
6633:
6632:
6586:
6585:
6561:, of a compact
6551:
6548:
6513:
6493:
6473:
6462:
6443:
6436:
6392:Conversely, if
6383:
6353:
6340:
6329:
6318:
6301:
6295:
6272:
6216:
6203:
6175:
6158:
6142:
6120:
6115:
6114:
6109:
6058:
6035:
6034:
6025:
6018:
6005:, known as the
6001:
5995:
5971:
5964:
5950:
5859:
5846:
5789:
5781:
5780:
5752:
5747:
5746:
5741:
5732:
5722:
5716:
5704:
5695:
5682:
5672:
5641:algebraic group
5505:
5500:
5499:
5455:
5448:
5403:diffeomorphisms
5355:
5344:
5333:
5256:
5251:
5250:
5206:Euclidean space
5190:
5173:
5162:
5045:pullback bundle
5040:smooth functors
4963:
4954:
4854:
4845:
4831:
4825:
4815:
4808:
4762:
4755:
4748:
4741:
4703:Hassler Whitney
4661:
4635:
4583:
4548:
4533:
4472:
4463:
4449:
4420:). We see that
4354:
4338:: the function
4279:
4277:
4269:
4249:
4223:
4172:
4171:
4152:
4151:
4132:
4131:
4124:
4117:
4106:pullback bundle
4103:
4096:
4089:
4082:
4072:covering a map
4071:
4064:
4057:
4052:
4050:
4043:
4036:
4031:
3983:locally trivial
3937:are said to be
3936:
3929:
3918:
3911:
3900:
3893:
3853:
3848:
3814:
3809:
3803:
3798:
3796:
3770:
3765:
3763:
3758:
3749:
3742:
3731:
3724:
3713:
3706:
3699:
3694:
3692:
3685:
3678:
3673:
3664:
3622:
3621:
3600:
3587:
3582:
3581:
3542:
3541:
3536:
3535:
3516:
3515:
3496:
3495:
3468:
3467:
3462:
3461:
3433:
3432:
3398:
3397:
3374:
3373:
3354:
3353:
3334:
3333:
3326:
3320:
3284:
3279:
3278:
3248:
3247:
3221:
3195:
3194:
3161:
3160:
3126:
3125:
3101:
3096:
3095:
3065:
3039:
3038:
3001:
3000:
2969:
2968:
2926:
2904:
2882:
2850:
2845:
2844:
2760:
2755:
2754:
2721:
2720:
2676:
2669:
2665:
2637:
2611:
2610:
2581:
2548:
2517:
2491:
2490:
2467:
2466:
2444:
2428:
2416:
2397:
2394:
2393:
2368:
2352:
2340:
2321:
2312:
2311:
2289:
2288:
2269:
2268:
2249:
2248:
2223:
2222:
2206:
2198:
2190:
2132:
2127:
2126:
2094:
2089:
2088:
2064:
2059:
2058:
2033:
2028:
2027:
2006:
2001:
2000:
1990:
1966:
1965:
1946:
1945:
1912:
1901:
1900:
1877:
1866:
1865:
1834:
1829:
1828:
1808:
1807:
1786:
1785:
1762:
1761:
1742:
1741:
1722:
1721:
1700:
1695:
1694:
1675:
1674:
1645:
1634:
1633:
1609:
1604:
1603:
1561:
1556:
1555:
1533:
1532:
1509:
1498:
1497:
1478:
1477:
1448:
1447:
1428:
1427:
1384:
1379:
1378:
1355:
1350:
1349:
1302:
1301:
1275:
1270:
1269:
1250:
1249:
1194:
1193:
1171:
1170:
1151:
1150:
1114:
1099:
1082:
1081:
1056:
1055:
1033:
1032:
1007:
1006:
984:
983:
964:
963:
921:
916:
915:
883:
882:
863:
862:
824:
823:
791:
790:
767:
766:
723:
722:
698:
693:
692:
673:
672:
649:
644:
639:
638:
615:
614:
593:
588:
587:
568:
567:
548:
547:
540:
524:complex numbers
512:locally trivial
489:tangent bundles
463:
462:
437:
436:
417:
416:
397:
396:
377:
376:
357:
356:
337:
336:
300:
299:
280:
279:
256:
255:
234:
233:
205:
204:
185:
184:
165:
164:
133:
132:
113:
112:
28:
23:
22:
15:
12:
11:
5:
7979:
7977:
7969:
7968:
7966:Vector bundles
7958:
7957:
7951:
7950:
7948:
7947:
7942:
7937:
7932:
7927:
7926:
7925:
7915:
7910:
7905:
7900:
7895:
7890:
7884:
7882:
7878:
7877:
7875:
7874:
7869:
7864:
7859:
7854:
7849:
7843:
7841:
7837:
7836:
7833:
7832:
7830:
7829:
7824:
7819:
7814:
7809:
7804:
7799:
7794:
7789:
7784:
7778:
7776:
7770:
7769:
7767:
7766:
7761:
7756:
7751:
7746:
7741:
7736:
7726:
7721:
7716:
7706:
7701:
7696:
7691:
7686:
7681:
7675:
7673:
7667:
7666:
7664:
7663:
7658:
7653:
7652:
7651:
7641:
7636:
7635:
7634:
7624:
7619:
7614:
7609:
7608:
7607:
7597:
7592:
7591:
7590:
7580:
7575:
7569:
7567:
7563:
7562:
7560:
7559:
7554:
7549:
7544:
7543:
7542:
7532:
7527:
7522:
7516:
7514:
7507:
7501:
7500:
7498:
7497:
7492:
7482:
7477:
7463:
7458:
7453:
7448:
7443:
7441:Parallelizable
7438:
7433:
7428:
7427:
7426:
7416:
7411:
7406:
7401:
7396:
7391:
7386:
7381:
7376:
7371:
7361:
7351:
7345:
7343:
7337:
7336:
7334:
7333:
7328:
7323:
7321:Lie derivative
7318:
7316:Integral curve
7313:
7308:
7303:
7302:
7301:
7291:
7286:
7285:
7284:
7277:Diffeomorphism
7274:
7268:
7266:
7260:
7259:
7257:
7256:
7251:
7246:
7241:
7236:
7231:
7226:
7221:
7216:
7210:
7208:
7199:
7198:
7196:
7195:
7190:
7185:
7180:
7175:
7170:
7165:
7160:
7155:
7154:
7153:
7148:
7138:
7137:
7136:
7125:
7123:
7122:Basic concepts
7119:
7118:
7108:
7106:
7105:
7098:
7091:
7083:
7077:
7076:
7071:
7062:
7042:
7041:External links
7039:
7037:
7036:
7030:
7017:
7011:
6996:
6990:
6971:
6965:
6945:
6939:
6922:
6919:(2.0 ed.)
6911:Hatcher, Allen
6907:
6901:
6880:
6878:
6875:
6872:
6871:
6859:
6857:, Example 3.6.
6846:
6845:
6843:
6840:
6839:
6838:
6833:
6828:
6821:
6818:
6817:
6816:
6810:
6802:
6799:
6798:
6797:
6792:
6787:
6782:
6763:
6760:
6758:
6755:
6670:
6667:
6664:
6661:
6658:
6655:
6652:
6649:
6646:
6643:
6640:
6620:
6617:
6614:
6611:
6608:
6605:
6602:
6599:
6596:
6593:
6583:exact sequence
6547:
6544:
6511:
6491:
6471:
6460:
6439:
6390:
6389:
6381:
6355:
6349:
6338:
6327:
6316:
6311:
6297:
6293:
6282:
6268:
6237:
6236:
6223:
6219:
6215:
6210:
6206:
6202:
6199:
6196:
6193:
6190:
6187:
6182:
6178:
6173:
6170:
6165:
6161:
6157:
6154:
6149:
6145:
6141:
6138:
6135:
6132:
6127:
6123:
6105:
6068:
6065:
6061:
6057:
6054:
6051:
6048:
6045:
6042:
6021:
5997:
5991:
5967:
5948:
5886:
5885:
5874:
5871:
5866:
5862:
5858:
5853:
5849:
5845:
5842:
5838:
5835:
5832:
5829:
5826:
5823:
5820:
5817:
5812:
5809:
5806:
5801:
5795:
5792:
5788:
5783:
5779:
5776:
5773:
5770:
5767:
5764:
5759:
5755:
5742:), defined as
5737:
5728:
5720:
5712:
5700:
5691:
5678:
5670:
5645:
5644:
5628:
5612:
5599:
5577:), which is a
5567:
5566:
5555:
5551:
5547:
5544:
5541:
5538:
5535:
5532:
5529:
5526:
5523:
5520:
5515:
5512:
5508:
5451:
5441:tangent bundle
5353:
5342:
5332:
5329:
5325:sphere bundles
5310:
5309:
5298:
5295:
5292:
5289:
5286:
5283:
5280:
5277:
5274:
5271:
5266:
5263:
5259:
5218:complex-linear
5189:
5186:
5028:
5027:
4977:
4959:
4950:
4884:
4850:
4841:
4829:
4821:
4813:
4806:
4785:
4782:tensor product
4765:
4760:
4753:
4746:
4739:
4657:
4633:
4582:
4579:
4568:if and only if
4544:
4529:
4468:
4459:
4445:
4436:of continuous
4179:
4159:
4139:
4123:
4120:
4115:
4101:
4094:
4087:
4080:
4069:
4062:
4055:
4048:
4041:
4034:
4014:
4013:
3963:trivialization
3961:) is called a
3934:
3927:
3919:) is called a
3916:
3909:
3898:
3891:
3851:
3837:
3836:
3812:
3801:
3794:
3784:
3783:
3782:
3768:
3761:
3747:
3740:
3729:
3722:
3711:
3704:
3697:
3690:
3683:
3676:
3663:
3660:
3635:
3632:
3629:
3607:
3603:
3599:
3594:
3590:
3569:
3566:
3563:
3560:
3555:
3550:
3547:
3544:
3523:
3503:
3481:
3476:
3473:
3470:
3457:for which the
3446:
3443:
3440:
3417:
3414:
3411:
3408:
3405:
3381:
3361:
3341:
3322:Main article:
3319:
3316:
3293:
3288:
3266:
3263:
3260:
3235:
3230:
3225:
3220:
3217:
3214:
3211:
3208:
3205:
3202:
3179:
3176:
3173:
3144:
3141:
3138:
3111:
3108:
3104:
3079:
3074:
3069:
3064:
3061:
3058:
3055:
3052:
3049:
3046:
3026:
3023:
3020:
3017:
3014:
3011:
3008:
2988:
2985:
2982:
2979:
2976:
2965:
2964:
2953:
2950:
2947:
2944:
2941:
2936:
2933:
2929:
2925:
2922:
2919:
2914:
2911:
2907:
2903:
2900:
2897:
2892:
2889:
2885:
2880:
2877:
2874:
2871:
2868:
2865:
2860:
2857:
2853:
2817:
2816:
2805:
2802:
2799:
2796:
2793:
2790:
2787:
2784:
2781:
2778:
2775:
2770:
2767:
2763:
2739:
2736:
2733:
2717:
2716:
2704:
2700:
2697:
2694:
2691:
2686:
2683:
2679:
2675:
2672:
2668:
2664:
2661:
2658:
2655:
2652:
2649:
2644:
2640:
2636:
2631:
2628:
2623:
2619:
2604:
2603:
2590:
2585:
2580:
2577:
2574:
2571:
2568:
2565:
2562:
2557:
2552:
2547:
2544:
2541:
2538:
2535:
2532:
2529:
2524:
2520:
2516:
2511:
2508:
2503:
2499:
2481:
2480:
2465:
2462:
2459:
2454:
2451:
2447:
2440:
2436:
2431:
2429:
2425:
2420:
2415:
2412:
2409:
2404:
2400:
2396:
2395:
2392:
2389:
2386:
2383:
2378:
2375:
2371:
2364:
2360:
2355:
2353:
2349:
2344:
2339:
2336:
2333:
2328:
2324:
2320:
2319:
2296:
2276:
2256:
2236:
2233:
2230:
2204:
2196:
2188:
2142:
2139:
2135:
2104:
2101:
2097:
2074:
2071:
2067:
2040:
2036:
2013:
2009:
1989:
1986:
1973:
1953:
1942:trivial bundle
1929:
1926:
1921:
1916:
1911:
1908:
1886:
1881:
1876:
1873:
1841:
1837:
1815:
1793:
1780:is called the
1769:
1749:
1729:
1707:
1703:
1682:
1652:
1648:
1644:
1641:
1616:
1612:
1588:
1585:
1582:
1579:
1576:
1571:
1568:
1564:
1540:
1518:
1513:
1508:
1505:
1485:
1455:
1435:
1424:
1423:
1411:
1408:
1405:
1402:
1399:
1394:
1391:
1387:
1364:
1359:
1330:
1327:
1324:
1321:
1318:
1315:
1312:
1309:
1298:
1284:
1279:
1257:
1234:
1231:
1228:
1225:
1222:
1219:
1216:
1213:
1210:
1207:
1204:
1201:
1178:
1158:
1147:
1146:
1135:
1132:
1129:
1124:
1121:
1117:
1113:
1108:
1103:
1098:
1095:
1092:
1089:
1063:
1053:natural number
1040:
1020:
1017:
1014:
1002:, there is an
991:
971:
960:
959:
948:
945:
942:
939:
936:
931:
928:
924:
890:
870:
859:
843:
840:
837:
834:
831:
814:
798:
774:
742:
739:
736:
733:
730:
705:
701:
680:
656:
652:
647:
622:
600:
596:
575:
555:
539:
536:
505:parallelizable
470:
450:
447:
444:
424:
404:
384:
364:
344:
322:
319:
316:
313:
310:
307:
287:
263:
241:
221:
218:
215:
212:
192:
172:
140:
120:
26:
24:
18:Vector bundles
14:
13:
10:
9:
6:
4:
3:
2:
7978:
7967:
7964:
7963:
7961:
7946:
7943:
7941:
7940:Supermanifold
7938:
7936:
7933:
7931:
7928:
7924:
7921:
7920:
7919:
7916:
7914:
7911:
7909:
7906:
7904:
7901:
7899:
7896:
7894:
7891:
7889:
7886:
7885:
7883:
7879:
7873:
7870:
7868:
7865:
7863:
7860:
7858:
7855:
7853:
7850:
7848:
7845:
7844:
7842:
7838:
7828:
7825:
7823:
7820:
7818:
7815:
7813:
7810:
7808:
7805:
7803:
7800:
7798:
7795:
7793:
7790:
7788:
7785:
7783:
7780:
7779:
7777:
7775:
7771:
7765:
7762:
7760:
7757:
7755:
7752:
7750:
7747:
7745:
7742:
7740:
7737:
7735:
7731:
7727:
7725:
7722:
7720:
7717:
7715:
7711:
7707:
7705:
7702:
7700:
7697:
7695:
7692:
7690:
7687:
7685:
7682:
7680:
7677:
7676:
7674:
7672:
7668:
7662:
7661:Wedge product
7659:
7657:
7654:
7650:
7647:
7646:
7645:
7642:
7640:
7637:
7633:
7630:
7629:
7628:
7625:
7623:
7620:
7618:
7615:
7613:
7610:
7606:
7605:Vector-valued
7603:
7602:
7601:
7598:
7596:
7593:
7589:
7586:
7585:
7584:
7581:
7579:
7576:
7574:
7571:
7570:
7568:
7564:
7558:
7555:
7553:
7550:
7548:
7545:
7541:
7538:
7537:
7536:
7535:Tangent space
7533:
7531:
7528:
7526:
7523:
7521:
7518:
7517:
7515:
7511:
7508:
7506:
7502:
7496:
7493:
7491:
7487:
7483:
7481:
7478:
7476:
7472:
7468:
7464:
7462:
7459:
7457:
7454:
7452:
7449:
7447:
7444:
7442:
7439:
7437:
7434:
7432:
7429:
7425:
7422:
7421:
7420:
7417:
7415:
7412:
7410:
7407:
7405:
7402:
7400:
7397:
7395:
7392:
7390:
7387:
7385:
7382:
7380:
7377:
7375:
7372:
7370:
7366:
7362:
7360:
7356:
7352:
7350:
7347:
7346:
7344:
7338:
7332:
7329:
7327:
7324:
7322:
7319:
7317:
7314:
7312:
7309:
7307:
7304:
7300:
7299:in Lie theory
7297:
7296:
7295:
7292:
7290:
7287:
7283:
7280:
7279:
7278:
7275:
7273:
7270:
7269:
7267:
7265:
7261:
7255:
7252:
7250:
7247:
7245:
7242:
7240:
7237:
7235:
7232:
7230:
7227:
7225:
7222:
7220:
7217:
7215:
7212:
7211:
7209:
7206:
7202:Main results
7200:
7194:
7191:
7189:
7186:
7184:
7183:Tangent space
7181:
7179:
7176:
7174:
7171:
7169:
7166:
7164:
7161:
7159:
7156:
7152:
7149:
7147:
7144:
7143:
7142:
7139:
7135:
7132:
7131:
7130:
7127:
7126:
7124:
7120:
7115:
7111:
7104:
7099:
7097:
7092:
7090:
7085:
7084:
7081:
7075:
7072:
7070:
7066:
7063:
7059:
7055:
7054:
7049:
7045:
7044:
7040:
7033:
7027:
7023:
7018:
7014:
7012:0-387-95448-1
7008:
7004:
7003:
6997:
6993:
6987:
6983:
6979:
6978:
6972:
6968:
6962:
6958:
6954:
6950:
6946:
6942:
6936:
6932:
6928:
6923:
6918:
6917:
6912:
6908:
6904:
6898:
6894:
6890:
6886:
6882:
6881:
6876:
6868:
6863:
6860:
6856:
6851:
6848:
6841:
6837:
6834:
6832:
6829:
6827:
6824:
6823:
6819:
6814:
6811:
6808:
6805:
6804:
6800:
6796:
6795:Stable bundle
6793:
6791:
6788:
6786:
6783:
6781:
6777:
6773:
6769:
6766:
6765:
6761:
6756:
6754:
6752:
6748:
6743:
6739:
6735:
6731:
6726:
6723:
6717:
6714:
6708:
6703:
6699:
6694:
6692:
6688:
6684:
6665:
6659:
6653:
6647:
6641:
6618:
6615:
6609:
6603:
6597:
6591:
6584:
6580:
6576:
6572:
6569:generated by
6568:
6567:abelian group
6564:
6558:
6554:
6545:
6543:
6541:
6537:
6533:
6529:
6525:
6521:
6517:
6509:
6505:
6501:
6497:
6489:
6485:
6481:
6477:
6470:
6466:
6459:
6455:
6451:
6447:
6442:
6434:
6430:
6426:
6422:
6418:
6413:
6411:
6407:
6403:
6399:
6395:
6387:
6380:
6376:
6372:
6368:
6364:
6360:
6356:
6352:
6348:
6344:
6337:
6333:
6326:
6322:
6315:
6312:
6309:
6305:
6300:
6291:
6287:
6283:
6280:
6276:
6271:
6266:
6262:
6258:
6257:
6256:
6254:
6250:
6246:
6242:
6221:
6213:
6208:
6197:
6194:
6188:
6180:
6176:
6171:
6168:
6163:
6159:
6152:
6147:
6143:
6139:
6133:
6125:
6121:
6113:
6112:
6111:
6108:
6104:
6100:
6096:
6092:
6088:
6084:
6066:
6063:
6059:
6049:
6046:
6043:
6033:
6029:
6024:
6016:
6012:
6008:
6004:
6000:
5994:
5990:
5986:
5981:
5979:
5975:
5970:
5962:
5958:
5954:
5947:
5944: := Ker(
5943:
5939:
5935:
5931:
5927:
5923:
5919:
5915:
5911:
5907:
5903:
5899:
5895:
5891:
5872:
5864:
5860:
5847:
5843:
5840:
5836:
5830:
5827:
5824:
5821:
5815:
5810:
5807:
5804:
5799:
5793:
5790:
5786:
5777:
5771:
5765:
5762:
5757:
5753:
5745:
5744:
5743:
5740:
5736:
5731:
5727:
5723:
5715:
5711:
5708:
5707:vertical lift
5703:
5699:
5694:
5690:
5686:
5681:
5677:
5673:
5666:
5662:
5658:
5654:
5650:
5642:
5638:
5637:
5632:
5629:
5626:
5622:
5621:
5616:
5613:
5610:
5606:
5605:
5600:
5597:
5596:vector bundle
5595:
5590:
5587:
5586:
5585:
5582:
5580:
5576:
5572:
5545:
5542:
5536:
5533:
5527:
5524:
5521:
5518:
5513:
5510:
5506:
5498:
5497:
5496:
5494:
5490:
5486:
5482:
5478:
5474:
5469:
5467:
5463:
5459:
5454:
5446:
5442:
5438:
5434:
5433:
5428:
5425:
5424:real analytic
5422:-bundles and
5421:
5418:
5414:
5413:
5408:
5404:
5400:
5396:
5392:
5388:
5384:
5380:
5376:
5372:
5368:
5360:
5356:
5349:
5345:
5337:
5330:
5328:
5326:
5322:
5321:fiber bundles
5317:
5315:
5293:
5287:
5284:
5278:
5275:
5272:
5269:
5264:
5261:
5257:
5249:
5248:
5247:
5244:
5243:
5242:Banach bundle
5238:
5234:
5229:
5227:
5223:
5219:
5215:
5211:
5207:
5203:
5199:
5195:
5187:
5185:
5183:
5179:
5172:
5168:
5161:
5157:
5153:
5149:
5148:compact space
5145:
5141:
5137:
5135:
5132: ×
5131:
5127:
5123:
5119:
5115:
5111:
5107:
5103:
5099:
5095:
5091:
5087:
5083:
5079:
5075:
5071:
5067:
5063:
5059:
5055:
5051:
5047:
5046:
5041:
5037:
5033:
5025:
5021:
5017:
5013:
5009:
5005:
5001:
4997:
4993:
4989:
4985:
4982:
4978:
4975:
4971:
4967:
4962:
4958:
4953:
4949:
4945:
4941:
4937:
4933:
4929:
4925:
4921:
4917:
4913:
4909:
4905:
4901:
4897:
4893:
4889:
4885:
4882:
4878:
4874:
4870:
4866:
4862:
4858:
4853:
4849:
4844:
4840:
4836:
4832:
4824:
4820:
4816:
4809:
4802:
4798:
4794:
4790:
4786:
4783:
4779:
4775:
4772:
4771:
4766:
4763:
4756:
4749:
4742:
4736:
4732:
4728:
4724:
4720:
4716:
4712:
4708:
4704:
4700:
4696:
4695:
4694:
4692:
4688:
4684:
4679:
4677:
4673:
4669:
4665:
4660:
4656:
4652:
4648:
4644:
4640:
4637:)*. Formally
4636:
4629:
4625:
4621:
4617:
4616:
4612:, called the
4611:
4607:
4603:
4599:
4594:
4592:
4588:
4580:
4578:
4576:
4573:
4569:
4565:
4560:
4558:
4554:
4550:
4547:
4540:
4535:
4532:
4527:
4523:
4519:
4514:
4512:
4508:
4504:
4500:
4496:
4492:
4488:
4484:
4480:
4476:
4471:
4465:
4462:
4457:
4453:
4448:
4443:
4439:
4435:
4431:
4427:
4423:
4419:
4415:
4411:
4407:
4403:
4399:
4395:
4391:
4386:
4384:
4380:
4376:
4372:
4369:of sections,
4368:
4365:addition and
4364:
4361:}). With the
4360:
4353:
4349:
4345:
4341:
4337:
4333:
4329:
4325:
4321:
4317:
4312:
4310:
4309:vector fields
4306:
4302:
4298:
4293:
4289:
4285:
4276:is such that
4275:
4272: ∘
4267:
4263:
4259:
4255:
4247:
4246:
4241:
4237:
4233:
4229:
4217:
4213:
4209:
4205:
4201:
4196:
4177:
4170:with section
4157:
4137:
4128:
4121:
4119:
4114:
4110:
4107:
4100:
4093:
4086:
4079:
4075:
4068:
4061:
4047:
4040:
4028:
4026:
4022:
4019:
4011:
4007:
4006:
4005:
4003:
3999:
3995:
3991:
3986:
3984:
3980:
3979:trivializable
3976:
3972:
3968:
3964:
3960:
3956:
3952:
3948:
3944:
3940:
3933:
3926:
3922:
3915:
3908:
3904:
3897:
3890:
3885:
3883:
3879:
3878:fiber bundles
3875:
3871:
3866:
3864:
3863:
3857:
3846:
3842:
3834:
3830:
3826:
3822:
3818:
3807:
3793:
3789:
3785:
3780:
3776:
3775:
3774:
3771: ∘
3757: ∘
3756:
3753:
3752:
3751:
3746:
3739:
3735:
3728:
3721:
3717:
3710:
3703:
3689:
3682:
3671:
3670:
3661:
3659:
3657:
3653:
3647:
3633:
3630:
3627:
3605:
3601:
3597:
3592:
3588:
3567:
3561:
3558:
3553:
3548:
3545:
3521:
3501:
3479:
3474:
3471:
3460:
3444:
3441:
3438:
3431:
3415:
3409:
3406:
3403:
3379:
3359:
3339:
3330:
3325:
3317:
3315:
3313:
3309:
3291:
3261:
3228:
3218:
3215:
3212:
3209:
3206:
3203:
3191:
3174:
3158:
3139:
3109:
3106:
3102:
3093:
3072:
3062:
3059:
3056:
3053:
3050:
3047:
3021:
3018:
3015:
3012:
3009:
3006:
2986:
2983:
2980:
2977:
2974:
2951:
2948:
2942:
2934:
2931:
2927:
2920:
2912:
2909:
2905:
2898:
2890:
2887:
2883:
2878:
2875:
2872:
2866:
2858:
2855:
2851:
2843:
2842:
2841:
2839:
2835:
2830:
2828:
2824:
2823:
2803:
2797:
2791:
2788:
2782:
2779:
2776:
2773:
2768:
2765:
2761:
2753:
2752:
2751:
2734:
2702:
2698:
2692:
2684:
2681:
2677:
2673:
2670:
2666:
2662:
2656:
2653:
2650:
2642:
2638:
2634:
2629:
2626:
2621:
2617:
2609:
2608:
2607:
2588:
2578:
2572:
2569:
2566:
2555:
2545:
2539:
2536:
2533:
2527:
2522:
2518:
2514:
2509:
2506:
2501:
2497:
2489:
2488:
2487:
2486:
2460:
2452:
2449:
2445:
2438:
2434:
2430:
2423:
2413:
2410:
2407:
2402:
2398:
2390:
2384:
2376:
2373:
2369:
2362:
2358:
2354:
2347:
2337:
2334:
2331:
2326:
2322:
2310:
2309:
2308:
2294:
2274:
2254:
2234:
2228:
2216:
2212:
2208:
2200:
2192:
2184:
2183:
2177:
2173:
2170:
2166:
2161:
2140:
2137:
2133:
2124:
2123:quadrilateral
2120:
2102:
2099:
2095:
2072:
2069:
2065:
2056:
2038:
2034:
2011:
2007:
1999:
1994:
1987:
1985:
1971:
1951:
1943:
1927:
1919:
1909:
1906:
1884:
1874:
1871:
1864:
1859:
1857:
1839:
1835:
1826:
1813:
1791:
1783:
1767:
1747:
1727:
1705:
1701:
1680:
1672:
1668:
1650:
1646:
1639:
1632:
1614:
1610:
1602:
1580:
1569:
1566:
1562:
1552:
1538:
1516:
1506:
1503:
1483:
1475:
1471:
1470:
1453:
1433:
1403:
1392:
1389:
1385:
1362:
1347:
1344:
1325:
1322:
1319:
1313:
1307:
1299:
1282:
1255:
1248:
1232:
1229:
1223:
1220:
1217:
1208:
1205:
1202:
1192:
1191:
1190:
1176:
1156:
1130:
1122:
1119:
1115:
1106:
1096:
1093:
1090:
1087:
1080:
1079:
1078:
1077:
1076:homeomorphism
1061:
1054:
1038:
1018:
1015:
1012:
1005:
989:
969:
940:
929:
926:
922:
914:
910:
907:
904:
888:
868:
860:
857:
841:
835:
832:
829:
822:
819:
815:
812:
796:
788:
772:
764:
763:
762:
761:consists of:
760:
740:
734:
731:
728:
721:
703:
699:
678:
654:
650:
645:
636:
620:
598:
594:
573:
553:
544:
537:
535:
533:
529:
525:
521:
517:
516:fiber bundles
513:
508:
506:
502:
498:
497:tangent space
494:
490:
486:
485:
468:
448:
445:
442:
422:
402:
382:
362:
342:
335:
320:
317:
311:
305:
285:
276:
274:
261:
239:
216:
210:
190:
183:of the space
170:
162:
158:
154:
138:
131:(for example
118:
111:
107:
106:vector spaces
103:
99:
95:
94:vector bundle
91:
83:
79:
76: ×
75:
71:
67:
63:
60: ×
59:
56:
52:
48:
45:
41:
37:
32:
19:
7867:Moving frame
7862:Morse theory
7852:Gauge theory
7763:
7644:Tensor field
7573:Closed/Exact
7552:Vector field
7520:Distribution
7461:Hypercomplex
7456:Quaternionic
7193:Vector field
7151:Smooth atlas
7069:MathOverflow
7051:
7021:
7001:
6976:
6952:
6926:
6915:
6892:
6862:
6855:Hatcher 2003
6850:
6831:Picard group
6813:Gauge theory
6780:line bundles
6768:Grassmannian
6741:
6727:
6721:
6715:
6712:
6706:
6695:
6556:
6552:
6549:
6539:
6535:
6531:
6527:
6523:
6519:
6515:
6507:
6503:
6499:
6495:
6487:
6483:
6479:
6476:push-forward
6468:
6464:
6457:
6453:
6445:
6440:
6432:
6428:
6424:
6420:
6416:
6414:
6409:
6405:
6401:
6397:
6393:
6391:
6385:
6378:
6370:
6365:is a smooth
6362:
6350:
6346:
6342:
6335:
6331:
6324:
6320:
6313:
6307:
6303:
6298:
6289:
6285:
6278:
6274:
6269:
6264:
6260:
6252:
6248:
6244:
6238:
6106:
6102:
6098:
6094:
6090:
6086:
6082:
6027:
6022:
6014:
6010:
6006:
6002:
5998:
5992:
5988:
5984:
5982:
5977:
5973:
5968:
5960:
5956:
5952:
5945:
5941:
5937:
5933:
5929:
5925:
5921:
5917:
5913:
5909:
5905:
5901:
5897:
5893:
5889:
5887:
5738:
5734:
5729:
5725:
5718:
5713:
5709:
5706:
5701:
5697:
5692:
5688:
5684:
5679:
5675:
5668:
5664:
5660:
5656:
5652:
5648:
5646:
5634:
5630:
5618:
5614:
5608:
5601:
5593:
5592:
5588:
5583:
5574:
5571:matrix group
5568:
5492:
5488:
5484:
5480:
5476:
5472:
5470:
5465:
5461:
5457:
5452:
5444:
5436:
5431:
5426:
5419:
5411:
5398:
5394:
5386:
5382:
5378:
5374:
5370:
5366:
5364:
5358:
5351:
5347:
5340:
5318:
5311:
5240:
5237:Banach space
5232:
5230:
5201:
5191:
5177:
5170:
5166:
5159:
5155:
5151:
5143:
5139:
5138:
5133:
5129:
5125:
5121:
5117:
5113:
5109:
5105:
5101:
5097:
5093:
5089:
5085:
5081:
5077:
5073:
5069:
5065:
5061:
5057:
5053:
5049:
5043:
5029:
5023:
5019:
5015:
5011:
5007:
5003:
4999:
4995:
4991:
4987:
4983:
4973:
4969:
4965:
4960:
4956:
4951:
4947:
4943:
4939:
4931:
4927:
4923:
4919:
4915:
4911:
4907:
4903:
4899:
4895:
4892:endomorphism
4887:
4880:
4876:
4872:
4868:
4864:
4860:
4851:
4847:
4842:
4838:
4834:
4827:
4822:
4818:
4811:
4804:
4800:
4796:
4792:
4788:
4777:
4773:
4768:
4758:
4751:
4744:
4737:
4730:
4726:
4722:
4718:
4714:
4710:
4706:
4698:
4690:
4686:
4682:
4680:
4671:
4667:
4658:
4654:
4650:
4646:
4645:, φ), where
4642:
4638:
4631:
4623:
4619:
4613:
4609:
4605:
4601:
4597:
4595:
4590:
4584:
4571:
4563:
4561:
4545:
4543:sheaves of O
4538:
4536:
4530:
4528:sheaves of O
4517:
4515:
4510:
4506:
4498:
4494:
4490:
4486:
4482:
4478:
4475:locally free
4469:
4466:
4460:
4455:
4451:
4446:
4441:
4425:
4421:
4417:
4413:
4409:
4405:
4401:
4397:
4393:
4389:
4387:
4382:
4374:
4370:
4358:
4347:
4343:
4339:
4336:zero section
4335:
4331:
4327:
4323:
4319:
4315:
4313:
4304:
4300:
4296:
4291:
4287:
4283:
4273:
4265:
4261:
4257:
4253:
4244:
4239:
4235:
4231:
4227:
4221:
4215:
4211:
4207:
4150:over a base
4112:
4108:
4098:
4091:
4084:
4077:
4073:
4066:
4059:
4045:
4038:
4029:
4017:
4015:
3997:
3994:identity map
3989:
3987:
3982:
3978:
3974:
3970:
3966:
3962:
3958:
3954:
3950:
3946:
3942:
3938:
3931:
3924:
3920:
3913:
3906:
3895:
3888:
3886:
3881:
3867:
3861:
3859:
3855:
3844:
3840:
3838:
3828:
3820:
3816:
3805:
3791:
3787:
3772:
3754:
3744:
3737:
3733:
3726:
3719:
3715:
3708:
3701:
3687:
3680:
3667:
3665:
3648:
3395:
3192:
3124:specifies a
3092:fiber bundle
2966:
2838:Čech cocycle
2831:
2826:
2820:
2818:
2718:
2605:
2482:
2220:
2210:
2202:
2194:
2186:
2181:
2175:
2171:
2165:Möbius strip
1941:
1860:
1856:line bundles
1805:
1781:
1554:Every fiber
1553:
1473:
1467:
1466:is called a
1425:
1148:
961:
909:vector space
855:
810:
786:
758:
756:
566:over a base
511:
509:
483:
277:
253:
93:
87:
80:(which is a
77:
73:
65:
61:
57:
50:
46:
36:Möbius strip
7812:Levi-Civita
7802:Generalized
7774:Connections
7724:Lie algebra
7656:Volume form
7557:Vector flow
7530:Pushforward
7525:Lie bracket
7424:Lie algebra
7389:G-structure
7178:Pushforward
7158:Submanifold
6949:Lang, Serge
6696:The famous
6577:modulo the
6375:codimension
6367:submanifold
6345:) whenever
5996: := vl
5615:holomorphic
4916:eigenbundle
4894:bundle Hom(
4701:(named for
4699:Whitney sum
4615:dual bundle
3923:, and then
3656:line bundle
3652:Möbius band
3459:restriction
3306:, there is
1346:isomorphism
811:total space
637:in a fibre
151:could be a
98:topological
90:mathematics
68:is an open
40:line bundle
7935:Stratifold
7893:Diffeology
7689:Associated
7490:Symplectic
7475:Riemannian
7404:Hyperbolic
7331:Submersion
7239:Hopf–Rinow
7173:Submersion
7168:Smooth map
6807:Connection
6702:Raoul Bott
6259:The flow (
6101:such that
5464:-manifold
5407:smoothness
5204:becomes a
5165:such that
5036:functorial
4936:eigenspace
4926:to be the
4789:Hom-bundle
4735:direct sum
4676:functorial
4664:dual space
4587:operations
4534:-modules.
4522:equivalent
4464:-modules.
3939:isomorphic
3874:bundle map
3839:Note that
3833:linear map
3797:, the map
3786:for every
3750:such that
3318:Subbundles
3308:associated
3090:defines a
1740:on all of
861:for every
821:surjection
818:continuous
787:base space
720:projection
586:. A point
298:such that
55:looks like
7817:Principal
7792:Ehresmann
7749:Subbundle
7739:Principal
7714:Fibration
7694:Cotangent
7566:Covectors
7419:Lie group
7399:Hermitian
7342:manifolds
7311:Immersion
7306:Foliation
7244:Noether's
7229:Frobenius
7224:De Rham's
7219:Darboux's
7110:Manifolds
7058:EMS Press
6867:Lang 1995
6687:KO-theory
6613:→
6607:→
6601:→
6595:→
6563:Hausdorff
6467:), where
6284:For each
6205:∇
6156:→
6056:↦
5900:, where (
5852:∞
5844:∈
5763:
5683:) at any
5579:Lie group
5537:
5531:→
5525:∩
5415:bundles,
5288:
5282:→
5276:∩
5270::
4857:bijection
4653:and φ ∈ (
4591:fiberwise
4557:cokernels
4432:over the
4400:) and α:
4363:pointwise
3847:(because
3631:∈
3598:⊂
3565:→
3546:π
3502:π
3472:π
3442:⊂
3413:→
3404:π
3324:Subbundle
3216:π
3060:π
3025:∅
3022:≠
3016:∩
3010:∩
2792:
2786:→
2780:∩
2774::
2719:for some
2639:φ
2635:∘
2627:−
2618:φ
2579:×
2570:∩
2561:→
2546:×
2537:∩
2528::
2519:φ
2515:∘
2507:−
2498:φ
2450:−
2446:π
2439:≅
2414:×
2408::
2399:φ
2374:−
2370:π
2363:≅
2338:×
2332::
2323:φ
2232:→
2141:β
2138:α
2103:β
2100:α
2073:β
2070:α
2039:β
2012:α
1998:open sets
1925:→
1910:×
1875:×
1643:→
1601:dimension
1567:−
1563:π
1507:×
1484:π
1454:φ
1390:−
1386:π
1314:φ
1311:↦
1209:φ
1206:∘
1203:π
1120:−
1116:π
1112:→
1097:×
1091::
1088:φ
1016:⊆
927:−
923:π
839:→
830:π
738:→
729:π
446:×
395:for each
84:instead).
42:over the
7960:Category
7913:Orbifold
7908:K-theory
7898:Diffiety
7622:Pullback
7436:Oriented
7414:Kenmotsu
7394:Hadamard
7340:Types of
7289:Geodesic
7114:Glossary
7016:see Ch.5
6951:(1995),
6913:(2003),
6891:(1978),
6757:See also
6579:relation
6546:K-theory
6359:zero set
5928:through
5604:analytic
4549:-modules
4295:for all
4245:sections
4002:commutes
3901:with an
3876:between
3870:category
3669:morphism
3430:subspace
2967:for all
2825:(or the
2435:→
2359:→
2247:of rank
1944:of rank
1631:function
1476:the map
1300:the map
1245:for all
1074:, and a
159:, or an
157:manifold
82:cylinder
44:1-sphere
7857:History
7840:Related
7754:Tangent
7732:)
7712:)
7679:Adjoint
7671:Bundles
7649:density
7547:Torsion
7513:Vectors
7505:Tensors
7488:)
7473:)
7469:,
7467:Pseudo−
7446:Poisson
7379:Finsler
7374:Fibered
7369:Contact
7367:)
7359:Complex
7357:)
7326:Section
7060:, 2001
6877:Sources
6474:is the
5955:is the
5924:) over
5460:) of a
5239:then a
4879:) over
4733:is the
4626:is the
4570:it has
4562:A rank
4553:kernels
4454:, then
4428:) is a
4350:to the
4204:surface
4104:to the
4021:abelian
3975:trivial
3903:inverse
3825:induced
3246:with a
2180:circle
2178:of the
2169:subsets
2053:may be
1760:, then
1474:locally
1247:vectors
911:on the
718:by the
484:trivial
334:for all
64:(where
7822:Vector
7807:Koszul
7787:Cartan
7782:Affine
7764:Vector
7759:Tensor
7744:Spinor
7734:Normal
7730:Stable
7684:Affine
7588:bundle
7540:bundle
7486:Almost
7409:Kähler
7365:Almost
7355:Almost
7349:Closed
7249:Sard's
7205:(list)
7028:
7009:
6988:
6963:
6937:
6899:
6751:smooth
6738:scheme
6631:then
6373:whose
5940:, and
5473:smooth
5379:smooth
5142:: Let
5140:Remark
4890:of an
4430:module
4200:normal
4025:kernel
4023:; the
3969:, and
3860:cover
3534:gives
3157:action
2215:charts
1343:linear
789:) and
635:origin
102:family
7930:Sheaf
7704:Fiber
7480:Rizza
7451:Prime
7282:Local
7272:Curve
7134:Atlas
6842:Notes
6736:on a
6510:and λ
6277:) of
6267:) → Φ
6243:∇ on
5602:real
5573:GL(k,
5393:, p:
5381:, if
5377:) is
5174:'
5163:'
5154:over
5146:be a
5076:over
4867:over
4833:) or
4725:over
4705:) or
4608:over
4585:Most
4503:tuple
4379:sheaf
4097:from
4076:from
3957:over
3949:over
3831:is a
3808:}) →
2055:glued
1964:over
1693:. If
1341:is a
1297:, and
913:fiber
461:over
110:space
96:is a
53:, it
38:is a
7797:Form
7699:Dual
7632:flow
7495:Tame
7471:Sub−
7384:Flat
7264:Maps
7026:ISBN
7007:ISBN
6986:ISBN
6961:ISBN
6935:ISBN
6897:ISBN
6778:for
6506:) →
6357:The
6354:= 0.
6334:) =
6306:) ∈
6093:and
5951:) ⊂
5647:The
5491:and
5479:and
5389:are
5385:and
5096:) ∈
5018:) =
4979:The
4791:Hom(
4787:The
4767:The
4757:and
4713:and
4697:The
4555:and
4434:ring
4314:Let
4290:) =
4051:and
3977:(or
3930:and
3823:)})
3732:and
2832:The
2483:the
2287:and
2174:and
2163:The
2026:and
1861:The
1782:rank
1377:and
1051:, a
906:real
520:real
155:, a
92:, a
7719:Jet
7067:on
6728:In
6700:of
6681:in
6573:of
6369:of
6361:of
6294:t→∞
6251:on
5906:p*p
5902:p*E
5894:p*E
5128:is
5112:to
5074:f*E
4946:):
4863:to
4810:to
4709:of
4689:on
4520:is
4513:.)
4440:on
4388:If
4346:of
4299:in
4252:on
4248:of
4238:of
4083:to
4018:not
3996:on
3965:of
3912:to
3894:to
3827:by
3790:in
3514:to
3494:of
2834:set
2207:=-1
1673:of
1665:is
1531:on
1268:in
1169:in
1031:of
982:in
881:in
613:in
522:or
491:of
415:in
355:in
104:of
88:In
70:arc
7962::
7710:Co
7056:,
7050:,
6980:,
6959:,
6933:,
6887:;
6770::
6753:.
6725:.
6685:.
6542:.
6538:→
6530:→
6526:×
6520:TE
6518:→
6516:TE
6514::
6508:TE
6502:×
6494::
6486:→
6482::
6465:TM
6463:,
6456:,
6454:TE
6444:,
6441:TE
6435:,
6433:TE
6429:TE
6423:,
6419:,
6412:.
6388:).
6323:)∘
6288:∈
6263:,
6097:∈
6026:,
6023:TE
6017:,
6015:TE
5980:.
5972:,
5969:TE
5963:,
5961:TE
5953:TE
5942:VE
5936:→
5932::
5920:,
5916:,
5908:,
5904:,
5898:VE
5896:→
5778::=
5754:vl
5724:→
5717::
5710:vl
5687:∈
5659:,
5655:,
5643:).
5627:),
5581:.
5534:GL
5468:.
5456:,
5453:TM
5447:,
5445:TM
5397:→
5373:,
5369:,
5354:UV
5343:UV
5285:GL
5169:⊕
5136:.
5124:×
5116:×
5104:⊕
5084:∈
5064:→
5060::
5052:→
5022:⊗
5020:E*
5014:,
5006:×
4994:×
4990:,
4984:E*
4955:→
4934:)-
4922:∈
4910:→
4906::
4898:,
4875:,
4846:,
4826:,
4795:,
4776:⊗
4743:⊕
4721:⊕
4685:,
4678:.
4672:E*
4649:∈
4639:E*
4622:∈
4606:E*
4593:.
4509:→
4493:→
4485:→
4481:×
4404:→
4385:.
4357:({
4326:.
4286:)(
4282:∘
4264:→
4260::
4230:→
4226::
4118:.
4065:→
4058::
4044:→
4037::
4004::
3985:.
3884:.
3865:.
3815:({
3804:({
3764:=
3743:→
3736::
3725:→
3718::
3707:→
3700::
3686:→
3679::
3666:A
3646:.
3255:GL
3190:.
3168:GL
3133:GL
2789:GL
2728:GL
2205:UV
2199:=1
2197:UV
2191:=1
2189:UV
1984:.
1551:.
1189:,
816:a
757:A
534:.
275:.
7728:(
7708:(
7484:(
7465:(
7363:(
7353:(
7116:)
7112:(
7102:e
7095:t
7088:v
7035:.
6995:.
6970:.
6921:.
6906:.
6869:.
6742:X
6722:X
6716:X
6713:S
6707:X
6669:]
6666:C
6663:[
6660:+
6657:]
6654:A
6651:[
6648:=
6645:]
6642:B
6639:[
6619:,
6616:0
6610:C
6604:B
6598:A
6592:0
6559:)
6557:X
6555:(
6553:K
6540:E
6536:E
6532:E
6528:E
6524:E
6512:*
6504:E
6500:E
6498:(
6496:T
6492:*
6488:M
6484:E
6480:p
6472:*
6469:p
6461:*
6458:p
6452:(
6446:E
6437:π
6425:M
6421:p
6417:E
6410:V
6406:E
6402:E
6398:V
6394:E
6386:V
6384:(
6382:v
6379:C
6371:E
6363:V
6351:v
6347:V
6343:V
6341:(
6339:v
6336:C
6332:V
6330:(
6328:v
6325:C
6321:V
6319:(
6317:v
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