5453:
5087:
1994:
3479:
2744:
5448:{\displaystyle {\begin{array}{ll}\Omega _{i}{}^{j}&=d(\Gamma ^{j}{}_{qi}\theta ^{q})+(\Gamma ^{j}{}_{pk}\theta ^{p})\wedge (\Gamma ^{k}{}_{qi}\theta ^{q})\\&\\&=\theta ^{p}\wedge \theta ^{q}\left(\partial _{p}\Gamma ^{j}{}_{qi}+\Gamma ^{j}{}_{pk}\Gamma ^{k}{}_{qi})\right)\\&\\&={\tfrac {1}{2}}\theta ^{p}\wedge \theta ^{q}R_{pqi}{}^{j}\end{array}}}
6770:
1754:
1022:
3244:
7520:
2506:
5057:
7039:
in the 1940s, provides a manner of organizing these many connection forms and the transformation laws connecting them into a single intrinsic form with a single rule for transformation. The disadvantage to this approach is that the forms are no longer defined on the manifold itself, but rather on a
7034:
The connection form, as introduced thus far, depends on a particular choice of frame. In the first definition, the frame is just a local basis of sections. To each frame, a connection form is given with a transformation law for passing from one frame to another. In the second definition, the
4878:
2307:
3875:
6596:
6358:
1989:{\displaystyle D\xi =\sum _{\alpha =1}^{k}D(e_{\alpha }\xi ^{\alpha }(\mathbf {e} ))=\sum _{\alpha =1}^{k}e_{\alpha }\otimes d\xi ^{\alpha }(\mathbf {e} )+\sum _{\alpha =1}^{k}\sum _{\beta =1}^{k}e_{\beta }\otimes \omega _{\alpha }^{\beta }\xi ^{\alpha }(\mathbf {e} ).}
6217:
6987:
7786:
5645:
2876:
2106:
4688:
878:
1451:
6468:
5745:
Given a metric connection with torsion, once can always find a single, unique connection that is torsion-free, this is the Levi-Civita connection. The difference between a
Riemannian connection and its associated Levi-Civita connection is the
4520:
3474:{\displaystyle \Gamma (E)\ {\stackrel {D}{\to }}\ \Gamma (E\otimes \Lambda ^{1}T^{*}M)\ {\stackrel {D}{\to }}\ \Gamma (E\otimes \Lambda ^{2}T^{*}M)\ {\stackrel {D}{\to }}\ \dots \ {\stackrel {D}{\to }}\ \Gamma (E\otimes \Lambda ^{n}T^{*}(M))}
8059:
1198:
1718:
2739:{\displaystyle \omega (\mathbf {e} _{q})=(\mathbf {e} _{p}^{-1}\mathbf {e} _{q})^{-1}d(\mathbf {e} _{p}^{-1}\mathbf {e} _{q})+(\mathbf {e} _{p}^{-1}\mathbf {e} _{q})^{-1}\omega (\mathbf {e} _{p})(\mathbf {e} _{p}^{-1}\mathbf {e} _{q}).}
1553:
7312:
3979:
4362:
4900:
783:
3147:
5998:
2410:
6573:
4706:
3044:
5730:
6862:
2968:
527:
can be extended as well; this defines the local frame. (Here the real numbers are used, although much of the development can be extended to modules over rings in general, and to vector spaces over complex numbers
7630:
3749:
7933:
7148:
623:
4052:
3738:
7278:
81:, rather than on an abstract principal bundle over it. Hence, despite their lack of tensoriality, connection forms continue to be used because of the relative ease of performing calculations with them. In
6765:{\displaystyle \omega _{\alpha }^{\beta }(\mathbf {e} \cdot g)=(g^{-1})_{\gamma }^{\beta }dg_{\alpha }^{\gamma }+(g^{-1})_{\gamma }^{\beta }\omega _{\delta }^{\gamma }(\mathbf {e} )g_{\alpha }^{\delta }.}
2205:
7035:
frames themselves carry some additional structure provided by a Lie group, and changes of frame are constrained to those that take their values in it. The language of principal bundles, pioneered by
6243:
3593:
6088:
4102:
6889:
824:
7657:
3221:
447:
5541:
1305:
2793:
2005:
4554:
1017:{\displaystyle \xi ={\mathbf {e} }{\begin{bmatrix}\xi ^{1}(\mathbf {e} )\\\xi ^{2}(\mathbf {e} )\\\vdots \\\xi ^{k}(\mathbf {e} )\end{bmatrix}}={\mathbf {e} }\,\xi (\mathbf {e} )}
77:
is a natural reinterpretation of the connection form as a tensorial object. On the other hand, the connection form has the advantage that it is a differential form defined on the
3646:
61:
in the first half of the 20th century as part of, and one of the principal motivations for, his method of moving frames. The connection form generally depends on a choice of a
525:
496:
4138:
1354:
6391:
870:
705:
548:
372:
7791:
The 1-form ω constructed in this way respects the transitions between overlapping sets, and therefore descends to give a globally defined 1-form on the principal bundle F
4423:
7956:
1106:
9253:
7304:
1638:
7515:{\displaystyle ((x,g_{U})\in U\times G)\sim ((x,g_{V})\in V\times G)\iff {\mathbf {e} }_{V}={\mathbf {e} }_{U}\cdot h_{UV}{\text{ and }}g_{U}=h_{UV}^{-1}(x)g_{V}.}
848:
683:
8444:
1462:
5052:{\displaystyle \Omega _{i}{}^{j}(\mathbf {e} )=d\omega _{i}{}^{j}(\mathbf {e} )+\sum _{k}\omega _{k}{}^{j}(\mathbf {e} )\wedge \omega _{i}{}^{k}(\mathbf {e} ).}
2470:, then it is possible to define a global connection form in terms of the patching data between the local connection forms on the overlap regions. In detail, a
1073:
1053:
663:
643:
467:
412:
392:
346:
326:
298:
270:
246:
223:
199:
3890:
4257:
9248:
8535:
713:
8559:
4873:{\displaystyle \nabla _{e_{i}}v=\langle Dv,e_{i}\rangle =\sum _{k}e_{k}\left(\nabla _{e_{i}}v^{k}+\sum _{j}\Gamma _{ij}^{k}(\mathbf {e} )v^{j}\right)}
3071:
69:
object. Various generalizations and reinterpretations of the connection form were formulated subsequent to Cartan's initial work. In particular, on a
8754:
2322:
9665:
3881:
2991:
5660:
5932:
6510:
8624:
8286:
6781:
8167:
In a non-holonomic frame, the expression of curvature is further complicated by the fact that the derivatives dθ must be taken into account.
3870:{\displaystyle \Theta ^{i}(\mathbf {e} )=d\theta ^{i}(\mathbf {e} )+\sum _{j}\omega _{j}^{i}(\mathbf {e} )\wedge \theta ^{j}(\mathbf {e} ).}
2902:
8850:
8304:
7558:
8903:
8431:
7882:
7089:
556:
9849:
9528:
9187:
3990:
3676:
9952:
9308:
8389:
8368:
8350:
8332:
8314:
8255:
7219:
2302:{\displaystyle {\mathbf {e} }'={\mathbf {e} }\,g,\quad {\text{i.e., }}\,e'_{\alpha }=\sum _{\beta }e_{\beta }g_{\alpha }^{\beta }.}
2881:
Unlike the connection form, the curvature behaves tensorially under a change of frame, which can be checked directly by using the
9730:
8952:
7007:
8544:
6353:{\displaystyle \nabla _{{\dot {\gamma }}(0)}e_{\alpha }=\sum _{\beta }e_{\beta }\omega _{\alpha }^{\beta }({\dot {\gamma }}(0))}
8935:
3608:
1326:
145:
9957:
6212:{\displaystyle \Gamma (\gamma )_{0}^{t}e_{\alpha }(\gamma (0))=\sum _{\beta }e_{\beta }(\gamma (t))g_{\alpha }^{\beta }(t)}
9581:
9513:
9147:
7533:
6078:-frame to another. Formally, along a curve γ, the following must hold locally (that is, for sufficiently small values of
1571:
1084:
74:
3542:
9606:
9132:
8855:
8629:
9844:
9177:
6982:{\displaystyle \omega ({\mathbf {e} }\cdot g)=g^{*}\omega _{\mathfrak {g}}+{\text{Ad}}_{g^{-1}}\omega (\mathbf {e} )}
5858:. The frames are unitary with respect to an invariant inner product on the spin space, and the group reduces to the
7781:{\displaystyle \omega _{(x,g)}=Ad_{g^{-1}}\pi _{1}^{*}\omega (\mathbf {e} _{U})+\pi _{2}^{*}\omega _{\mathbf {g} }.}
4072:
2147:
to a suitable global object, it is necessary to examine how it behaves when a different choice of basic sections of
9967:
9655:
9475:
9182:
9152:
8860:
8816:
8797:
8564:
8508:
8243:
8077:
1090:
165:
90:
791:
9972:
9327:
8719:
8584:
9829:
7798:. It can be shown that ω is a principal connection in the sense that it reproduces the generators of the right
5640:{\displaystyle \Theta ^{i}(\mathbf {e} )=d\theta ^{i}+\sum _{j}\omega _{j}^{i}(\mathbf {e} )\wedge \theta ^{j}.}
9911:
9783:
9490:
9104:
8969:
8661:
8503:
5463:
3162:
2871:{\displaystyle \Omega (\mathbf {e} )=d\omega (\mathbf {e} )+\omega (\mathbf {e} )\wedge \omega (\mathbf {e} ).}
2423:
manner, since the rule for passing from one frame to another involves the derivatives of the transition matrix
2101:{\displaystyle D\xi (\mathbf {e} )=d\xi (\mathbf {e} )+\omega \xi (\mathbf {e} )=(d+\omega )\xi (\mathbf {e} )}
43:
417:
4683:{\displaystyle Dv=\sum _{k}e_{k}\otimes (dv^{k})+\sum _{j,k}e_{k}\otimes \omega _{j}^{k}(\mathbf {e} )v^{j}.}
1245:
9881:
9568:
9485:
9455:
8801:
8771:
8695:
8685:
8641:
8471:
8424:
5821:
273:
226:
97:
78:
8569:
2131:
sufficient to completely determine the connection locally on the open set over which the basis of sections
9962:
9839:
9695:
9650:
9142:
8761:
8656:
8476:
7019:
5782:
4195:
1208:
277:
125:
3618:
9921:
9876:
9356:
9301:
8791:
8786:
4187:
3238:
to a local section to define this expression). Thus the curvature measures the failure of the sequence
1098:
1035:. The tetrad specifically relates the local frame to an explicit coordinate system on the base manifold
148:. In many cases, connection forms are considered on vector bundles with additional structure: that of a
35:
1446:{\displaystyle D:\Gamma (E\otimes \Lambda ^{*}T^{*}M)\rightarrow \Gamma (E\otimes \Lambda ^{*}T^{*}M)}
501:
472:
300:. It is always possible to construct a local frame, as vector bundles are always defined in terms of
9896:
9824:
9710:
9576:
9538:
9470:
9122:
9060:
8908:
8612:
8602:
8574:
8549:
8459:
8398:
8377:
8089:
7284:
6463:{\displaystyle De_{\alpha }=\sum _{\beta }e_{\beta }\otimes \omega _{\alpha }^{\beta }(\mathbf {e} )}
5078:
301:
105:
9773:
9596:
9586:
9435:
9420:
9376:
9260:
9233:
8942:
8820:
8805:
8734:
8493:
5487:
4515:{\displaystyle \omega _{i}^{j}(\mathbf {e} )=\sum _{k}\Gamma ^{j}{}_{ki}(\mathbf {e} )\theta ^{k}.}
4248:
4113:
1224:
1204:
121:
113:
853:
688:
531:
351:
9906:
9763:
9616:
9430:
9366:
9202:
9157:
9054:
8925:
8729:
8554:
8417:
8265:
6999:
6071:
5483:
4063:
3525:
3489:
1028:
8739:
8054:{\displaystyle \langle X,({\mathbf {e} }\cdot g)^{*}\omega \rangle =\langle (X),\omega \rangle }
5478:
in the tangent bundle with zero torsion. To describe the torsion, note that the vector bundle
1193:{\displaystyle D:\Gamma (E)\rightarrow \Gamma (E\otimes T^{*}M)=\Gamma (E)\otimes \Omega ^{1}M}
9901:
9809:
9670:
9645:
9460:
9371:
9351:
9137:
9117:
9112:
9019:
8930:
8744:
8724:
8579:
8518:
8385:
8364:
8346:
8328:
8310:
8282:
8251:
8239:
8099:
8094:
7036:
6059:
5790:
5763:
5747:
5475:
4693:
One can recover the Levi-Civita connection, in the usual sense, from this by contracting with
4172:
4153:
3054:
2882:
1713:{\displaystyle De_{\alpha }=\sum _{\beta =1}^{k}e_{\beta }\otimes \omega _{\alpha }^{\beta }.}
51:
4190:. Given a Riemannian connection, one can always find a unique, equivalent connection that is
9916:
9814:
9591:
9558:
9543:
9425:
9294:
9275:
9069:
9024:
8947:
8918:
8776:
8709:
8704:
8699:
8689:
8481:
8464:
8274:
8226:
5841:
5817:
5794:
4388:
1548:{\displaystyle D(v\wedge \alpha )=(Dv)\wedge \alpha +(-1)^{{\text{deg}}\,v}v\wedge d\alpha }
1337:
1032:
305:
117:
70:
8296:
7289:
4171:. One may then define a connection that is compatible with this bundle metric, this is the
9886:
9834:
9778:
9758:
9548:
9415:
9386:
9218:
8957:
8913:
8679:
8292:
8145:, chapter 4, for a complete account of the Levi-Civita connection from this point of view.
5067:
4414:
833:
668:
109:
3974:{\displaystyle \Theta ^{i}(\mathbf {e} \,g)=\sum _{j}g_{j}^{i}\Theta ^{j}(\mathbf {e} ).}
3662:
A solder form and the associated torsion may both be described in terms of a local frame
4357:{\displaystyle \nabla _{e_{i}}e_{j}=\sum _{k=1}^{n}\Gamma _{ij}^{k}(\mathbf {e} )e_{k}.}
3049:
is independent of the choice of frame. In particular, Ω is a vector-valued two-form on
9926:
9891:
9871:
9788:
9621:
9611:
9601:
9523:
9495:
9480:
9465:
9381:
9084:
9009:
8979:
8877:
8870:
8810:
8781:
8651:
8646:
8607:
8104:
5855:
4191:
4180:
2774:
1058:
1038:
648:
628:
452:
397:
377:
331:
311:
283:
255:
231:
208:
184:
141:
133:
58:
9946:
9863:
9768:
9680:
9553:
9270:
9094:
9089:
9074:
9064:
9014:
8991:
8865:
8825:
8766:
8714:
8513:
5845:
4168:
4157:
3485:
202:
101:
8158:, II.7 for a complete account of the Levi-Civita connection from this point of view.
5797:, since this group preserves the orthonormality of frames. Other examples include:
9931:
9735:
9720:
9685:
9533:
9518:
9197:
9192:
9034:
9001:
8974:
8882:
8523:
7950:) transforms in the required manner by using the Leibniz rule, and the adjunction:
6031:
5482:
is the tangent bundle. This carries a canonical solder form (sometimes called the
176:
149:
86:
62:
47:
5758:
A more specific type of connection form can be constructed when the vector bundle
3516:
is sometimes equipped with an additional piece of data besides its connection: a
778:{\displaystyle \xi =\sum _{\alpha =1}^{k}e_{\alpha }\xi ^{\alpha }(\mathbf {e} )}
17:
9819:
9793:
9715:
9404:
9343:
9040:
9029:
8986:
8887:
8488:
6368:
5866:
5801:
The usual frames, considered in the preceding section, have structural group GL(
5513:
corresponding to the identity endomorphism of the tangent spaces. In the frame
3517:
3142:{\displaystyle \Omega \in \Gamma (\Lambda ^{2}T^{*}M\otimes {\text{Hom}}(E,E)).}
2765:, does not depend on the choice of basis section used to define the connection.
2312:
Applying the exterior connection to both sides gives the transformation law for
137:
31:
3984:
The frame-independent torsion may also be recovered from the frame components:
2405:{\displaystyle \omega (\mathbf {e} \,g)=g^{-1}dg+g^{-1}\omega (\mathbf {e} )g.}
9700:
9265:
9223:
9049:
8962:
8594:
8498:
8278:
5859:
5092:
4384:
2978:
7552:, which respects the equivalence relation on the overlap regions. First let
3039:{\displaystyle \Omega ={\mathbf {e} }\Omega (\mathbf {e} ){\mathbf {e} }^{*}}
9675:
9626:
9079:
9044:
8749:
8636:
8217:
Chern S. S.; Moser, J.K. (1974), "Real hypersurfaces in complex manifolds",
5775:
4066:
relate the torsion to the curvature. The first
Bianchi identity states that
153:
108:
of differential forms. The connection form is not tensorial because under a
5725:{\displaystyle \Theta ^{i}=\Gamma ^{i}{}_{kj}\theta ^{k}\wedge \theta ^{j}}
4525:
In terms of the connection form, the exterior connection on a vector field
1723:
In terms of the connection form, the exterior connection of any section of
1344:
satisfying this compatibility property, there exists a unique extension of
498:
can thereby be extended to the entire local trivialization, and a basis on
7213:
can be realized in terms of gluing data among the sets of the open cover:
5993:{\displaystyle e_{\alpha }'=\sum _{\beta }e_{\beta }g_{\alpha }^{\beta }.}
9705:
9690:
9243:
9238:
9228:
8619:
8440:
6568:{\displaystyle e_{\alpha }'=\sum _{\beta }e_{\beta }g_{\alpha }^{\beta }}
1625:
4209:
A local frame on the tangent bundle is an ordered list of vector fields
1332:, thus regarding it as a differential operator on the tensor product of
9399:
9361:
8409:
8231:
6857:{\displaystyle \omega ({\mathbf {e} }\cdot g)=g^{-1}dg+g^{-1}\omega g.}
3670:. If θ is a solder form, then it decomposes into the frame components
2115:
and ω refer to the component-wise derivative with respect to the frame
82:
7083:. These are related on the intersections of overlapping open sets by
6022:
if a preferred class of frames is specified, all of which are locally
4167:, then the metric can be extended to the entire vector bundle, as the
2963:{\displaystyle \Omega (\mathbf {e} \,g)=g^{-1}\Omega (\mathbf {e} )g.}
9725:
9317:
8835:
7625:{\displaystyle \pi _{1}:U\times G\to U,\quad \pi _{2}:U\times G\to G}
5851:
2420:
2119:, and a matrix of 1-forms, respectively, acting on the components of
66:
4888:
The curvature 2-form of the Levi-Civita connection is the matrix (Ω
4247:
that are linearly independent at every point of their domain. The
2753:
ensures in particular that the exterior connection of a section of
2187:
is a different choice of local basis. Then there is an invertible
1594:
arises when applying the exterior connection to a particular frame
7928:{\displaystyle \omega ({\mathbf {e} })={\mathbf {e} }^{*}\omega .}
2973:
One interpretation of this transformation law is as follows. Let
8213:, Institute for Advanced Study, mimeographed lecture notes, 1951.
7143:{\displaystyle {\mathbf {e} }_{V}={\mathbf {e} }_{U}\cdot h_{UV}}
618:{\displaystyle \mathbf {e} =(e_{\alpha })_{\alpha =1,2,\dots ,k}}
8361:
A Comprehensive introduction to differential geometry (Volume 3)
8343:
A Comprehensive introduction to differential geometry (Volume 2)
4047:{\displaystyle \Theta =\sum _{i}e_{i}\Theta ^{i}(\mathbf {e} ).}
3733:{\displaystyle \theta =\sum _{i}\theta ^{i}(\mathbf {e} )e_{i}.}
3606:. If a solder form is given, then it is possible to define the
2896:
is a change of frame, then the curvature two-form transforms by
685:
is a local section, defined over the same open set as the frame
9290:
8413:
112:, the connection form transforms in a manner that involves the
7273:{\displaystyle F_{G}E=\left.\coprod _{U}U\times G\right/\sim }
6492:
The curvature form of a compatible connection is, moreover, a
3880:
Much like the curvature, it can be shown that Θ behaves as a
5535:, or in terms of the frame components of the solder form by
645:. This frame can be used to express locally any section of
7263:
7240:
3612:
of the connection (in terms of the exterior connection) as
85:, connection forms are also used broadly in the context of
9286:
5474:
The Levi-Civita connection is characterized as the unique
6883:-valued (locally defined) function. With this in mind,
1223:
to be a connection, it must be correctly coupled to the
8273:, Universitext (Sixth ed.), Springer, Heidelberg,
7809:, and equivariantly intertwines the right action on T(F
1321:
Sometimes it is convenient to extend the definition of
5889:) a given Lie subgroup of the general linear group of
5382:
900:
7959:
7885:
7824:
Connection forms associated to a principal connection
7660:
7561:
7315:
7292:
7222:
7092:
6892:
6784:
6599:
6513:
6394:
6246:
6091:
5935:
5663:
5544:
5090:
4903:
4709:
4557:
4426:
4260:
4116:
4075:
3993:
3893:
3752:
3679:
3621:
3545:
3247:
3165:
3074:
2994:
2905:
2796:
2509:
2325:
2208:
2008:
1757:
1641:
1465:
1357:
1340:
of differential forms. Given an exterior connection
1248:
1109:
1061:
1041:
881:
856:
836:
794:
716:
691:
671:
651:
631:
559:
534:
504:
475:
455:
420:
400:
380:
354:
334:
314:
286:
258:
234:
211:
187:
9862:
9802:
9751:
9744:
9636:
9567:
9504:
9448:
9395:
9342:
9335:
9211:
9170:
9103:
9000:
8896:
8843:
8834:
8670:
8593:
8532:
8452:
3588:{\displaystyle \theta _{x}:T_{x}M\rightarrow E_{x}}
2500:that satisfy the following compatibility condition
8053:
7927:
7844:gives rise to a collection of connection forms on
7780:
7624:
7514:
7298:
7272:
7142:
6981:
6856:
6764:
6567:
6462:
6352:
6211:
5992:
5844:is given, then the structure group reduces to the
5724:
5639:
5447:
5051:
4872:
4682:
4514:
4356:
4132:
4096:
4046:
3973:
3869:
3732:
3640:
3587:
3473:
3215:
3141:
3038:
2962:
2870:
2738:
2404:
2301:
2100:
1988:
1712:
1547:
1445:
1299:
1192:
1067:
1047:
1016:
864:
842:
818:
777:
699:
677:
657:
637:
617:
542:
519:
490:
461:
441:
406:
386:
366:
340:
320:
292:
264:
240:
217:
193:
57:Historically, connection forms were introduced by
8188:See for instance Kobayashi and Nomizu, Volume II.
5793:at each point. The structure group is then the
1727:can now be expressed. For example, suppose that
1598:. Upon applying the exterior connection to the
8122:
5766:. This amounts to a preferred class of frames
8323:Kobayashi, Shoshichi; Nomizu, Katsumi (1996),
8303:Kobayashi, Shoshichi; Nomizu, Katsumi (1996),
7044:The principal connection for a connection form
6867:To interpret each of these terms, recall that
6586:-valued function defined on an open subset of
4107:while the second Bianchi identity states that
4097:{\displaystyle D\Theta =\Omega \wedge \theta }
2173:) to indicate the dependence on the choice of
9302:
8425:
8:
8325:Foundations of Differential Geometry, Vol. 2
8306:Foundations of Differential Geometry, Vol. 1
8048:
8004:
7998:
7960:
6381:With this observation, the connection form ω
5877:be a given vector bundle of fibre dimension
4755:
4733:
819:{\displaystyle \xi ^{\alpha }(\mathbf {e} )}
7635:be the projection maps. Now, for a point (
2757:, when regarded abstractly as a section of
9748:
9339:
9309:
9295:
9287:
8840:
8432:
8418:
8410:
8403:Differential analysis on complex manifolds
8382:Differential analysis on complex manifolds
8267:Riemannian geometry and geometric analysis
7403:
7399:
6026:-related to each other. In formal terms,
5528:The torsion of the connection is given by
3508:is equal to the dimension of the manifold
414:is locally trivial, that is isomorphic to
8230:
8016:
7989:
7973:
7972:
7958:
7913:
7907:
7906:
7893:
7892:
7884:
7768:
7767:
7757:
7752:
7736:
7731:
7718:
7713:
7698:
7693:
7665:
7659:
7598:
7566:
7560:
7503:
7481:
7473:
7460:
7451:
7442:
7429:
7423:
7422:
7412:
7406:
7405:
7375:
7332:
7314:
7291:
7247:
7227:
7221:
7131:
7118:
7112:
7111:
7101:
7095:
7094:
7091:
6971:
6954:
6949:
6944:
6933:
6932:
6922:
6900:
6899:
6891:
6836:
6814:
6792:
6791:
6783:
6753:
6748:
6736:
6727:
6722:
6712:
6707:
6694:
6678:
6673:
6660:
6655:
6642:
6618:
6609:
6604:
6598:
6559:
6554:
6544:
6534:
6518:
6512:
6452:
6443:
6438:
6425:
6415:
6402:
6393:
6327:
6326:
6317:
6312:
6302:
6292:
6279:
6253:
6252:
6251:
6245:
6194:
6189:
6161:
6151:
6120:
6110:
6105:
6090:
5981:
5976:
5966:
5956:
5940:
5934:
5716:
5703:
5690:
5688:
5681:
5668:
5662:
5654:is holonomic, this expression reduces to
5628:
5613:
5604:
5599:
5589:
5576:
5558:
5549:
5543:
5435:
5433:
5420:
5410:
5397:
5381:
5352:
5350:
5343:
5330:
5328:
5321:
5305:
5303:
5296:
5286:
5271:
5258:
5233:
5220:
5218:
5211:
5192:
5179:
5177:
5170:
5151:
5138:
5136:
5129:
5108:
5106:
5099:
5091:
5089:
5038:
5029:
5027:
5020:
5005:
4996:
4994:
4987:
4977:
4962:
4953:
4951:
4944:
4926:
4917:
4915:
4908:
4902:
4859:
4847:
4838:
4830:
4820:
4807:
4795:
4790:
4775:
4765:
4749:
4719:
4714:
4708:
4671:
4659:
4650:
4645:
4632:
4616:
4600:
4581:
4571:
4556:
4503:
4491:
4479:
4477:
4470:
4460:
4445:
4436:
4431:
4425:
4345:
4333:
4324:
4316:
4306:
4295:
4282:
4270:
4265:
4259:
4117:
4115:
4074:
4033:
4024:
4014:
4004:
3992:
3960:
3951:
3941:
3936:
3926:
3912:
3907:
3898:
3892:
3856:
3847:
3832:
3823:
3818:
3808:
3793:
3784:
3766:
3757:
3751:
3721:
3709:
3700:
3690:
3678:
3637:
3620:
3579:
3563:
3550:
3544:
3453:
3443:
3416:
3411:
3409:
3408:
3391:
3386:
3384:
3383:
3368:
3358:
3331:
3326:
3324:
3323:
3308:
3298:
3271:
3266:
3264:
3263:
3246:
3216:{\displaystyle \Omega (v)=D(Dv)=D^{2}v\,}
3212:
3203:
3164:
3156:, the curvature endomorphism is given by
3113:
3101:
3091:
3073:
3030:
3024:
3023:
3014:
3002:
3001:
2993:
2946:
2931:
2917:
2912:
2904:
2857:
2840:
2823:
2803:
2795:
2724:
2719:
2709:
2704:
2699:
2686:
2681:
2665:
2655:
2650:
2640:
2635:
2630:
2614:
2609:
2599:
2594:
2589:
2573:
2563:
2558:
2548:
2543:
2538:
2522:
2517:
2508:
2388:
2373:
2351:
2337:
2332:
2324:
2290:
2285:
2275:
2265:
2249:
2244:
2239:
2231:
2225:
2224:
2211:
2210:
2207:
2090:
2058:
2038:
2018:
2007:
1975:
1966:
1956:
1951:
1938:
1928:
1917:
1907:
1896:
1881:
1872:
1856:
1846:
1835:
1817:
1808:
1798:
1782:
1771:
1756:
1701:
1696:
1683:
1673:
1662:
1649:
1640:
1527:
1522:
1521:
1464:
1431:
1421:
1390:
1380:
1356:
1247:
1215:is the bundle of differential 1-forms on
1181:
1147:
1108:
1060:
1040:
1006:
999:
993:
992:
973:
964:
941:
932:
916:
907:
895:
889:
888:
880:
857:
855:
835:
808:
799:
793:
767:
758:
748:
738:
727:
715:
693:
692:
690:
670:
650:
630:
585:
575:
560:
558:
536:
535:
533:
511:
507:
506:
503:
482:
478:
477:
474:
454:
433:
429:
428:
419:
399:
379:
353:
333:
313:
285:
257:
233:
210:
186:
9666:Covariance and contravariance of vectors
7056:is a vector bundle with structure group
442:{\displaystyle U\times \mathbb {R} ^{k}}
308:of a manifold. That is, given any point
144:, there is an additional invariant: the
42:is a manner of organizing the data of a
8175:
8173:
8155:
8130:
8115:
5062:For simplicity, suppose that the frame
3743:The components of the torsion are then
1300:{\displaystyle D(fv)=v\otimes (df)+fDv}
1031:, such frame fields are referred to as
140:identifying the vector bundle with the
7198:. In detail, using the fact that the
6590:, the connection form transforms via
4186:, the metric connection is called the
132:invariant of a connection form is its
8126:
7816:) with the adjoint representation of
5789:, one works with frames that form an
5781:. For example, in the presence of a
4251:define the Levi-Civita connection by
96:A connection form associates to each
7:
8327:(New ed.), Wiley-Interscience,
8309:(New ed.), Wiley-Interscience,
8142:
5816:The holomorphic tangent bundle of a
3152:In terms of the exterior connection
872:. As a matrix equation, this reads
348:, there exists an open neighborhood
6934:
5742:is symmetric on its lower indices.
5650:Assuming again for simplicity that
4144:Example: the Levi-Civita connection
3641:{\displaystyle \Theta =D\theta .\,}
3512:. In this case, the vector bundle
2123:. Conversely, a matrix of 1-forms
9529:Tensors in curvilinear coordinates
6248:
6092:
5824:). Here the structure group is GL
5678:
5665:
5546:
5340:
5318:
5293:
5283:
5208:
5167:
5126:
5096:
4905:
4827:
4787:
4711:
4467:
4313:
4262:
4121:
4085:
4079:
4021:
3994:
3948:
3895:
3754:
3622:
3440:
3427:
3355:
3342:
3295:
3282:
3248:
3166:
3088:
3081:
3075:
3008:
2995:
2940:
2906:
2797:
2457:is equipped with a trivialization
1418:
1405:
1377:
1364:
1178:
1162:
1131:
1116:
25:
7544:-valued one-form on each product
7186:-frames taken over each point of
6010:. Informally, the vector bundle
5904:, then a matrix-valued function (
3500:Suppose that the fibre dimension
1999:Taking components on both sides,
1574:on the sheaf of graded modules Γ(
1075:being established by the atlas).
394:for which the vector bundle over
8248:Principles of algebraic geometry
8017:
7974:
7908:
7894:
7864:. Then the pullback of ω along
7769:
7732:
7424:
7407:
7113:
7096:
6972:
6901:
6793:
6737:
6619:
6453:
5614:
5559:
5039:
5006:
4963:
4927:
4848:
4660:
4492:
4446:
4334:
4034:
3961:
3908:
3857:
3833:
3794:
3767:
3710:
3598:is a linear isomorphism for all
3025:
3015:
3003:
2947:
2913:
2858:
2841:
2824:
2804:
2720:
2700:
2682:
2651:
2631:
2610:
2590:
2559:
2539:
2518:
2389:
2333:
2226:
2212:
2091:
2059:
2039:
2019:
1976:
1882:
1818:
1007:
994:
974:
942:
917:
890:
858:
809:
768:
561:
520:{\displaystyle \mathbb {R} ^{k}}
491:{\displaystyle \mathbb {R} ^{k}}
469:. The vector space structure on
8211:Topics in Differential Geometry
7593:
5865:Holomorphic tangent bundles on
5736:which vanishes if and only if Γ
5486:, especially in the context of
4417:), then the connection form is
4243:, defined on an open subset of
2238:
8472:Differentiable/Smooth manifold
8039:
8033:
8030:
8027:
8013:
8007:
7986:
7969:
7899:
7889:
7742:
7727:
7678:
7666:
7616:
7584:
7496:
7490:
7400:
7396:
7381:
7362:
7359:
7353:
7338:
7319:
7316:
6976:
6968:
6912:
6896:
6804:
6788:
6741:
6733:
6704:
6687:
6652:
6635:
6629:
6615:
6457:
6449:
6347:
6344:
6338:
6323:
6270:
6264:
6206:
6200:
6182:
6179:
6173:
6167:
6141:
6138:
6132:
6126:
6102:
6095:
5854:on a manifold equipped with a
5618:
5610:
5563:
5555:
5361:
5239:
5204:
5198:
5163:
5157:
5122:
5043:
5035:
5010:
5002:
4967:
4959:
4931:
4923:
4852:
4844:
4664:
4656:
4606:
4590:
4496:
4488:
4450:
4442:
4338:
4330:
4038:
4030:
3965:
3957:
3916:
3904:
3861:
3853:
3837:
3829:
3798:
3790:
3771:
3763:
3714:
3706:
3572:
3468:
3465:
3459:
3430:
3412:
3387:
3377:
3345:
3327:
3317:
3285:
3267:
3257:
3251:
3193:
3184:
3175:
3169:
3133:
3130:
3118:
3084:
3019:
3011:
2951:
2943:
2921:
2909:
2862:
2854:
2845:
2837:
2828:
2820:
2808:
2800:
2730:
2695:
2692:
2677:
2662:
2626:
2620:
2585:
2570:
2534:
2528:
2513:
2393:
2385:
2341:
2329:
2095:
2087:
2081:
2069:
2063:
2055:
2043:
2035:
2023:
2015:
1980:
1972:
1886:
1878:
1825:
1822:
1814:
1791:
1518:
1508:
1496:
1487:
1481:
1469:
1440:
1408:
1402:
1399:
1367:
1314:is the exterior derivative of
1282:
1273:
1261:
1252:
1171:
1165:
1156:
1134:
1128:
1125:
1119:
1011:
1003:
978:
970:
946:
938:
921:
913:
813:
805:
772:
764:
582:
568:
120:, in much the same way as the
1:
9582:Exterior covariant derivative
9514:Tensor (intrinsic definition)
8123:Griffiths & Harris (1978)
7832:-connection ω in a principal
6475:compatible with the structure
6070:provided that the associated
4133:{\displaystyle \,D\Omega =0.}
2491:) of 1-forms defined on each
1562:is homogeneous of degree deg
1085:Exterior covariant derivative
665:. For example, suppose that
9607:Raising and lowering indices
7540:as follows, by specifying a
6477:if the matrix of one-forms ω
4148:As an example, suppose that
2111:where it is understood that
865:{\displaystyle \mathbf {e} }
700:{\displaystyle \mathbb {e} }
543:{\displaystyle \mathbb {C} }
367:{\displaystyle U\subseteq M}
9845:Gluon field strength tensor
9178:Classification of manifolds
6775:Or, using matrix products:
6042:with the natural action of
5077:. Then, employing now the
4175:. For the special case of
2981:corresponding to the frame
1239:is a smooth function, then
9989:
9656:Cartan formalism (physics)
9476:Penrose graphical notation
6229:(which may also depend on
5809:is the fibre dimension of
2772:
1082:
1055:(the coordinate system on
174:
166:Connection (vector bundle)
163:
91:gauge covariant derivative
9328:Glossary of tensor theory
9324:
9254:over commutative algebras
8359:Spivak, Michael (1999b),
8341:Spivak, Michael (1999a),
8279:10.1007/978-3-642-21298-7
7040:larger principal bundle.
5848:acting on unitary frames.
5774:, which are related by a
3884:under a change in frame:
2444:} is an open covering of
1211:of a vector bundle, and Ω
171:Frames on a vector bundle
9953:Connection (mathematics)
9912:Gregorio Ricci-Curbastro
9784:Riemann curvature tensor
9491:Van der Waerden notation
8970:Riemann curvature tensor
7828:Conversely, a principal
6504:Under a change of frame
6363:where the coefficients ω
6062:with the structure of a
5464:Riemann curvature tensor
3536:) such that the mapping
2783:of a connection form in
2478:is a system of matrices
2419:fails to transform in a
2415:Note in particular that
9882:Elwin Bruno Christoffel
9815:Angular momentum tensor
9486:Tetrad (index notation)
9456:Abstract index notation
8250:, John Wiley and sons,
7190:. This is a principal
6233:). Differentiation at
6038:whose typical fibre is
5926:to produce a new frame
5822:almost complex manifold
2751:compatibility condition
2431:Global connection forms
227:differentiable manifold
136:. In the presence of a
79:differentiable manifold
9696:Levi-Civita connection
8762:Manifold with boundary
8477:Differential structure
8055:
7929:
7860:is a local section of
7782:
7626:
7516:
7300:
7274:
7144:
7064:} be an open cover of
7020:adjoint representation
6983:
6858:
6766:
6569:
6485:) takes its values in
6464:
6354:
6213:
6054:Compatible connections
5994:
5900:) is a local frame of
5726:
5641:
5490:) that is the section
5449:
5053:
4874:
4684:
4516:
4358:
4311:
4198:on the tangent bundle
4196:Levi-Civita connection
4134:
4098:
4048:
3975:
3871:
3734:
3642:
3589:
3526:vector-valued one-form
3524:is a globally defined
3475:
3217:
3143:
3040:
2964:
2872:
2740:
2406:
2303:
2102:
1990:
1933:
1912:
1851:
1787:
1714:
1678:
1549:
1447:
1301:
1231:is a local section of
1194:
1069:
1049:
1018:
866:
844:
820:
779:
743:
701:
679:
659:
639:
619:
544:
521:
492:
463:
443:
408:
388:
368:
342:
322:
294:
266:
242:
219:
195:
126:Levi-Civita connection
46:using the language of
9958:Differential geometry
9922:Jan Arnoldus Schouten
9877:Augustin-Louis Cauchy
9357:Differential geometry
8363:, Publish or Perish,
8345:, Publish or Perish,
8264:Jost, Jürgen (2011),
8056:
7938:Changing frames by a
7930:
7783:
7627:
7517:
7301:
7299:{\displaystyle \sim }
7275:
7145:
6984:
6859:
6767:
6570:
6465:
6355:
6214:
6074:maps always send one
6034:with structure group
5995:
5727:
5642:
5517:, the solder form is
5450:
5081:on repeated indices,
5054:
4875:
4685:
4517:
4359:
4291:
4188:Riemannian connection
4135:
4099:
4049:
3976:
3872:
3735:
3643:
3590:
3496:Soldering and torsion
3476:
3218:
3144:
3041:
2965:
2873:
2741:
2407:
2304:
2103:
1991:
1913:
1892:
1831:
1767:
1715:
1658:
1550:
1448:
1302:
1195:
1099:differential operator
1070:
1050:
1019:
867:
845:
821:
780:
723:
702:
680:
660:
640:
620:
545:
522:
493:
464:
444:
409:
389:
369:
343:
328:on the base manifold
323:
302:local trivializations
295:
267:
243:
220:
196:
36:differential geometry
9897:Carl Friedrich Gauss
9830:stress–energy tensor
9825:Cauchy stress tensor
9577:Covariant derivative
9539:Antisymmetric tensor
9471:Multi-index notation
8909:Covariant derivative
8460:Topological manifold
8197:See Chern and Moser.
8090:Ehresmann connection
7957:
7883:
7872:-valued one-form on
7658:
7559:
7313:
7290:
7285:equivalence relation
7220:
7090:
7026:on its Lie algebra.
6890:
6782:
6597:
6511:
6392:
6244:
6089:
6046:as a subgroup of GL(
6003:Two such frames are
5933:
5661:
5542:
5088:
5079:summation convention
4901:
4707:
4555:
4424:
4258:
4114:
4073:
3991:
3891:
3882:contravariant tensor
3750:
3677:
3651:The torsion Θ is an
3619:
3543:
3245:
3163:
3072:
2992:
2903:
2885:. Specifically, if
2794:
2507:
2323:
2206:
2195:matrix of functions
2006:
1755:
1639:
1463:
1355:
1246:
1227:. Specifically, if
1203:where Γ denotes the
1107:
1079:Exterior connections
1059:
1039:
879:
854:
843:{\displaystyle \xi }
834:
792:
714:
689:
678:{\displaystyle \xi }
669:
649:
629:
625:be a local frame on
557:
532:
502:
473:
453:
418:
398:
378:
352:
332:
312:
304:, in analogy to the
284:
256:
232:
209:
185:
118:transition functions
75:principal connection
27:Math/physics concept
9774:Nonmetricity tensor
9629:(2nd-order tensors)
9597:Hodge star operator
9587:Exterior derivative
9436:Transport phenomena
9421:Continuum mechanics
9377:Multilinear algebra
8943:Exterior derivative
8545:Atiyah–Singer index
8494:Riemannian manifold
8384:, Springer-Verlag,
7762:
7723:
7489:
7014:along the function
6758:
6732:
6717:
6683:
6665:
6614:
6564:
6526:
6448:
6322:
6199:
6115:
5986:
5948:
5609:
5525:is the dual basis.
5488:classical mechanics
4843:
4655:
4441:
4329:
4249:Christoffel symbols
3946:
3828:
3053:with values in the
2985:. Then the 2-form
2717:
2648:
2607:
2556:
2295:
2257:
2143:In order to extend
1961:
1706:
1607:, it is the unique
1566:. In other words,
1225:exterior derivative
205:of fibre dimension
122:Christoffel symbols
114:exterior derivative
34:, and specifically
9907:Tullio Levi-Civita
9850:Metric tensor (GR)
9764:Levi-Civita symbol
9617:Tensor contraction
9431:General relativity
9367:Euclidean geometry
9249:Secondary calculus
9203:Singularity theory
9158:Parallel transport
8926:De Rham cohomology
8565:Generalized Stokes
8240:Griffiths, Phillip
8232:10.1007/BF02392146
8051:
7946:, one sees that ω(
7925:
7778:
7748:
7709:
7622:
7512:
7469:
7296:
7270:
7252:
7182:be the set of all
7140:
7000:Maurer-Cartan form
6979:
6854:
6762:
6744:
6718:
6703:
6669:
6651:
6600:
6565:
6550:
6539:
6514:
6496:-valued two-form.
6460:
6434:
6420:
6350:
6308:
6297:
6209:
6185:
6156:
6101:
6072:parallel transport
5990:
5972:
5961:
5936:
5722:
5637:
5595:
5594:
5484:canonical one-form
5445:
5443:
5391:
5049:
4982:
4870:
4826:
4825:
4770:
4680:
4641:
4627:
4576:
4512:
4465:
4427:
4354:
4312:
4130:
4094:
4064:Bianchi identities
4058:Bianchi identities
4044:
4009:
3971:
3932:
3931:
3867:
3814:
3813:
3730:
3695:
3655:-valued 2-form on
3638:
3585:
3490:de Rham cohomology
3471:
3213:
3139:
3065:). Symbolically,
3036:
2960:
2868:
2781:curvature two-form
2736:
2698:
2629:
2588:
2537:
2402:
2299:
2281:
2270:
2245:
2151:is chosen. Write
2098:
1986:
1947:
1710:
1692:
1545:
1443:
1297:
1190:
1065:
1045:
1029:general relativity
1014:
983:
862:
840:
816:
775:
697:
675:
655:
635:
615:
540:
517:
488:
459:
439:
404:
384:
364:
338:
318:
290:
262:
238:
215:
191:
65:, and so is not a
52:differential forms
9968:Maps of manifolds
9940:
9939:
9902:Hermann Grassmann
9858:
9857:
9810:Moment of inertia
9671:Differential form
9646:Affine connection
9461:Einstein notation
9444:
9443:
9372:Exterior calculus
9352:Coordinate system
9284:
9283:
9166:
9165:
8931:Differential form
8585:Whitney embedding
8519:Differential form
8288:978-3-642-21297-0
8100:Affine connection
8095:Cartan connection
7942:-valued function
7454:
7243:
7157:-valued function
7037:Charles Ehresmann
7030:Principal bundles
6947:
6530:
6411:
6374:of the Lie group
6335:
6288:
6261:
6147:
5952:
5791:orthonormal basis
5748:contorsion tensor
5585:
5476:metric connection
5390:
4973:
4816:
4761:
4612:
4567:
4456:
4173:metric connection
4154:Riemannian metric
4000:
3922:
3804:
3686:
3488:(in the sense of
3426:
3421:
3407:
3401:
3396:
3382:
3341:
3336:
3322:
3281:
3276:
3262:
3116:
3055:endomorphism ring
2261:
2242:
1525:
1068:{\displaystyle M}
1048:{\displaystyle M}
658:{\displaystyle E}
638:{\displaystyle E}
462:{\displaystyle U}
407:{\displaystyle U}
387:{\displaystyle x}
341:{\displaystyle M}
321:{\displaystyle x}
293:{\displaystyle E}
265:{\displaystyle E}
241:{\displaystyle M}
218:{\displaystyle k}
194:{\displaystyle E}
18:Connection 1-form
16:(Redirected from
9980:
9973:Smooth functions
9917:Bernhard Riemann
9749:
9592:Exterior product
9559:Two-point tensor
9544:Symmetric tensor
9426:Electromagnetism
9340:
9311:
9304:
9297:
9288:
9276:Stratified space
9234:Fréchet manifold
8948:Interior product
8841:
8538:
8434:
8427:
8420:
8411:
8406:
8394:
8373:
8355:
8337:
8319:
8299:
8272:
8260:
8235:
8234:
8198:
8195:
8189:
8186:
8180:
8177:
8168:
8165:
8159:
8152:
8146:
8139:
8133:
8120:
8060:
8058:
8057:
8052:
8020:
7994:
7993:
7978:
7977:
7934:
7932:
7931:
7926:
7918:
7917:
7912:
7911:
7898:
7897:
7848:. Suppose that
7787:
7785:
7784:
7779:
7774:
7773:
7772:
7761:
7756:
7741:
7740:
7735:
7722:
7717:
7708:
7707:
7706:
7705:
7682:
7681:
7631:
7629:
7628:
7623:
7603:
7602:
7571:
7570:
7521:
7519:
7518:
7513:
7508:
7507:
7488:
7480:
7465:
7464:
7455:
7452:
7450:
7449:
7434:
7433:
7428:
7427:
7417:
7416:
7411:
7410:
7380:
7379:
7337:
7336:
7305:
7303:
7302:
7297:
7279:
7277:
7276:
7271:
7266:
7262:
7251:
7232:
7231:
7202:-frames are all
7149:
7147:
7146:
7141:
7139:
7138:
7123:
7122:
7117:
7116:
7106:
7105:
7100:
7099:
7072:-frames on each
7018:, and Ad is the
6988:
6986:
6985:
6980:
6975:
6964:
6963:
6962:
6961:
6948:
6945:
6939:
6938:
6937:
6927:
6926:
6905:
6904:
6863:
6861:
6860:
6855:
6844:
6843:
6822:
6821:
6797:
6796:
6771:
6769:
6768:
6763:
6757:
6752:
6740:
6731:
6726:
6716:
6711:
6702:
6701:
6682:
6677:
6664:
6659:
6650:
6649:
6622:
6613:
6608:
6574:
6572:
6571:
6566:
6563:
6558:
6549:
6548:
6538:
6522:
6469:
6467:
6466:
6461:
6456:
6447:
6442:
6430:
6429:
6419:
6407:
6406:
6359:
6357:
6356:
6351:
6337:
6336:
6328:
6321:
6316:
6307:
6306:
6296:
6284:
6283:
6274:
6273:
6263:
6262:
6254:
6222:for some matrix
6218:
6216:
6215:
6210:
6198:
6193:
6166:
6165:
6155:
6125:
6124:
6114:
6109:
6058:A connection is
5999:
5997:
5996:
5991:
5985:
5980:
5971:
5970:
5960:
5944:
5873:In general, let
5842:hermitian metric
5818:complex manifold
5795:orthogonal group
5754:Structure groups
5731:
5729:
5728:
5723:
5721:
5720:
5708:
5707:
5698:
5697:
5689:
5686:
5685:
5673:
5672:
5646:
5644:
5643:
5638:
5633:
5632:
5617:
5608:
5603:
5593:
5581:
5580:
5562:
5554:
5553:
5534:
5520:
5512:
5454:
5452:
5451:
5446:
5444:
5440:
5439:
5434:
5431:
5430:
5415:
5414:
5402:
5401:
5392:
5383:
5376:
5373:
5372:
5368:
5364:
5360:
5359:
5351:
5348:
5347:
5338:
5337:
5329:
5326:
5325:
5313:
5312:
5304:
5301:
5300:
5291:
5290:
5276:
5275:
5263:
5262:
5249:
5246:
5245:
5238:
5237:
5228:
5227:
5219:
5216:
5215:
5197:
5196:
5187:
5186:
5178:
5175:
5174:
5156:
5155:
5146:
5145:
5137:
5134:
5133:
5113:
5112:
5107:
5104:
5103:
5076:
5058:
5056:
5055:
5050:
5042:
5034:
5033:
5028:
5025:
5024:
5009:
5001:
5000:
4995:
4992:
4991:
4981:
4966:
4958:
4957:
4952:
4949:
4948:
4930:
4922:
4921:
4916:
4913:
4912:
4879:
4877:
4876:
4871:
4869:
4865:
4864:
4863:
4851:
4842:
4837:
4824:
4812:
4811:
4802:
4801:
4800:
4799:
4780:
4779:
4769:
4754:
4753:
4726:
4725:
4724:
4723:
4689:
4687:
4686:
4681:
4676:
4675:
4663:
4654:
4649:
4637:
4636:
4626:
4605:
4604:
4586:
4585:
4575:
4547:
4521:
4519:
4518:
4513:
4508:
4507:
4495:
4487:
4486:
4478:
4475:
4474:
4464:
4449:
4440:
4435:
4389:cotangent bundle
4363:
4361:
4360:
4355:
4350:
4349:
4337:
4328:
4323:
4310:
4305:
4287:
4286:
4277:
4276:
4275:
4274:
4242:
4232:
4139:
4137:
4136:
4131:
4103:
4101:
4100:
4095:
4053:
4051:
4050:
4045:
4037:
4029:
4028:
4019:
4018:
4008:
3980:
3978:
3977:
3972:
3964:
3956:
3955:
3945:
3940:
3930:
3911:
3903:
3902:
3876:
3874:
3873:
3868:
3860:
3852:
3851:
3836:
3827:
3822:
3812:
3797:
3789:
3788:
3770:
3762:
3761:
3739:
3737:
3736:
3731:
3726:
3725:
3713:
3705:
3704:
3694:
3647:
3645:
3644:
3639:
3594:
3592:
3591:
3586:
3584:
3583:
3568:
3567:
3555:
3554:
3480:
3478:
3477:
3472:
3458:
3457:
3448:
3447:
3424:
3423:
3422:
3420:
3415:
3410:
3405:
3399:
3398:
3397:
3395:
3390:
3385:
3380:
3373:
3372:
3363:
3362:
3339:
3338:
3337:
3335:
3330:
3325:
3320:
3313:
3312:
3303:
3302:
3279:
3278:
3277:
3275:
3270:
3265:
3260:
3222:
3220:
3219:
3214:
3208:
3207:
3148:
3146:
3145:
3140:
3117:
3114:
3106:
3105:
3096:
3095:
3045:
3043:
3042:
3037:
3035:
3034:
3029:
3028:
3018:
3007:
3006:
2969:
2967:
2966:
2961:
2950:
2939:
2938:
2916:
2877:
2875:
2874:
2869:
2861:
2844:
2827:
2807:
2745:
2743:
2742:
2737:
2729:
2728:
2723:
2716:
2708:
2703:
2691:
2690:
2685:
2673:
2672:
2660:
2659:
2654:
2647:
2639:
2634:
2619:
2618:
2613:
2606:
2598:
2593:
2581:
2580:
2568:
2567:
2562:
2555:
2547:
2542:
2527:
2526:
2521:
2411:
2409:
2408:
2403:
2392:
2381:
2380:
2359:
2358:
2336:
2308:
2306:
2305:
2300:
2294:
2289:
2280:
2279:
2269:
2253:
2243:
2240:
2230:
2229:
2220:
2216:
2215:
2186:
2107:
2105:
2104:
2099:
2094:
2062:
2042:
2022:
1995:
1993:
1992:
1987:
1979:
1971:
1970:
1960:
1955:
1943:
1942:
1932:
1927:
1911:
1906:
1885:
1877:
1876:
1861:
1860:
1850:
1845:
1821:
1813:
1812:
1803:
1802:
1786:
1781:
1719:
1717:
1716:
1711:
1705:
1700:
1688:
1687:
1677:
1672:
1654:
1653:
1586:Connection forms
1554:
1552:
1551:
1546:
1532:
1531:
1526:
1523:
1452:
1450:
1449:
1444:
1436:
1435:
1426:
1425:
1395:
1394:
1385:
1384:
1338:exterior algebra
1306:
1304:
1303:
1298:
1199:
1197:
1196:
1191:
1186:
1185:
1152:
1151:
1074:
1072:
1071:
1066:
1054:
1052:
1051:
1046:
1023:
1021:
1020:
1015:
1010:
998:
997:
988:
987:
977:
969:
968:
945:
937:
936:
920:
912:
911:
894:
893:
871:
869:
868:
863:
861:
849:
847:
846:
841:
825:
823:
822:
817:
812:
804:
803:
784:
782:
781:
776:
771:
763:
762:
753:
752:
742:
737:
706:
704:
703:
698:
696:
684:
682:
681:
676:
664:
662:
661:
656:
644:
642:
641:
636:
624:
622:
621:
616:
614:
613:
580:
579:
564:
550:in particular.)
549:
547:
546:
541:
539:
526:
524:
523:
518:
516:
515:
510:
497:
495:
494:
489:
487:
486:
481:
468:
466:
465:
460:
448:
446:
445:
440:
438:
437:
432:
413:
411:
410:
405:
393:
391:
390:
385:
373:
371:
370:
365:
347:
345:
344:
339:
327:
325:
324:
319:
299:
297:
296:
291:
271:
269:
268:
263:
247:
245:
244:
239:
224:
222:
221:
216:
200:
198:
197:
192:
71:principal bundle
63:coordinate frame
21:
9988:
9987:
9983:
9982:
9981:
9979:
9978:
9977:
9943:
9942:
9941:
9936:
9887:Albert Einstein
9854:
9835:Einstein tensor
9798:
9779:Ricci curvature
9759:Kronecker delta
9745:Notable tensors
9740:
9661:Connection form
9638:
9632:
9563:
9549:Tensor operator
9506:
9500:
9440:
9416:Computer vision
9409:
9391:
9387:Tensor calculus
9331:
9320:
9315:
9285:
9280:
9219:Banach manifold
9212:Generalizations
9207:
9162:
9099:
8996:
8958:Ricci curvature
8914:Cotangent space
8892:
8830:
8672:
8666:
8625:Exponential map
8589:
8534:
8528:
8448:
8438:
8405:, Prentice–Hall
8397:
8392:
8376:
8371:
8358:
8353:
8340:
8335:
8322:
8317:
8302:
8289:
8270:
8263:
8258:
8238:
8216:
8206:
8201:
8196:
8192:
8187:
8183:
8178:
8171:
8166:
8162:
8153:
8149:
8140:
8136:
8121:
8117:
8113:
8086:
8068:is a vector on
7985:
7955:
7954:
7905:
7881:
7880:
7826:
7812:
7805:
7794:
7763:
7730:
7694:
7689:
7661:
7656:
7655:
7594:
7562:
7557:
7556:
7528:
7499:
7456:
7453: and
7438:
7421:
7404:
7371:
7328:
7311:
7310:
7288:
7287:
7242:
7239:
7223:
7218:
7217:
7209:
7178:
7163:
7127:
7110:
7093:
7088:
7087:
7082:
7046:
7032:
6997:
6950:
6943:
6928:
6918:
6888:
6887:
6832:
6810:
6780:
6779:
6690:
6638:
6595:
6594:
6540:
6509:
6508:
6502:
6500:Change of frame
6480:
6421:
6398:
6390:
6389:
6384:
6366:
6298:
6275:
6247:
6242:
6241:
6228:
6157:
6116:
6087:
6086:
6056:
6016:structure of a
5962:
5931:
5930:
5925:
5919:may act on the
5910:
5899:
5835:
5827:
5764:structure group
5756:
5741:
5712:
5699:
5687:
5677:
5664:
5659:
5658:
5624:
5572:
5545:
5540:
5539:
5529:
5518:
5495:
5472:
5442:
5441:
5432:
5416:
5406:
5393:
5374:
5370:
5369:
5349:
5339:
5327:
5317:
5302:
5292:
5282:
5281:
5277:
5267:
5254:
5247:
5243:
5242:
5229:
5217:
5207:
5188:
5176:
5166:
5147:
5135:
5125:
5114:
5105:
5095:
5086:
5085:
5071:
5026:
5016:
4993:
4983:
4950:
4940:
4914:
4904:
4899:
4898:
4893:
4886:
4855:
4803:
4791:
4786:
4785:
4781:
4771:
4745:
4715:
4710:
4705:
4704:
4699:
4667:
4628:
4596:
4577:
4553:
4552:
4543:
4535:
4526:
4499:
4476:
4466:
4422:
4421:
4415:Kronecker delta
4412:
4403:
4383:}, denotes the
4341:
4278:
4266:
4261:
4256:
4255:
4234:
4222:
4210:
4156:. If one has a
4146:
4112:
4111:
4071:
4070:
4060:
4020:
4010:
3989:
3988:
3947:
3894:
3889:
3888:
3843:
3780:
3753:
3748:
3747:
3717:
3696:
3675:
3674:
3617:
3616:
3575:
3559:
3546:
3541:
3540:
3498:
3449:
3439:
3364:
3354:
3304:
3294:
3243:
3242:
3234:(we can extend
3199:
3161:
3160:
3097:
3087:
3070:
3069:
3022:
2990:
2989:
2927:
2901:
2900:
2792:
2791:
2777:
2771:
2718:
2680:
2661:
2649:
2608:
2569:
2557:
2516:
2505:
2504:
2499:
2490:
2472:connection form
2465:
2456:
2443:
2433:
2369:
2347:
2321:
2320:
2271:
2209:
2204:
2203:
2184:
2168:
2159:
2141:
2139:Change of frame
2004:
2003:
1962:
1934:
1868:
1852:
1804:
1794:
1753:
1752:
1744:
1736:
1679:
1645:
1637:
1636:
1623:
1606:
1592:connection form
1588:
1517:
1461:
1460:
1427:
1417:
1386:
1376:
1353:
1352:
1244:
1243:
1177:
1143:
1105:
1104:
1087:
1081:
1057:
1056:
1037:
1036:
982:
981:
960:
957:
956:
950:
949:
928:
925:
924:
903:
896:
877:
876:
852:
851:
832:
831:
795:
790:
789:
754:
744:
712:
711:
687:
686:
667:
666:
647:
646:
627:
626:
581:
571:
555:
554:
530:
529:
505:
500:
499:
476:
471:
470:
451:
450:
427:
416:
415:
396:
395:
376:
375:
350:
349:
330:
329:
310:
309:
282:
281:
254:
253:
230:
229:
207:
206:
183:
182:
179:
173:
168:
162:
154:structure group
110:change of basis
40:connection form
28:
23:
22:
15:
12:
11:
5:
9986:
9984:
9976:
9975:
9970:
9965:
9960:
9955:
9945:
9944:
9938:
9937:
9935:
9934:
9929:
9927:Woldemar Voigt
9924:
9919:
9914:
9909:
9904:
9899:
9894:
9892:Leonhard Euler
9889:
9884:
9879:
9874:
9868:
9866:
9864:Mathematicians
9860:
9859:
9856:
9855:
9853:
9852:
9847:
9842:
9837:
9832:
9827:
9822:
9817:
9812:
9806:
9804:
9800:
9799:
9797:
9796:
9791:
9789:Torsion tensor
9786:
9781:
9776:
9771:
9766:
9761:
9755:
9753:
9746:
9742:
9741:
9739:
9738:
9733:
9728:
9723:
9718:
9713:
9708:
9703:
9698:
9693:
9688:
9683:
9678:
9673:
9668:
9663:
9658:
9653:
9648:
9642:
9640:
9634:
9633:
9631:
9630:
9624:
9622:Tensor product
9619:
9614:
9612:Symmetrization
9609:
9604:
9602:Lie derivative
9599:
9594:
9589:
9584:
9579:
9573:
9571:
9565:
9564:
9562:
9561:
9556:
9551:
9546:
9541:
9536:
9531:
9526:
9524:Tensor density
9521:
9516:
9510:
9508:
9502:
9501:
9499:
9498:
9496:Voigt notation
9493:
9488:
9483:
9481:Ricci calculus
9478:
9473:
9468:
9466:Index notation
9463:
9458:
9452:
9450:
9446:
9445:
9442:
9441:
9439:
9438:
9433:
9428:
9423:
9418:
9412:
9410:
9408:
9407:
9402:
9396:
9393:
9392:
9390:
9389:
9384:
9382:Tensor algebra
9379:
9374:
9369:
9364:
9362:Dyadic algebra
9359:
9354:
9348:
9346:
9337:
9333:
9332:
9325:
9322:
9321:
9316:
9314:
9313:
9306:
9299:
9291:
9282:
9281:
9279:
9278:
9273:
9268:
9263:
9258:
9257:
9256:
9246:
9241:
9236:
9231:
9226:
9221:
9215:
9213:
9209:
9208:
9206:
9205:
9200:
9195:
9190:
9185:
9180:
9174:
9172:
9168:
9167:
9164:
9163:
9161:
9160:
9155:
9150:
9145:
9140:
9135:
9130:
9125:
9120:
9115:
9109:
9107:
9101:
9100:
9098:
9097:
9092:
9087:
9082:
9077:
9072:
9067:
9057:
9052:
9047:
9037:
9032:
9027:
9022:
9017:
9012:
9006:
9004:
8998:
8997:
8995:
8994:
8989:
8984:
8983:
8982:
8972:
8967:
8966:
8965:
8955:
8950:
8945:
8940:
8939:
8938:
8928:
8923:
8922:
8921:
8911:
8906:
8900:
8898:
8894:
8893:
8891:
8890:
8885:
8880:
8875:
8874:
8873:
8863:
8858:
8853:
8847:
8845:
8838:
8832:
8831:
8829:
8828:
8823:
8813:
8808:
8794:
8789:
8784:
8779:
8774:
8772:Parallelizable
8769:
8764:
8759:
8758:
8757:
8747:
8742:
8737:
8732:
8727:
8722:
8717:
8712:
8707:
8702:
8692:
8682:
8676:
8674:
8668:
8667:
8665:
8664:
8659:
8654:
8652:Lie derivative
8649:
8647:Integral curve
8644:
8639:
8634:
8633:
8632:
8622:
8617:
8616:
8615:
8608:Diffeomorphism
8605:
8599:
8597:
8591:
8590:
8588:
8587:
8582:
8577:
8572:
8567:
8562:
8557:
8552:
8547:
8541:
8539:
8530:
8529:
8527:
8526:
8521:
8516:
8511:
8506:
8501:
8496:
8491:
8486:
8485:
8484:
8479:
8469:
8468:
8467:
8456:
8454:
8453:Basic concepts
8450:
8449:
8439:
8437:
8436:
8429:
8422:
8414:
8408:
8407:
8395:
8390:
8374:
8369:
8356:
8351:
8338:
8333:
8320:
8315:
8300:
8287:
8261:
8256:
8244:Harris, Joseph
8236:
8214:
8209:Chern, S.-S.,
8205:
8202:
8200:
8199:
8190:
8181:
8169:
8160:
8156:Spivak (1999a)
8147:
8134:
8131:Spivak (1999a)
8114:
8112:
8109:
8108:
8107:
8105:Curvature form
8102:
8097:
8092:
8085:
8082:
8062:
8061:
8050:
8047:
8044:
8041:
8038:
8035:
8032:
8029:
8026:
8023:
8019:
8015:
8012:
8009:
8006:
8003:
8000:
7997:
7992:
7988:
7984:
7981:
7976:
7971:
7968:
7965:
7962:
7936:
7935:
7924:
7921:
7916:
7910:
7904:
7901:
7896:
7891:
7888:
7825:
7822:
7810:
7803:
7792:
7789:
7788:
7777:
7771:
7766:
7760:
7755:
7751:
7747:
7744:
7739:
7734:
7729:
7726:
7721:
7716:
7712:
7704:
7701:
7697:
7692:
7688:
7685:
7680:
7677:
7674:
7671:
7668:
7664:
7633:
7632:
7621:
7618:
7615:
7612:
7609:
7606:
7601:
7597:
7592:
7589:
7586:
7583:
7580:
7577:
7574:
7569:
7565:
7526:
7523:
7522:
7511:
7506:
7502:
7498:
7495:
7492:
7487:
7484:
7479:
7476:
7472:
7468:
7463:
7459:
7448:
7445:
7441:
7437:
7432:
7426:
7420:
7415:
7409:
7402:
7398:
7395:
7392:
7389:
7386:
7383:
7378:
7374:
7370:
7367:
7364:
7361:
7358:
7355:
7352:
7349:
7346:
7343:
7340:
7335:
7331:
7327:
7324:
7321:
7318:
7306:is defined by
7295:
7281:
7280:
7269:
7265:
7261:
7258:
7255:
7250:
7246:
7241:
7238:
7235:
7230:
7226:
7207:
7176:
7161:
7151:
7150:
7137:
7134:
7130:
7126:
7121:
7115:
7109:
7104:
7098:
7080:
7045:
7042:
7031:
7028:
7002:for the group
6993:
6990:
6989:
6978:
6974:
6970:
6967:
6960:
6957:
6953:
6942:
6936:
6931:
6925:
6921:
6917:
6914:
6911:
6908:
6903:
6898:
6895:
6865:
6864:
6853:
6850:
6847:
6842:
6839:
6835:
6831:
6828:
6825:
6820:
6817:
6813:
6809:
6806:
6803:
6800:
6795:
6790:
6787:
6773:
6772:
6761:
6756:
6751:
6747:
6743:
6739:
6735:
6730:
6725:
6721:
6715:
6710:
6706:
6700:
6697:
6693:
6689:
6686:
6681:
6676:
6672:
6668:
6663:
6658:
6654:
6648:
6645:
6641:
6637:
6634:
6631:
6628:
6625:
6621:
6617:
6612:
6607:
6603:
6576:
6575:
6562:
6557:
6553:
6547:
6543:
6537:
6533:
6529:
6525:
6521:
6517:
6501:
6498:
6478:
6471:
6470:
6459:
6455:
6451:
6446:
6441:
6437:
6433:
6428:
6424:
6418:
6414:
6410:
6405:
6401:
6397:
6382:
6364:
6361:
6360:
6349:
6346:
6343:
6340:
6334:
6331:
6325:
6320:
6315:
6311:
6305:
6301:
6295:
6291:
6287:
6282:
6278:
6272:
6269:
6266:
6260:
6257:
6250:
6226:
6220:
6219:
6208:
6205:
6202:
6197:
6192:
6188:
6184:
6181:
6178:
6175:
6172:
6169:
6164:
6160:
6154:
6150:
6146:
6143:
6140:
6137:
6134:
6131:
6128:
6123:
6119:
6113:
6108:
6104:
6100:
6097:
6094:
6055:
6052:
6001:
6000:
5989:
5984:
5979:
5975:
5969:
5965:
5959:
5955:
5951:
5947:
5943:
5939:
5923:
5908:
5897:
5871:
5870:
5863:
5856:spin structure
5849:
5840:). In case a
5833:
5825:
5814:
5755:
5752:
5737:
5734:
5733:
5719:
5715:
5711:
5706:
5702:
5696:
5693:
5684:
5680:
5676:
5671:
5667:
5648:
5647:
5636:
5631:
5627:
5623:
5620:
5616:
5612:
5607:
5602:
5598:
5592:
5588:
5584:
5579:
5575:
5571:
5568:
5565:
5561:
5557:
5552:
5548:
5521:, where again
5471:
5468:
5456:
5455:
5438:
5429:
5426:
5423:
5419:
5413:
5409:
5405:
5400:
5396:
5389:
5386:
5380:
5377:
5375:
5371:
5367:
5363:
5358:
5355:
5346:
5342:
5336:
5333:
5324:
5320:
5316:
5311:
5308:
5299:
5295:
5289:
5285:
5280:
5274:
5270:
5266:
5261:
5257:
5253:
5250:
5248:
5244:
5241:
5236:
5232:
5226:
5223:
5214:
5210:
5206:
5203:
5200:
5195:
5191:
5185:
5182:
5173:
5169:
5165:
5162:
5159:
5154:
5150:
5144:
5141:
5132:
5128:
5124:
5121:
5118:
5115:
5111:
5102:
5098:
5094:
5093:
5060:
5059:
5048:
5045:
5041:
5037:
5032:
5023:
5019:
5015:
5012:
5008:
5004:
4999:
4990:
4986:
4980:
4976:
4972:
4969:
4965:
4961:
4956:
4947:
4943:
4939:
4936:
4933:
4929:
4925:
4920:
4911:
4907:
4889:
4885:
4882:
4881:
4880:
4868:
4862:
4858:
4854:
4850:
4846:
4841:
4836:
4833:
4829:
4823:
4819:
4815:
4810:
4806:
4798:
4794:
4789:
4784:
4778:
4774:
4768:
4764:
4760:
4757:
4752:
4748:
4744:
4741:
4738:
4735:
4732:
4729:
4722:
4718:
4713:
4697:
4691:
4690:
4679:
4674:
4670:
4666:
4662:
4658:
4653:
4648:
4644:
4640:
4635:
4631:
4625:
4622:
4619:
4615:
4611:
4608:
4603:
4599:
4595:
4592:
4589:
4584:
4580:
4574:
4570:
4566:
4563:
4560:
4539:
4531:
4523:
4522:
4511:
4506:
4502:
4498:
4494:
4490:
4485:
4482:
4473:
4469:
4463:
4459:
4455:
4452:
4448:
4444:
4439:
4434:
4430:
4408:
4399:
4365:
4364:
4353:
4348:
4344:
4340:
4336:
4332:
4327:
4322:
4319:
4315:
4309:
4304:
4301:
4298:
4294:
4290:
4285:
4281:
4273:
4269:
4264:
4218:
4194:. This is the
4181:tangent bundle
4145:
4142:
4141:
4140:
4129:
4126:
4123:
4120:
4105:
4104:
4093:
4090:
4087:
4084:
4081:
4078:
4059:
4056:
4055:
4054:
4043:
4040:
4036:
4032:
4027:
4023:
4017:
4013:
4007:
4003:
3999:
3996:
3982:
3981:
3970:
3967:
3963:
3959:
3954:
3950:
3944:
3939:
3935:
3929:
3925:
3921:
3918:
3915:
3910:
3906:
3901:
3897:
3878:
3877:
3866:
3863:
3859:
3855:
3850:
3846:
3842:
3839:
3835:
3831:
3826:
3821:
3817:
3811:
3807:
3803:
3800:
3796:
3792:
3787:
3783:
3779:
3776:
3773:
3769:
3765:
3760:
3756:
3741:
3740:
3729:
3724:
3720:
3716:
3712:
3708:
3703:
3699:
3693:
3689:
3685:
3682:
3649:
3648:
3636:
3633:
3630:
3627:
3624:
3596:
3595:
3582:
3578:
3574:
3571:
3566:
3562:
3558:
3553:
3549:
3497:
3494:
3482:
3481:
3470:
3467:
3464:
3461:
3456:
3452:
3446:
3442:
3438:
3435:
3432:
3429:
3419:
3414:
3404:
3394:
3389:
3379:
3376:
3371:
3367:
3361:
3357:
3353:
3350:
3347:
3344:
3334:
3329:
3319:
3316:
3311:
3307:
3301:
3297:
3293:
3290:
3287:
3284:
3274:
3269:
3259:
3256:
3253:
3250:
3224:
3223:
3211:
3206:
3202:
3198:
3195:
3192:
3189:
3186:
3183:
3180:
3177:
3174:
3171:
3168:
3150:
3149:
3138:
3135:
3132:
3129:
3126:
3123:
3120:
3112:
3109:
3104:
3100:
3094:
3090:
3086:
3083:
3080:
3077:
3047:
3046:
3033:
3027:
3021:
3017:
3013:
3010:
3005:
3000:
2997:
2971:
2970:
2959:
2956:
2953:
2949:
2945:
2942:
2937:
2934:
2930:
2926:
2923:
2920:
2915:
2911:
2908:
2883:Poincaré lemma
2879:
2878:
2867:
2864:
2860:
2856:
2853:
2850:
2847:
2843:
2839:
2836:
2833:
2830:
2826:
2822:
2819:
2816:
2813:
2810:
2806:
2802:
2799:
2787:is defined by
2775:Curvature form
2773:Main article:
2770:
2767:
2747:
2746:
2735:
2732:
2727:
2722:
2715:
2712:
2707:
2702:
2697:
2694:
2689:
2684:
2679:
2676:
2671:
2668:
2664:
2658:
2653:
2646:
2643:
2638:
2633:
2628:
2625:
2622:
2617:
2612:
2605:
2602:
2597:
2592:
2587:
2584:
2579:
2576:
2572:
2566:
2561:
2554:
2551:
2546:
2541:
2536:
2533:
2530:
2525:
2520:
2515:
2512:
2495:
2486:
2461:
2452:
2439:
2432:
2429:
2413:
2412:
2401:
2398:
2395:
2391:
2387:
2384:
2379:
2376:
2372:
2368:
2365:
2362:
2357:
2354:
2350:
2346:
2343:
2340:
2335:
2331:
2328:
2310:
2309:
2298:
2293:
2288:
2284:
2278:
2274:
2268:
2264:
2260:
2256:
2252:
2248:
2237:
2234:
2228:
2223:
2219:
2214:
2164:
2155:
2140:
2137:
2109:
2108:
2097:
2093:
2089:
2086:
2083:
2080:
2077:
2074:
2071:
2068:
2065:
2061:
2057:
2054:
2051:
2048:
2045:
2041:
2037:
2034:
2031:
2028:
2025:
2021:
2017:
2014:
2011:
1997:
1996:
1985:
1982:
1978:
1974:
1969:
1965:
1959:
1954:
1950:
1946:
1941:
1937:
1931:
1926:
1923:
1920:
1916:
1910:
1905:
1902:
1899:
1895:
1891:
1888:
1884:
1880:
1875:
1871:
1867:
1864:
1859:
1855:
1849:
1844:
1841:
1838:
1834:
1830:
1827:
1824:
1820:
1816:
1811:
1807:
1801:
1797:
1793:
1790:
1785:
1780:
1777:
1774:
1770:
1766:
1763:
1760:
1740:
1732:
1721:
1720:
1709:
1704:
1699:
1695:
1691:
1686:
1682:
1676:
1671:
1668:
1665:
1661:
1657:
1652:
1648:
1644:
1619:
1602:
1587:
1584:
1556:
1555:
1544:
1541:
1538:
1535:
1530:
1520:
1516:
1513:
1510:
1507:
1504:
1501:
1498:
1495:
1492:
1489:
1486:
1483:
1480:
1477:
1474:
1471:
1468:
1454:
1453:
1442:
1439:
1434:
1430:
1424:
1420:
1416:
1413:
1410:
1407:
1404:
1401:
1398:
1393:
1389:
1383:
1379:
1375:
1372:
1369:
1366:
1363:
1360:
1336:with the full
1308:
1307:
1296:
1293:
1290:
1287:
1284:
1281:
1278:
1275:
1272:
1269:
1266:
1263:
1260:
1257:
1254:
1251:
1201:
1200:
1189:
1184:
1180:
1176:
1173:
1170:
1167:
1164:
1161:
1158:
1155:
1150:
1146:
1142:
1139:
1136:
1133:
1130:
1127:
1124:
1121:
1118:
1115:
1112:
1083:Main article:
1080:
1077:
1064:
1044:
1025:
1024:
1013:
1009:
1005:
1002:
996:
991:
986:
980:
976:
972:
967:
963:
959:
958:
955:
952:
951:
948:
944:
940:
935:
931:
927:
926:
923:
919:
915:
910:
906:
902:
901:
899:
892:
887:
884:
860:
839:
815:
811:
807:
802:
798:
786:
785:
774:
770:
766:
761:
757:
751:
747:
741:
736:
733:
730:
726:
722:
719:
695:
674:
654:
634:
612:
609:
606:
603:
600:
597:
594:
591:
588:
584:
578:
574:
570:
567:
563:
538:
514:
509:
485:
480:
458:
449:projecting to
436:
431:
426:
423:
403:
383:
363:
360:
357:
337:
317:
289:
278:local sections
272:is an ordered
261:
237:
214:
190:
175:Main article:
172:
169:
161:
160:Vector bundles
158:
142:tangent bundle
134:curvature form
89:, through the
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
9985:
9974:
9971:
9969:
9966:
9964:
9963:Fiber bundles
9961:
9959:
9956:
9954:
9951:
9950:
9948:
9933:
9930:
9928:
9925:
9923:
9920:
9918:
9915:
9913:
9910:
9908:
9905:
9903:
9900:
9898:
9895:
9893:
9890:
9888:
9885:
9883:
9880:
9878:
9875:
9873:
9870:
9869:
9867:
9865:
9861:
9851:
9848:
9846:
9843:
9841:
9838:
9836:
9833:
9831:
9828:
9826:
9823:
9821:
9818:
9816:
9813:
9811:
9808:
9807:
9805:
9801:
9795:
9792:
9790:
9787:
9785:
9782:
9780:
9777:
9775:
9772:
9770:
9769:Metric tensor
9767:
9765:
9762:
9760:
9757:
9756:
9754:
9750:
9747:
9743:
9737:
9734:
9732:
9729:
9727:
9724:
9722:
9719:
9717:
9714:
9712:
9709:
9707:
9704:
9702:
9699:
9697:
9694:
9692:
9689:
9687:
9684:
9682:
9681:Exterior form
9679:
9677:
9674:
9672:
9669:
9667:
9664:
9662:
9659:
9657:
9654:
9652:
9649:
9647:
9644:
9643:
9641:
9635:
9628:
9625:
9623:
9620:
9618:
9615:
9613:
9610:
9608:
9605:
9603:
9600:
9598:
9595:
9593:
9590:
9588:
9585:
9583:
9580:
9578:
9575:
9574:
9572:
9570:
9566:
9560:
9557:
9555:
9554:Tensor bundle
9552:
9550:
9547:
9545:
9542:
9540:
9537:
9535:
9532:
9530:
9527:
9525:
9522:
9520:
9517:
9515:
9512:
9511:
9509:
9503:
9497:
9494:
9492:
9489:
9487:
9484:
9482:
9479:
9477:
9474:
9472:
9469:
9467:
9464:
9462:
9459:
9457:
9454:
9453:
9451:
9447:
9437:
9434:
9432:
9429:
9427:
9424:
9422:
9419:
9417:
9414:
9413:
9411:
9406:
9403:
9401:
9398:
9397:
9394:
9388:
9385:
9383:
9380:
9378:
9375:
9373:
9370:
9368:
9365:
9363:
9360:
9358:
9355:
9353:
9350:
9349:
9347:
9345:
9341:
9338:
9334:
9330:
9329:
9323:
9319:
9312:
9307:
9305:
9300:
9298:
9293:
9292:
9289:
9277:
9274:
9272:
9271:Supermanifold
9269:
9267:
9264:
9262:
9259:
9255:
9252:
9251:
9250:
9247:
9245:
9242:
9240:
9237:
9235:
9232:
9230:
9227:
9225:
9222:
9220:
9217:
9216:
9214:
9210:
9204:
9201:
9199:
9196:
9194:
9191:
9189:
9186:
9184:
9181:
9179:
9176:
9175:
9173:
9169:
9159:
9156:
9154:
9151:
9149:
9146:
9144:
9141:
9139:
9136:
9134:
9131:
9129:
9126:
9124:
9121:
9119:
9116:
9114:
9111:
9110:
9108:
9106:
9102:
9096:
9093:
9091:
9088:
9086:
9083:
9081:
9078:
9076:
9073:
9071:
9068:
9066:
9062:
9058:
9056:
9053:
9051:
9048:
9046:
9042:
9038:
9036:
9033:
9031:
9028:
9026:
9023:
9021:
9018:
9016:
9013:
9011:
9008:
9007:
9005:
9003:
8999:
8993:
8992:Wedge product
8990:
8988:
8985:
8981:
8978:
8977:
8976:
8973:
8971:
8968:
8964:
8961:
8960:
8959:
8956:
8954:
8951:
8949:
8946:
8944:
8941:
8937:
8936:Vector-valued
8934:
8933:
8932:
8929:
8927:
8924:
8920:
8917:
8916:
8915:
8912:
8910:
8907:
8905:
8902:
8901:
8899:
8895:
8889:
8886:
8884:
8881:
8879:
8876:
8872:
8869:
8868:
8867:
8866:Tangent space
8864:
8862:
8859:
8857:
8854:
8852:
8849:
8848:
8846:
8842:
8839:
8837:
8833:
8827:
8824:
8822:
8818:
8814:
8812:
8809:
8807:
8803:
8799:
8795:
8793:
8790:
8788:
8785:
8783:
8780:
8778:
8775:
8773:
8770:
8768:
8765:
8763:
8760:
8756:
8753:
8752:
8751:
8748:
8746:
8743:
8741:
8738:
8736:
8733:
8731:
8728:
8726:
8723:
8721:
8718:
8716:
8713:
8711:
8708:
8706:
8703:
8701:
8697:
8693:
8691:
8687:
8683:
8681:
8678:
8677:
8675:
8669:
8663:
8660:
8658:
8655:
8653:
8650:
8648:
8645:
8643:
8640:
8638:
8635:
8631:
8630:in Lie theory
8628:
8627:
8626:
8623:
8621:
8618:
8614:
8611:
8610:
8609:
8606:
8604:
8601:
8600:
8598:
8596:
8592:
8586:
8583:
8581:
8578:
8576:
8573:
8571:
8568:
8566:
8563:
8561:
8558:
8556:
8553:
8551:
8548:
8546:
8543:
8542:
8540:
8537:
8533:Main results
8531:
8525:
8522:
8520:
8517:
8515:
8514:Tangent space
8512:
8510:
8507:
8505:
8502:
8500:
8497:
8495:
8492:
8490:
8487:
8483:
8480:
8478:
8475:
8474:
8473:
8470:
8466:
8463:
8462:
8461:
8458:
8457:
8455:
8451:
8446:
8442:
8435:
8430:
8428:
8423:
8421:
8416:
8415:
8412:
8404:
8400:
8396:
8393:
8391:0-387-90419-0
8387:
8383:
8379:
8375:
8372:
8370:0-914098-72-1
8366:
8362:
8357:
8354:
8352:0-914098-71-3
8348:
8344:
8339:
8336:
8334:0-471-15732-5
8330:
8326:
8321:
8318:
8316:0-471-15733-3
8312:
8308:
8307:
8301:
8298:
8294:
8290:
8284:
8280:
8276:
8269:
8268:
8262:
8259:
8257:0-471-05059-8
8253:
8249:
8245:
8241:
8237:
8233:
8228:
8224:
8220:
8215:
8212:
8208:
8207:
8203:
8194:
8191:
8185:
8182:
8179:Wells (1973).
8176:
8174:
8170:
8164:
8161:
8157:
8151:
8148:
8144:
8138:
8135:
8132:
8128:
8124:
8119:
8116:
8110:
8106:
8103:
8101:
8098:
8096:
8093:
8091:
8088:
8087:
8083:
8081:
8079:
8075:
8071:
8067:
8045:
8042:
8036:
8024:
8021:
8010:
8001:
7995:
7990:
7982:
7979:
7966:
7963:
7953:
7952:
7951:
7949:
7945:
7941:
7922:
7919:
7914:
7902:
7886:
7879:
7878:
7877:
7875:
7871:
7867:
7863:
7859:
7855:
7851:
7847:
7843:
7839:
7835:
7831:
7823:
7821:
7819:
7815:
7808:
7801:
7797:
7775:
7764:
7758:
7753:
7749:
7745:
7737:
7724:
7719:
7714:
7710:
7702:
7699:
7695:
7690:
7686:
7683:
7675:
7672:
7669:
7662:
7654:
7653:
7652:
7650:
7646:
7642:
7638:
7619:
7613:
7610:
7607:
7604:
7599:
7595:
7590:
7587:
7581:
7578:
7575:
7572:
7567:
7563:
7555:
7554:
7553:
7551:
7547:
7543:
7539:
7537:
7531:
7509:
7504:
7500:
7493:
7485:
7482:
7477:
7474:
7470:
7466:
7461:
7457:
7446:
7443:
7439:
7435:
7430:
7418:
7413:
7393:
7390:
7387:
7384:
7376:
7372:
7368:
7365:
7356:
7350:
7347:
7344:
7341:
7333:
7329:
7325:
7322:
7309:
7308:
7307:
7293:
7286:
7267:
7259:
7256:
7253:
7248:
7244:
7236:
7233:
7228:
7224:
7216:
7215:
7214:
7212:
7205:
7201:
7197:
7194:-bundle over
7193:
7189:
7185:
7181:
7173:
7171:
7167:
7160:
7156:
7135:
7132:
7128:
7124:
7119:
7107:
7102:
7086:
7085:
7084:
7079:
7076:, denoted by
7075:
7071:
7068:, along with
7067:
7063:
7059:
7055:
7051:
7048:Suppose that
7043:
7041:
7038:
7029:
7027:
7025:
7021:
7017:
7013:
7009:
7005:
7001:
6996:
6965:
6958:
6955:
6951:
6940:
6929:
6923:
6919:
6915:
6909:
6906:
6893:
6886:
6885:
6884:
6882:
6878:
6874:
6870:
6851:
6848:
6845:
6840:
6837:
6833:
6829:
6826:
6823:
6818:
6815:
6811:
6807:
6801:
6798:
6785:
6778:
6777:
6776:
6759:
6754:
6749:
6745:
6728:
6723:
6719:
6713:
6708:
6698:
6695:
6691:
6684:
6679:
6674:
6670:
6666:
6661:
6656:
6646:
6643:
6639:
6632:
6626:
6623:
6610:
6605:
6601:
6593:
6592:
6591:
6589:
6585:
6581:
6560:
6555:
6551:
6545:
6541:
6535:
6531:
6527:
6523:
6519:
6515:
6507:
6506:
6505:
6499:
6497:
6495:
6490:
6488:
6484:
6476:
6444:
6439:
6435:
6431:
6426:
6422:
6416:
6412:
6408:
6403:
6399:
6395:
6388:
6387:
6386:
6379:
6377:
6373:
6370:
6341:
6332:
6329:
6318:
6313:
6309:
6303:
6299:
6293:
6289:
6285:
6280:
6276:
6267:
6258:
6255:
6240:
6239:
6238:
6236:
6232:
6225:
6203:
6195:
6190:
6186:
6176:
6170:
6162:
6158:
6152:
6148:
6144:
6135:
6129:
6121:
6117:
6111:
6106:
6098:
6085:
6084:
6083:
6081:
6077:
6073:
6069:
6065:
6061:
6053:
6051:
6049:
6045:
6041:
6037:
6033:
6029:
6025:
6021:
6019:
6013:
6009:
6007:
5987:
5982:
5977:
5973:
5967:
5963:
5957:
5953:
5949:
5945:
5941:
5937:
5929:
5928:
5927:
5922:
5918:
5914:
5907:
5903:
5896:
5892:
5888:
5884:
5880:
5876:
5868:
5864:
5861:
5857:
5853:
5850:
5847:
5846:unitary group
5843:
5839:
5831:
5823:
5819:
5815:
5812:
5808:
5804:
5800:
5799:
5798:
5796:
5792:
5788:
5784:
5780:
5777:
5773:
5769:
5765:
5761:
5753:
5751:
5749:
5743:
5740:
5717:
5713:
5709:
5704:
5700:
5694:
5691:
5682:
5674:
5669:
5657:
5656:
5655:
5653:
5634:
5629:
5625:
5621:
5605:
5600:
5596:
5590:
5586:
5582:
5577:
5573:
5569:
5566:
5550:
5538:
5537:
5536:
5533:
5526:
5524:
5516:
5511:
5507:
5503:
5499:
5493:
5489:
5485:
5481:
5477:
5469:
5467:
5465:
5461:
5436:
5427:
5424:
5421:
5417:
5411:
5407:
5403:
5398:
5394:
5387:
5384:
5378:
5365:
5356:
5353:
5344:
5334:
5331:
5322:
5314:
5309:
5306:
5297:
5287:
5278:
5272:
5268:
5264:
5259:
5255:
5251:
5234:
5230:
5224:
5221:
5212:
5201:
5193:
5189:
5183:
5180:
5171:
5160:
5152:
5148:
5142:
5139:
5130:
5119:
5116:
5109:
5100:
5084:
5083:
5082:
5080:
5074:
5069:
5065:
5046:
5030:
5021:
5017:
5013:
4997:
4988:
4984:
4978:
4974:
4970:
4954:
4945:
4941:
4937:
4934:
4918:
4909:
4897:
4896:
4895:
4892:
4883:
4866:
4860:
4856:
4839:
4834:
4831:
4821:
4817:
4813:
4808:
4804:
4796:
4792:
4782:
4776:
4772:
4766:
4762:
4758:
4750:
4746:
4742:
4739:
4736:
4730:
4727:
4720:
4716:
4703:
4702:
4701:
4696:
4677:
4672:
4668:
4651:
4646:
4642:
4638:
4633:
4629:
4623:
4620:
4617:
4613:
4609:
4601:
4597:
4593:
4587:
4582:
4578:
4572:
4568:
4564:
4561:
4558:
4551:
4550:
4549:
4546:
4542:
4538:
4534:
4529:
4509:
4504:
4500:
4483:
4480:
4471:
4461:
4457:
4453:
4437:
4432:
4428:
4420:
4419:
4418:
4416:
4411:
4407:
4402:
4398:
4394:
4390:
4386:
4382:
4379:= 1, 2, ...,
4378:
4374:
4370:
4351:
4346:
4342:
4325:
4320:
4317:
4307:
4302:
4299:
4296:
4292:
4288:
4283:
4279:
4271:
4267:
4254:
4253:
4252:
4250:
4246:
4241:
4237:
4230:
4227:= 1, 2, ...,
4226:
4221:
4217:
4213:
4207:
4205:
4201:
4197:
4193:
4189:
4185:
4182:
4178:
4174:
4170:
4169:bundle metric
4166:
4162:
4159:
4158:vector bundle
4155:
4151:
4143:
4127:
4124:
4118:
4110:
4109:
4108:
4091:
4088:
4082:
4076:
4069:
4068:
4067:
4065:
4057:
4041:
4025:
4015:
4011:
4005:
4001:
3997:
3987:
3986:
3985:
3968:
3952:
3942:
3937:
3933:
3927:
3923:
3919:
3913:
3899:
3887:
3886:
3885:
3883:
3864:
3848:
3844:
3840:
3824:
3819:
3815:
3809:
3805:
3801:
3785:
3781:
3777:
3774:
3758:
3746:
3745:
3744:
3727:
3722:
3718:
3701:
3697:
3691:
3687:
3683:
3680:
3673:
3672:
3671:
3669:
3665:
3660:
3658:
3654:
3634:
3631:
3628:
3625:
3615:
3614:
3613:
3611:
3610:
3605:
3601:
3580:
3576:
3569:
3564:
3560:
3556:
3551:
3547:
3539:
3538:
3537:
3535:
3531:
3527:
3523:
3519:
3515:
3511:
3507:
3503:
3495:
3493:
3491:
3487:
3486:chain complex
3462:
3454:
3450:
3444:
3436:
3433:
3417:
3402:
3392:
3374:
3369:
3365:
3359:
3351:
3348:
3332:
3314:
3309:
3305:
3299:
3291:
3288:
3272:
3254:
3241:
3240:
3239:
3237:
3233:
3229:
3209:
3204:
3200:
3196:
3190:
3187:
3181:
3178:
3172:
3159:
3158:
3157:
3155:
3136:
3127:
3124:
3121:
3110:
3107:
3102:
3098:
3092:
3078:
3068:
3067:
3066:
3064:
3060:
3056:
3052:
3031:
2998:
2988:
2987:
2986:
2984:
2980:
2976:
2957:
2954:
2935:
2932:
2928:
2924:
2918:
2899:
2898:
2897:
2895:
2892:
2888:
2884:
2865:
2851:
2848:
2834:
2831:
2817:
2814:
2811:
2790:
2789:
2788:
2786:
2782:
2776:
2768:
2766:
2764:
2760:
2756:
2752:
2733:
2725:
2713:
2710:
2705:
2687:
2674:
2669:
2666:
2656:
2644:
2641:
2636:
2623:
2615:
2603:
2600:
2595:
2582:
2577:
2574:
2564:
2552:
2549:
2544:
2531:
2523:
2510:
2503:
2502:
2501:
2498:
2494:
2489:
2485:
2481:
2477:
2473:
2469:
2464:
2460:
2455:
2451:
2447:
2442:
2438:
2430:
2428:
2426:
2422:
2418:
2399:
2396:
2382:
2377:
2374:
2370:
2366:
2363:
2360:
2355:
2352:
2348:
2344:
2338:
2326:
2319:
2318:
2317:
2315:
2296:
2291:
2286:
2282:
2276:
2272:
2266:
2262:
2258:
2254:
2250:
2246:
2235:
2232:
2221:
2217:
2202:
2201:
2200:
2198:
2194:
2190:
2183:
2180:Suppose that
2178:
2176:
2172:
2167:
2163:
2158:
2154:
2150:
2146:
2138:
2136:
2134:
2130:
2126:
2122:
2118:
2114:
2084:
2078:
2075:
2072:
2066:
2052:
2049:
2046:
2032:
2029:
2026:
2012:
2009:
2002:
2001:
2000:
1983:
1967:
1963:
1957:
1952:
1948:
1944:
1939:
1935:
1929:
1924:
1921:
1918:
1914:
1908:
1903:
1900:
1897:
1893:
1889:
1873:
1869:
1865:
1862:
1857:
1853:
1847:
1842:
1839:
1836:
1832:
1828:
1809:
1805:
1799:
1795:
1788:
1783:
1778:
1775:
1772:
1768:
1764:
1761:
1758:
1751:
1750:
1749:
1747:
1743:
1739:
1735:
1730:
1726:
1707:
1702:
1697:
1693:
1689:
1684:
1680:
1674:
1669:
1666:
1663:
1659:
1655:
1650:
1646:
1642:
1635:
1634:
1633:
1631:
1627:
1622:
1618:
1614:
1610:
1605:
1601:
1597:
1593:
1585:
1583:
1581:
1577:
1573:
1569:
1565:
1561:
1542:
1539:
1536:
1533:
1528:
1514:
1511:
1505:
1502:
1499:
1493:
1490:
1484:
1478:
1475:
1472:
1466:
1459:
1458:
1457:
1437:
1432:
1428:
1422:
1414:
1411:
1396:
1391:
1387:
1381:
1373:
1370:
1361:
1358:
1351:
1350:
1349:
1347:
1343:
1339:
1335:
1331:
1330:-valued forms
1329:
1325:to arbitrary
1324:
1319:
1317:
1313:
1294:
1291:
1288:
1285:
1279:
1276:
1270:
1267:
1264:
1258:
1255:
1249:
1242:
1241:
1240:
1238:
1234:
1230:
1226:
1222:
1218:
1214:
1210:
1206:
1187:
1182:
1174:
1168:
1159:
1153:
1148:
1144:
1140:
1137:
1122:
1113:
1110:
1103:
1102:
1101:
1100:
1097:is a type of
1096:
1092:
1086:
1078:
1076:
1062:
1042:
1034:
1030:
1000:
989:
984:
965:
961:
953:
933:
929:
908:
904:
897:
885:
882:
875:
874:
873:
850:in the frame
837:
829:
800:
796:
759:
755:
749:
745:
739:
734:
731:
728:
724:
720:
717:
710:
709:
708:
672:
652:
632:
610:
607:
604:
601:
598:
595:
592:
589:
586:
576:
572:
565:
551:
512:
483:
456:
434:
424:
421:
401:
381:
361:
358:
355:
335:
315:
307:
303:
287:
279:
275:
259:
251:
235:
228:
212:
204:
203:vector bundle
188:
178:
170:
167:
159:
157:
155:
151:
147:
143:
139:
135:
131:
127:
123:
119:
115:
111:
107:
103:
102:vector bundle
99:
94:
92:
88:
84:
80:
76:
72:
68:
64:
60:
55:
53:
49:
48:moving frames
45:
41:
37:
33:
19:
9932:Hermann Weyl
9736:Vector space
9721:Pseudotensor
9686:Fiber bundle
9660:
9639:abstractions
9534:Mixed tensor
9519:Tensor field
9326:
9198:Moving frame
9193:Morse theory
9183:Gauge theory
9127:
8975:Tensor field
8904:Closed/Exact
8883:Vector field
8851:Distribution
8792:Hypercomplex
8787:Quaternionic
8524:Vector field
8482:Smooth atlas
8402:
8381:
8360:
8342:
8324:
8305:
8266:
8247:
8222:
8218:
8210:
8193:
8184:
8163:
8150:
8137:
8127:Wells (1980)
8118:
8076:denotes the
8073:
8069:
8065:
8063:
7947:
7943:
7939:
7937:
7873:
7869:
7865:
7861:
7857:
7853:
7849:
7845:
7841:
7837:
7833:
7829:
7827:
7817:
7813:
7806:
7799:
7795:
7790:
7648:
7644:
7640:
7636:
7634:
7549:
7545:
7541:
7535:
7529:
7524:
7282:
7210:
7203:
7199:
7195:
7191:
7187:
7183:
7179:
7174:
7169:
7165:
7158:
7154:
7152:
7077:
7073:
7069:
7065:
7061:
7057:
7053:
7049:
7047:
7033:
7023:
7015:
7011:
7003:
6994:
6991:
6880:
6876:
6872:
6868:
6866:
6774:
6587:
6583:
6579:
6577:
6503:
6493:
6491:
6486:
6482:
6474:
6472:
6380:
6375:
6371:
6362:
6234:
6230:
6223:
6221:
6079:
6075:
6067:
6063:
6057:
6047:
6043:
6039:
6035:
6032:fibre bundle
6027:
6023:
6017:
6015:
6011:
6005:
6004:
6002:
5920:
5916:
5912:
5905:
5901:
5894:
5890:
5886:
5882:
5878:
5874:
5872:
5867:CR manifolds
5837:
5829:
5810:
5806:
5802:
5786:
5778:
5771:
5767:
5759:
5757:
5744:
5738:
5735:
5651:
5649:
5531:
5527:
5522:
5514:
5509:
5505:
5501:
5497:
5491:
5479:
5473:
5459:
5457:
5072:
5063:
5061:
4890:
4887:
4694:
4692:
4548:is given by
4544:
4540:
4536:
4532:
4527:
4524:
4409:
4405:
4400:
4396:
4392:
4391:, such that
4380:
4376:
4372:
4368:
4366:
4244:
4239:
4235:
4228:
4224:
4219:
4215:
4211:
4208:
4203:
4199:
4192:torsion-free
4183:
4176:
4164:
4160:
4149:
4147:
4106:
4061:
3983:
3879:
3742:
3667:
3663:
3661:
3656:
3652:
3650:
3607:
3603:
3599:
3597:
3533:
3529:
3521:
3513:
3509:
3505:
3501:
3499:
3483:
3235:
3231:
3227:
3225:
3153:
3151:
3062:
3058:
3050:
3048:
2982:
2974:
2972:
2893:
2890:
2886:
2880:
2784:
2780:
2778:
2762:
2758:
2754:
2750:
2748:
2496:
2492:
2487:
2483:
2479:
2475:
2471:
2467:
2462:
2458:
2453:
2449:
2445:
2440:
2436:
2434:
2424:
2416:
2414:
2313:
2311:
2196:
2192:
2188:
2181:
2179:
2174:
2170:
2165:
2161:
2156:
2152:
2148:
2144:
2142:
2135:is defined.
2132:
2128:
2124:
2120:
2116:
2112:
2110:
1998:
1745:
1741:
1737:
1733:
1728:
1724:
1722:
1629:
1620:
1616:
1612:
1608:
1603:
1599:
1595:
1591:
1589:
1579:
1575:
1567:
1563:
1559:
1557:
1455:
1345:
1341:
1333:
1327:
1322:
1320:
1315:
1311:
1309:
1236:
1232:
1228:
1220:
1216:
1212:
1202:
1094:
1088:
1026:
827:
826:denotes the
787:
552:
249:
180:
177:Frame bundle
150:fiber bundle
146:torsion form
129:
95:
87:gauge theory
56:
39:
29:
9872:Élie Cartan
9820:Spin tensor
9794:Weyl tensor
9752:Mathematics
9716:Multivector
9507:definitions
9405:Engineering
9344:Mathematics
9143:Levi-Civita
9133:Generalized
9105:Connections
9055:Lie algebra
8987:Volume form
8888:Vector flow
8861:Pushforward
8856:Lie bracket
8755:Lie algebra
8720:G-structure
8509:Pushforward
8489:Submanifold
8399:Wells, R.O.
8378:Wells, R.O.
8225:: 219–271,
8143:Jost (2011)
8078:pushforward
7802:action on F
7538:-connection
7532:, define a
7206:-related, F
7164:defined on
7008:pulled back
6385:defined by
6369:Lie algebra
6367:are in the
6066:-bundle on
4894:) given by
3522:solder form
3518:solder form
2448:, and each
2241:i.e.,
250:local frame
138:solder form
128:. The main
59:Élie Cartan
32:mathematics
9947:Categories
9701:Linear map
9569:Operations
9266:Stratifold
9224:Diffeology
9020:Associated
8821:Symplectic
8806:Riemannian
8735:Hyperbolic
8662:Submersion
8570:Hopf–Rinow
8504:Submersion
8499:Smooth map
8219:Acta Math.
8204:References
7868:defines a
7534:principal
7283:where the
6060:compatible
5860:spin group
5762:carries a
5070:, so that
4385:dual basis
4179:being the
4152:carries a
2979:dual basis
2199:such that
1632:such that
1572:derivation
1456:such that
1091:connection
828:components
164:See also:
44:connection
9840:EM tensor
9676:Dimension
9627:Transpose
9148:Principal
9123:Ehresmann
9080:Subbundle
9070:Principal
9045:Fibration
9025:Cotangent
8897:Covectors
8750:Lie group
8730:Hermitian
8673:manifolds
8642:Immersion
8637:Foliation
8575:Noether's
8560:Frobenius
8555:De Rham's
8550:Darboux's
8441:Manifolds
8049:⟩
8046:ω
8022:⋅
8005:⟨
7999:⟩
7996:ω
7991:∗
7980:⋅
7961:⟨
7920:ω
7915:∗
7887:ω
7765:ω
7759:∗
7750:π
7725:ω
7720:∗
7711:π
7700:−
7663:ω
7617:→
7611:×
7596:π
7585:→
7579:×
7564:π
7483:−
7436:⋅
7401:⟺
7391:×
7385:∈
7357:∼
7348:×
7342:∈
7294:∼
7268:∼
7257:×
7245:∐
7153:for some
7125:⋅
6966:ω
6956:−
6930:ω
6924:∗
6907:⋅
6894:ω
6846:ω
6838:−
6816:−
6799:⋅
6786:ω
6755:δ
6750:α
6729:γ
6724:δ
6720:ω
6714:β
6709:γ
6696:−
6680:γ
6675:α
6662:β
6657:γ
6644:−
6624:⋅
6611:β
6606:α
6602:ω
6561:β
6556:α
6546:β
6536:β
6532:∑
6520:α
6445:β
6440:α
6436:ω
6432:⊗
6427:β
6417:β
6413:∑
6404:α
6333:˙
6330:γ
6319:β
6314:α
6310:ω
6304:β
6294:β
6290:∑
6281:α
6259:˙
6256:γ
6249:∇
6237:=0 gives
6196:β
6191:α
6171:γ
6163:β
6153:β
6149:∑
6130:γ
6122:α
6099:γ
6093:Γ
5983:β
5978:α
5968:β
5958:β
5954:∑
5942:α
5776:Lie group
5714:θ
5710:∧
5701:θ
5679:Γ
5666:Θ
5626:θ
5622:∧
5597:ω
5587:∑
5574:θ
5547:Θ
5408:θ
5404:∧
5395:θ
5341:Γ
5319:Γ
5294:Γ
5284:∂
5269:θ
5265:∧
5256:θ
5231:θ
5209:Γ
5202:∧
5190:θ
5168:Γ
5149:θ
5127:Γ
5097:Ω
5068:holonomic
5018:ω
5014:∧
4985:ω
4975:∑
4942:ω
4906:Ω
4884:Curvature
4828:Γ
4818:∑
4788:∇
4763:∑
4756:⟩
4734:⟨
4712:∇
4643:ω
4639:⊗
4614:∑
4588:⊗
4569:∑
4501:θ
4468:Γ
4458:∑
4429:ω
4314:Γ
4293:∑
4263:∇
4122:Ω
4092:θ
4089:∧
4086:Ω
4080:Θ
4022:Θ
4002:∑
3995:Θ
3949:Θ
3924:∑
3896:Θ
3845:θ
3841:∧
3816:ω
3806:∑
3782:θ
3755:Θ
3698:θ
3688:∑
3681:θ
3632:θ
3623:Θ
3573:→
3548:θ
3455:∗
3441:Λ
3437:⊗
3428:Γ
3413:→
3403:…
3388:→
3370:∗
3356:Λ
3352:⊗
3343:Γ
3328:→
3310:∗
3296:Λ
3292:⊗
3283:Γ
3268:→
3249:Γ
3167:Ω
3111:⊗
3103:∗
3089:Λ
3082:Γ
3079:∈
3076:Ω
3032:∗
3009:Ω
2996:Ω
2941:Ω
2933:−
2907:Ω
2852:ω
2849:∧
2835:ω
2818:ω
2798:Ω
2769:Curvature
2711:−
2675:ω
2667:−
2642:−
2601:−
2575:−
2550:−
2511:ω
2421:tensorial
2383:ω
2375:−
2353:−
2327:ω
2292:β
2287:α
2277:β
2267:β
2263:∑
2251:α
2085:ξ
2079:ω
2053:ξ
2050:ω
2033:ξ
2013:ξ
1968:α
1964:ξ
1958:β
1953:α
1949:ω
1945:⊗
1940:β
1919:β
1915:∑
1898:α
1894:∑
1874:α
1870:ξ
1863:⊗
1858:α
1837:α
1833:∑
1810:α
1806:ξ
1800:α
1773:α
1769:∑
1762:ξ
1703:β
1698:α
1694:ω
1690:⊗
1685:β
1664:β
1660:∑
1651:α
1626:one-forms
1543:α
1537:∧
1512:−
1503:α
1500:∧
1479:α
1476:∧
1433:∗
1423:∗
1419:Λ
1415:⊗
1406:Γ
1403:→
1392:∗
1382:∗
1378:Λ
1374:⊗
1365:Γ
1271:⊗
1207:of local
1179:Ω
1175:⊗
1163:Γ
1149:∗
1141:⊗
1132:Γ
1129:→
1117:Γ
1001:ξ
962:ξ
954:⋮
930:ξ
905:ξ
883:ξ
838:ξ
801:α
797:ξ
760:α
756:ξ
750:α
729:α
725:∑
718:ξ
673:ξ
605:…
587:α
577:α
425:×
359:⊆
130:tensorial
67:tensorial
9706:Manifold
9691:Geodesic
9449:Notation
9244:Orbifold
9239:K-theory
9229:Diffiety
8953:Pullback
8767:Oriented
8745:Kenmotsu
8725:Hadamard
8671:Types of
8620:Geodesic
8445:Glossary
8401:(1980),
8380:(1973),
8246:(1978),
8084:See also
7852: :
7836:-bundle
7060:. Let {
6871: :
6524:′
6014:has the
6008:-related
5946:′
5805:) where
4233:, where
3484:to be a
2255:′
2218:′
2129:a priori
1748:. Then
1615:matrix (
1209:sections
707:. Then
124:for the
9803:Physics
9637:Related
9400:Physics
9318:Tensors
9188:History
9171:Related
9085:Tangent
9063:)
9043:)
9010:Adjoint
9002:Bundles
8980:density
8878:Torsion
8844:Vectors
8836:Tensors
8819:)
8804:)
8800:,
8798:Pseudo−
8777:Poisson
8710:Finsler
8705:Fibered
8700:Contact
8698:)
8690:Complex
8688:)
8657:Section
8297:2829653
7006:, here
6998:is the
6992:where ω
6020:-bundle
5893:. If (
5852:Spinors
5519:{{{1}}}
5470:Torsion
5462:is the
4387:of the
3609:torsion
2977:be the
1219:. For
1033:tetrads
225:over a
152:with a
116:of the
83:physics
9731:Vector
9726:Spinor
9711:Matrix
9505:Tensor
9153:Vector
9138:Koszul
9118:Cartan
9113:Affine
9095:Vector
9090:Tensor
9075:Spinor
9065:Normal
9061:Stable
9015:Affine
8919:bundle
8871:bundle
8817:Almost
8740:Kähler
8696:Almost
8686:Almost
8680:Closed
8580:Sard's
8536:(list)
8388:
8367:
8349:
8331:
8313:
8295:
8285:
8254:
8072:, and
8064:where
7651:, set
6578:where
5832:) ⊂ GL
5783:metric
5458:where
4238:= dim
3528:θ ∈ Ω(
3425:
3406:
3400:
3381:
3340:
3321:
3280:
3261:
1558:where
1310:where
1235:, and
788:where
106:matrix
9651:Basis
9336:Scope
9261:Sheaf
9035:Fiber
8811:Rizza
8782:Prime
8613:Local
8603:Curve
8465:Atlas
8271:(PDF)
8111:Notes
7175:Let F
6879:is a
6582:is a
6030:is a
5885:⊂ GL(
5504:) = T
5496:Hom(T
4413:(the
4163:over
3520:. A
2749:This
1624:) of
1570:is a
1205:sheaf
306:atlas
274:basis
248:. A
201:be a
100:of a
98:basis
9128:Form
9030:Dual
8963:flow
8826:Tame
8802:Sub−
8715:Flat
8595:Maps
8386:ISBN
8365:ISBN
8347:ISBN
8329:ISBN
8311:ISBN
8283:ISBN
8252:ISBN
8154:See
8141:See
7643:) ∈
7525:On F
5881:and
5820:(or
5530:Θ =
4404:) =
4062:The
3226:for
3057:Hom(
2779:The
2435:If {
1590:The
553:Let
252:for
181:Let
73:, a
50:and
38:, a
9050:Jet
8275:doi
8227:doi
8223:133
7022:of
7010:to
6473:is
6082:):
6050:).
5911:):
5785:in
5770:on
5508:⊗ T
5500:, T
5494:of
5075:= 0
5066:is
4530:= Σ
4371:= {
4367:If
4214:= (
4202:of
3666:of
3504:of
3492:).
3115:Hom
2761:⊗ Ω
2474:on
2466:of
2127:is
1731:= Σ
1628:on
1582:).
1578:⊗ Ω
1524:deg
1093:in
1027:In
830:of
374:of
280:of
276:of
30:In
9949::
9041:Co
8293:MR
8291:,
8281:,
8242:;
8221:,
8172:^
8129:,
8125:,
8080:.
7876::
7856:→
7820:.
7647:×
7548:×
7172:.
7168:∩
7162:UV
7052:→
6946:Ad
6875:→
6489:.
6378:.
5915:→
5834:2n
5750:.
5739:kj
5532:Dθ
5466:.
5073:dθ
4700::
4375:|
4223:|
4206:.
4200:TM
4184:TM
4128:0.
3659:.
3602:∈
3230:∈
2889:→
2427:.
2316::
2191:×
2177:.
2160:=
1611:×
1348::
1318:.
1312:df
1089:A
156:.
104:a
93:.
54:.
9310:e
9303:t
9296:v
9059:(
9039:(
8815:(
8796:(
8694:(
8684:(
8447:)
8443:(
8433:e
8426:t
8419:v
8277::
8229::
8074:d
8070:M
8066:X
8043:,
8040:)
8037:X
8034:(
8031:]
8028:)
8025:g
8018:e
8014:(
8011:d
8008:[
8002:=
7987:)
7983:g
7975:e
7970:(
7967:,
7964:X
7948:e
7944:g
7940:G
7923:.
7909:e
7903:=
7900:)
7895:e
7890:(
7874:M
7870:g
7866:e
7862:P
7858:P
7854:M
7850:e
7846:M
7842:M
7840:→
7838:P
7834:G
7830:G
7818:G
7814:E
7811:G
7807:E
7804:G
7800:G
7796:E
7793:G
7776:.
7770:g
7754:2
7746:+
7743:)
7738:U
7733:e
7728:(
7715:1
7703:1
7696:g
7691:d
7687:A
7684:=
7679:)
7676:g
7673:,
7670:x
7667:(
7649:G
7645:U
7641:g
7639:,
7637:x
7620:G
7614:G
7608:U
7605::
7600:2
7591:,
7588:U
7582:G
7576:U
7573::
7568:1
7550:G
7546:U
7542:g
7536:G
7530:E
7527:G
7510:.
7505:V
7501:g
7497:)
7494:x
7491:(
7486:1
7478:V
7475:U
7471:h
7467:=
7462:U
7458:g
7447:V
7444:U
7440:h
7431:U
7425:e
7419:=
7414:V
7408:e
7397:)
7394:G
7388:V
7382:)
7377:V
7373:g
7369:,
7366:x
7363:(
7360:(
7354:)
7351:G
7345:U
7339:)
7334:U
7330:g
7326:,
7323:x
7320:(
7317:(
7264:/
7260:G
7254:U
7249:U
7237:=
7234:E
7229:G
7225:F
7211:E
7208:G
7204:G
7200:G
7196:M
7192:G
7188:M
7184:G
7180:E
7177:G
7170:V
7166:U
7159:h
7155:G
7136:V
7133:U
7129:h
7120:U
7114:e
7108:=
7103:V
7097:e
7081:U
7078:e
7074:U
7070:G
7066:M
7062:U
7058:G
7054:M
7050:E
7024:G
7016:g
7012:M
7004:G
6995:g
6977:)
6973:e
6969:(
6959:1
6952:g
6941:+
6935:g
6920:g
6916:=
6913:)
6910:g
6902:e
6897:(
6881:G
6877:G
6873:M
6869:g
6852:.
6849:g
6841:1
6834:g
6830:+
6827:g
6824:d
6819:1
6812:g
6808:=
6805:)
6802:g
6794:e
6789:(
6760:.
6746:g
6742:)
6738:e
6734:(
6705:)
6699:1
6692:g
6688:(
6685:+
6671:g
6667:d
6653:)
6647:1
6640:g
6636:(
6633:=
6630:)
6627:g
6620:e
6616:(
6588:M
6584:G
6580:g
6552:g
6542:e
6528:=
6516:e
6494:g
6487:g
6483:e
6481:(
6479:α
6458:)
6454:e
6450:(
6423:e
6409:=
6400:e
6396:D
6383:α
6376:G
6372:g
6365:α
6348:)
6345:)
6342:0
6339:(
6324:(
6300:e
6286:=
6277:e
6271:)
6268:0
6265:(
6235:t
6231:t
6227:α
6224:g
6207:)
6204:t
6201:(
6187:g
6183:)
6180:)
6177:t
6174:(
6168:(
6159:e
6145:=
6142:)
6139:)
6136:0
6133:(
6127:(
6118:e
6112:t
6107:0
6103:)
6096:(
6080:t
6076:G
6068:E
6064:G
6048:k
6044:G
6040:R
6036:G
6028:E
6024:G
6018:G
6012:E
6006:G
5988:.
5974:g
5964:e
5950:=
5938:e
5924:α
5921:e
5917:G
5913:M
5909:i
5906:g
5902:E
5898:α
5895:e
5891:R
5887:k
5883:G
5879:k
5875:E
5869:.
5862:.
5838:R
5836:(
5830:C
5828:(
5826:n
5813:.
5811:E
5807:k
5803:k
5787:E
5779:G
5772:E
5768:e
5760:E
5732:,
5718:j
5705:k
5695:j
5692:k
5683:i
5675:=
5670:i
5652:e
5635:.
5630:j
5619:)
5615:e
5611:(
5606:i
5601:j
5591:j
5583:+
5578:i
5570:d
5567:=
5564:)
5560:e
5556:(
5551:i
5523:θ
5515:e
5510:M
5506:M
5502:M
5498:M
5492:θ
5480:E
5460:R
5437:j
5428:i
5425:q
5422:p
5418:R
5412:q
5399:p
5388:2
5385:1
5379:=
5366:)
5362:)
5357:i
5354:q
5345:k
5335:k
5332:p
5323:j
5315:+
5310:i
5307:q
5298:j
5288:p
5279:(
5273:q
5260:p
5252:=
5240:)
5235:q
5225:i
5222:q
5213:k
5205:(
5199:)
5194:p
5184:k
5181:p
5172:j
5164:(
5161:+
5158:)
5153:q
5143:i
5140:q
5131:j
5123:(
5120:d
5117:=
5110:j
5101:i
5064:e
5047:.
5044:)
5040:e
5036:(
5031:k
5022:i
5011:)
5007:e
5003:(
4998:j
4989:k
4979:k
4971:+
4968:)
4964:e
4960:(
4955:j
4946:i
4938:d
4935:=
4932:)
4928:e
4924:(
4919:j
4910:i
4891:i
4867:)
4861:j
4857:v
4853:)
4849:e
4845:(
4840:k
4835:j
4832:i
4822:j
4814:+
4809:k
4805:v
4797:i
4793:e
4783:(
4777:k
4773:e
4767:k
4759:=
4751:i
4747:e
4743:,
4740:v
4737:D
4731:=
4728:v
4721:i
4717:e
4698:i
4695:e
4678:.
4673:j
4669:v
4665:)
4661:e
4657:(
4652:k
4647:j
4634:k
4630:e
4624:k
4621:,
4618:j
4610:+
4607:)
4602:k
4598:v
4594:d
4591:(
4583:k
4579:e
4573:k
4565:=
4562:v
4559:D
4545:v
4541:i
4537:e
4533:i
4528:v
4510:.
4505:k
4497:)
4493:e
4489:(
4484:i
4481:k
4472:j
4462:k
4454:=
4451:)
4447:e
4443:(
4438:j
4433:i
4410:j
4406:δ
4401:j
4397:e
4395:(
4393:θ
4381:n
4377:i
4373:θ
4369:θ
4352:.
4347:k
4343:e
4339:)
4335:e
4331:(
4326:k
4321:j
4318:i
4308:n
4303:1
4300:=
4297:k
4289:=
4284:j
4280:e
4272:i
4268:e
4245:M
4240:M
4236:n
4231:)
4229:n
4225:i
4220:i
4216:e
4212:e
4204:M
4177:E
4165:M
4161:E
4150:M
4125:=
4119:D
4083:=
4077:D
4042:.
4039:)
4035:e
4031:(
4026:i
4016:i
4012:e
4006:i
3998:=
3969:.
3966:)
3962:e
3958:(
3953:j
3943:i
3938:j
3934:g
3928:j
3920:=
3917:)
3914:g
3909:e
3905:(
3900:i
3865:.
3862:)
3858:e
3854:(
3849:j
3838:)
3834:e
3830:(
3825:i
3820:j
3810:j
3802:+
3799:)
3795:e
3791:(
3786:i
3778:d
3775:=
3772:)
3768:e
3764:(
3759:i
3728:.
3723:i
3719:e
3715:)
3711:e
3707:(
3702:i
3692:i
3684:=
3668:E
3664:e
3657:M
3653:E
3635:.
3629:D
3626:=
3604:M
3600:x
3581:x
3577:E
3570:M
3565:x
3561:T
3557::
3552:x
3534:E
3532:,
3530:M
3514:E
3510:M
3506:E
3502:k
3469:)
3466:)
3463:M
3460:(
3451:T
3445:n
3434:E
3431:(
3418:D
3393:D
3378:)
3375:M
3366:T
3360:2
3349:E
3346:(
3333:D
3318:)
3315:M
3306:T
3300:1
3289:E
3286:(
3273:D
3258:)
3255:E
3252:(
3236:v
3232:E
3228:v
3210:v
3205:2
3201:D
3197:=
3194:)
3191:v
3188:D
3185:(
3182:D
3179:=
3176:)
3173:v
3170:(
3154:D
3137:.
3134:)
3131:)
3128:E
3125:,
3122:E
3119:(
3108:M
3099:T
3093:2
3085:(
3063:E
3061:,
3059:E
3051:M
3026:e
3020:)
3016:e
3012:(
3004:e
2999:=
2983:e
2975:e
2958:.
2955:g
2952:)
2948:e
2944:(
2936:1
2929:g
2925:=
2922:)
2919:g
2914:e
2910:(
2894:g
2891:e
2887:e
2866:.
2863:)
2859:e
2855:(
2846:)
2842:e
2838:(
2832:+
2829:)
2825:e
2821:(
2815:d
2812:=
2809:)
2805:e
2801:(
2785:E
2763:M
2759:E
2755:E
2734:.
2731:)
2726:q
2721:e
2714:1
2706:p
2701:e
2696:(
2693:)
2688:p
2683:e
2678:(
2670:1
2663:)
2657:q
2652:e
2645:1
2637:p
2632:e
2627:(
2624:+
2621:)
2616:q
2611:e
2604:1
2596:p
2591:e
2586:(
2583:d
2578:1
2571:)
2565:q
2560:e
2553:1
2545:p
2540:e
2535:(
2532:=
2529:)
2524:q
2519:e
2514:(
2497:p
2493:U
2488:p
2484:e
2482:(
2480:ω
2476:M
2468:E
2463:p
2459:e
2454:p
2450:U
2446:M
2441:p
2437:U
2425:g
2417:ω
2400:.
2397:g
2394:)
2390:e
2386:(
2378:1
2371:g
2367:+
2364:g
2361:d
2356:1
2349:g
2345:=
2342:)
2339:g
2334:e
2330:(
2314:ω
2297:.
2283:g
2273:e
2259:=
2247:e
2236:,
2233:g
2227:e
2222:=
2213:e
2197:g
2193:k
2189:k
2185:′
2182:e
2175:e
2171:e
2169:(
2166:α
2162:ω
2157:α
2153:ω
2149:E
2145:ω
2133:e
2125:ω
2121:ξ
2117:e
2113:d
2096:)
2092:e
2088:(
2082:)
2076:+
2073:d
2070:(
2067:=
2064:)
2060:e
2056:(
2047:+
2044:)
2040:e
2036:(
2030:d
2027:=
2024:)
2020:e
2016:(
2010:D
1984:.
1981:)
1977:e
1973:(
1936:e
1930:k
1925:1
1922:=
1909:k
1904:1
1901:=
1890:+
1887:)
1883:e
1879:(
1866:d
1854:e
1848:k
1843:1
1840:=
1829:=
1826:)
1823:)
1819:e
1815:(
1796:e
1792:(
1789:D
1784:k
1779:1
1776:=
1765:=
1759:D
1746:ξ
1742:α
1738:e
1734:α
1729:ξ
1725:E
1708:.
1681:e
1675:k
1670:1
1667:=
1656:=
1647:e
1643:D
1630:M
1621:α
1617:ω
1613:k
1609:k
1604:α
1600:e
1596:e
1580:M
1576:E
1568:D
1564:v
1560:v
1540:d
1534:v
1529:v
1519:)
1515:1
1509:(
1506:+
1497:)
1494:v
1491:D
1488:(
1485:=
1482:)
1473:v
1470:(
1467:D
1441:)
1438:M
1429:T
1412:E
1409:(
1400:)
1397:M
1388:T
1371:E
1368:(
1362::
1359:D
1346:D
1342:D
1334:E
1328:E
1323:D
1316:f
1295:v
1292:D
1289:f
1286:+
1283:)
1280:f
1277:d
1274:(
1268:v
1265:=
1262:)
1259:v
1256:f
1253:(
1250:D
1237:f
1233:E
1229:v
1221:D
1217:M
1213:M
1188:M
1183:1
1172:)
1169:E
1166:(
1160:=
1157:)
1154:M
1145:T
1138:E
1135:(
1126:)
1123:E
1120:(
1114::
1111:D
1095:E
1063:M
1043:M
1012:)
1008:e
1004:(
995:e
990:=
985:]
979:)
975:e
971:(
966:k
947:)
943:e
939:(
934:2
922:)
918:e
914:(
909:1
898:[
891:e
886:=
859:e
814:)
810:e
806:(
773:)
769:e
765:(
746:e
740:k
735:1
732:=
721:=
694:e
653:E
633:E
611:k
608:,
602:,
599:2
596:,
593:1
590:=
583:)
573:e
569:(
566:=
562:e
537:C
513:k
508:R
484:k
479:R
457:U
435:k
430:R
422:U
402:U
382:x
362:M
356:U
336:M
316:x
288:E
260:E
236:M
213:k
189:E
20:)
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