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{{dablink|This article is about connections on principal bundles. See [[connection (mathematics)]] for other types of connections in mathematics.}}
In [[mathematics]], a '''connection''' is a device that defines a notion of [[parallel transport]] on the bundle; that is, a way to "connect" or identify fibers over nearby points. A '''principal ''G''-connection''' on a [[principal bundle|principal G-bundle]] ''P'' over a [[smooth manifold]] ''M'' is a particular type of connection which is compatible with the [[group action|action]] of the group ''G''.


A principal connection can be viewed as a special case of the notion of an [[Ehresmann connection]], and is sometimes called a principal Ehresmann connection. It gives rise to (Ehresmann) connections on any [[fiber bundle]] associated to ''P'' via the [[associated bundle]] construction. In particular, on any [[associated vector bundle]] the principal connection induces a [[covariant derivative]], an operator that can differentiate [[section (fiber bundle)|sections]] of that bundle along [[tangent vector|tangent directions]] in the base manifold. Principal connections generalize to arbitrary principal bundles the concept of a [[linear connection]] on the [[frame bundle]] of a [[smooth manifold]].


==Formal definition==
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Let ''π'':''P''→''M'' be a smooth [[principal bundle|principal ''G''-bundle]] over a [[smooth manifold]] ''M''. Then a '''principal ''G''-connection''' on ''P'' is a differential 1-form on ''P'' [[Lie algebra valued form|with values in the Lie algebra]] <math>\mathfrak g</math> of ''G'' which is '''''G-equivariant''''' and '''''reproduces''''' the '''''Lie algebra generators''''' of the '''''fundamental vector fields''''' on ''P''.
 
In other words, it is an element ''ω'' of <math>\Omega^1(P,\mathfrak g)\cong C^\infty(P, T^*P)\otimes\mathfrak g</math> such that
# <math>\hbox{ad}_g(R_g^*\omega)=\omega</math> where ''R''<sub>''g''</sub> denotes right multiplication by ''g'', and <math>\operatorname{ad}_g</math> is the [[adjoint representation]] of ''G'' (explicitly, <math>\operatorname{ad}_gX = \frac{d}{dt}g\exp(tX)g^{-1}\bigl|_{t=0}</math>);
# if <math>\xi\in \mathfrak g</math> and ''X''<sub>''ξ''</sub> is [[fundamental vector field|the vector field on ''P'' associated to ''ξ'' by differentiating the ''G'' action on ''P'']], then ''ω''(''X''<sub>''ξ''</sub>) = ''ξ'' (identically on ''P'').
 
Sometimes the term ''principal G-connection'' refers to the pair (''P'',''ω'') and ''ω'' itself is called the '''[[connection form]]''' or '''connection 1-form''' of the principal connection.
 
===Relation to Ehresmann connections===
A principal G-connection ''ω'' on ''P'' determines an [[Ehresmann connection]] on ''P'' in the following way. First note that the fundamental vector fields generating the ''G'' action on ''P'' provide a bundle isomorphism (covering the identity of ''P'') from the [[Fiber bundle|bundle]] ''VP'' to <math>P\times\mathfrak g</math>, where ''VP''&nbsp;=&nbsp;ker(d''&pi;'') is the kernel of the [[Pushforward (differential)|tangent mapping]] <math>{\mathrm d}\pi\colon TP\to TM</math> which is called the [[vertical bundle]] of ''P''. It follows that ''ω'' determines uniquely a bundle map ''v'':''TP''→''V'' which is the identity on ''V''. Such a projection ''v'' is uniquely determined by its kernel, which is a smooth subbundle ''H'' of ''TP'' (called the [[horizontal bundle]]) such that ''TP''=''V''⊕''H''. This is an Ehresmann connection.
 
Conversely, an Ehresmann connection ''H''⊂''TP'' (or ''v'':''TP''→''V'') on ''P'' defines a principal ''G''-connection ''ω'' if and only if it is ''G''-equivariant in the sense that <math>H_{pg}=\mathrm d(R_g)_p(H_{p})</math>.
 
===Form in a local trivialization===
A local trivialization of a principal bundle ''P'' is given by a section ''s'' of ''P'' over an open subset ''U'' of ''M''. Then the [[pullback (differential geometry)|pullback]] ''s''<sup>*</sup>''ω'' of a principal connection is a 1-form on ''U'' with values in <math>\mathfrak g</math>.
If the section ''s'' is replaced by a new section ''sg'', defined by (''sg'')(''x'') = ''s''(''x'')''g''(''x''), where ''g'':''M''→''G'' is a smooth map, then (''sg'')<sup>*</sup>''ω'' = Ad(g)<sup>-1</sup> ''s''<sup>*</sup>''ω''+''g''<sup>-1</sup>d''g''. The principal connection is uniquely determined by this family of <math>\mathfrak g</math>-valued 1-forms, and these 1-forms are also called '''connection forms''' or '''connection 1-forms''', particularly in older or more physics-oriented literature.
 
===Bundle of principal connections===
The group ''G'' acts on the [[tangent bundle]] ''TP'' by right translation.  The [[quotient space]] ''TP''/''G'' is also a manifold, and inherits the structure of a [[fibre bundle]] over ''TM'' which shall be denoted ''dπ'':''TP''/''G''→''TM''. Let ρ:''TP''/''G''→''M'' be the projection onto ''M''.  The fibres of the bundle ''TP''/''G'' under the projection ρ carry an additive structure.
 
The bundle ''TP''/''G'' is called the '''bundle of principal connections''' {{harv|Kobayashi|1957}}. A [[section (fiber bundle)|section]] Γ of dπ:''TP''/''G''→''TM'' such that Γ : ''TM'' → ''TP''/''G'' is a linear morphism of vector bundles over ''M'', can be identified with a principal connection in ''P''.  Conversely, a principal connection as defined above gives rise to such a section Γ of ''TP''/''G''.
 
Finally, let Γ be a principal connection in this sense. Let ''q'':''TP''→''TP''/''G'' be the quotient map.  The horizontal distribution of the connection is the bundle
 
:<math>H = q^{-1}\Gamma(TM) \subset TP.</math>
 
===Affine property===
If ''ω'' and ''ω' '' are principal connections on a principal bundle ''P'', then the difference ''ω' ''- ''ω'' is a <math>\mathfrak g</math>-valued 1-form on ''P'' which is not only ''G''-equivariant, but '''horizontal''' in the sense that it vanishes on any section of the vertical bundle ''V'' of ''P''. Hence it is '''basic''' and so is determined by a ''1''-form on ''M'' with values in the [[adjoint bundle]]
:<math>\mathfrak g_P:=P\times_G\mathfrak g.</math>
Conversely, any such one form defines (via pullback) a ''G''-equivariant horizontal 1-form on ''P'', and the space of principal ''G''-connections is an [[affine space]] for this space of 1-forms.
 
==Induced covariant and exterior derivatives==
For any [[linear representation]] ''W'' of ''G'' there is an [[associated vector bundle]] <math> P\times_G W</math> over ''M'', and a principal connection induces a [[Connection (vector bundle)|covariant derivative]] on any such vector bundle. This covariant derivative can be defined using the fact that the space of sections of <math> P\times_G W</math> over ''M'' is isomorphic to the space of ''G''-equivariant ''W''-valued functions on ''P''. More generally, the space of ''k''-forms [[vector-valued differential form|with values in]] <math> P\times_G W</math> is identified with the space of ''G''-equivariant and horizontal ''W''-valued ''k''-forms on ''P''. If ''α'' is such a ''k''-form, then its [[exterior derivative]] d''α'', although ''G''-equivariant, is no longer horizontal. However, the combination d''α''+''ω''Λ''α'' is. This defines an [[exterior covariant derivative]] d<sup>''ω''</sup> from <math> P\times_G W</math>-valued ''k''-forms on ''M'' to <math> P\times_G W</math>-valued (''k''+1)-forms on ''M''. In particular, when ''k''=0, we obtain a covariant derivative on <math> P\times_G W</math>.
 
==Curvature form==
The [[curvature form]] of a principal ''G''-connection ''ω'' is the <math>\mathfrak g</math>-valued 2-form Ω defined by
:<math>\Omega=d\omega +\tfrac12 [\omega\wedge\omega].</math>
It is ''G''-equivariant and horizontal, hence corresponds to a 2-form on ''M'' with values in <math>\mathfrak g_P</math>. The identification of the curvature with this quantity is sometimes called the ''second structure equation''.
 
==Connections on frame bundles and torsion==
If the principal bundle ''P'' is the [[frame bundle]], or (more generally) if it has a [[solder form]], then the connection is an example of an [[affine connection]], and the curvature is not the only invariant, since the additional structure of the solder form ''θ'', which is an equivariant '''R'''<sup>n</sup>-valued 1-form on ''P'', should be taken into account. In particular, the [[Torsion (differential geometry)|torsion form]] on ''P'', is an '''R'''<sup>n</sup>-valued 2-form Θ defined by
:<math> \Theta=\mathrm d\theta+\omega\wedge\theta. </math>
Θ is ''G''-equivariant and horizontal, and so it descends to a tangent-valued 2-form on ''M'', called the ''torsion''. This equation is sometimes called the ''first structure equation''.
 
==References==
*{{citation | first = Shoshichi | last = Kobayashi | title = Theory of Connections | journal = Ann. Mat. Pura Appl. | year = 1957 | volume = 43 | pages = 119–194 | doi = 10.1007/BF02411907}}
* {{citation | last1=Kobayashi|first1=Shoshichi|last2=Nomizu|first2=Katsumi | title = [[Foundations of Differential Geometry]]|volume=Vol. 1| publisher=[[Wiley Interscience]] | year=1996|edition=New|isbn=0-471-15733-3}}
* {{citation|last1 = Kolář|first1=Ivan|last2=Michor|first2=Peter|last3=Slovák|first3=Jan|url=http://www.emis.de/monographs/KSM/kmsbookh.pdf|format=PDF|title=Natural operators in differential geometry|year = 1993|publisher = Springer-Verlag}}
 
[[Category:Connection (mathematics)]]
[[Category:Fiber bundles]]

Latest revision as of 18:24, 6 August 2014


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