Editing Connection (principal bundle) (section)

==Formal definition==
[[File:Principal bundle connection form projection.png|thumb|300px|A principal bundle connection form <math>\omega</math> may be thought of as a projection operator on the tangent bundle <math>TP</math> of the principal bundle <math>P</math>. The kernel of the connection form is given by the horizontal subspaces for the associated [[Ehresmann connection]].]]
[[File:Ehresmann connection.png|thumb|300px|A connection is equivalently specified by a choice of horizontal subspace <math>H_p\subset T_pP</math> for every tangent space to the principal bundle <math>P</math>.]]
[[File:Equivariance of Principal bundle connection.png|thumb|300px|A principal bundle connection is required to be compatible with the right group action of <math>G</math> on <math>P</math>. This can be visualized as the right multiplication <math>R_g</math> taking the horizontal subspaces into each other. This equivariance of the horizontal subspaces <math>H\subset TP</math> interpreted in terms of the connection form <math>\omega</math> leads to its characteristic equivariance properties.]]
Let <math>\pi : P \to M</math> be a smooth [[principal bundle|principal ''G''-bundle]] over a [[smooth manifold]] <math>M</math>. Then a '''principal''' <math>G</math>'''-connection''' on <math>P</math> is a differential 1-form on <math>P</math> [[Lie algebra valued form|with values in the Lie algebra]] <math>\mathfrak g</math> of <math>G</math> which is <math>G</math>'''-equivariant''' and '''reproduces''' the '''Lie algebra generators''' of the '''fundamental vector fields''' on <math>P</math>.

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 <math>R_g</math> denotes right multiplication by <math>g</math>, and <math>\operatorname{Ad}_g</math> is the [[adjoint representation]] on <math> \mathfrak g</math> (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 <math>X_\xi</math> is [[fundamental vector field|the vector field on ''P'' associated to ''ξ'' by differentiating the ''G'' action on ''P'']], then <math>\omega(X_\xi)=\xi</math> (identically on <math>P</math>).

Sometimes the term ''principal <math>G</math>-connection'' refers to the pair <math>(P,\omega)</math> and <math>\omega</math> itself is called the '''[[connection form]]''' or '''connection 1-form''' of the principal connection.

=== Computational remarks ===
Most known non-trivial computations of principal ''<math>G</math>''-connections are done with [[homogeneous space]]s because of the triviality of the (co)tangent bundle. (For example, let <math>G \to H \to H/G</math>, be a principal ''<math>G</math>''-bundle over <math> H/G</math>.) This means that 1-forms on the total space are canonically isomorphic to <math>C^\infty(H,\mathfrak{g}^*)</math>, where <math> \mathfrak{g}^*</math> is the dual lie algebra, hence ''<math>G</math>''-connections are in bijection with <math>C^\infty(H,\mathfrak{g}^*\otimes \mathfrak{g})^G</math>.

===Relation to Ehresmann connections===
A principal ''<math>G</math>''-connection <math>\omega</math> on <math>P</math> determines an [[Ehresmann connection]] on <math>P</math> in the following way. First note that the fundamental vector fields generating the <math>G</math> action on <math>P</math> provide a bundle isomorphism (covering the identity of <math>P</math>) from the [[Fiber bundle|bundle]] <math>V</math> to <math>P\times\mathfrak g</math>, where <math>V=\ker(d\pi)</math> 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 <math>P</math>. It follows that <math>\omega</math> determines uniquely a bundle map <math>v:TP\rightarrow V</math> which is the identity on <math>V</math>. Such a projection <math>v</math> is uniquely determined by its kernel, which is a smooth subbundle <math>H</math> of <math>TP</math> (called the [[horizontal bundle]]) such that <math>TP=V\oplus H</math>. This is an Ehresmann connection.

Conversely, an Ehresmann connection <math>H\subset TP</math> (or <math>v:TP\rightarrow V</math>) on <math>P</math> defines a principal <math>G</math>-connection <math>\omega</math> if and only if it is <math>G</math>-equivariant in the sense that <math>H_{pg}=\mathrm d(R_g)_p(H_{p})</math>.

===Pull back via trivializing section===
A trivializing section of a principal bundle ''<math>P</math>'' is given by a section ''s'' of ''<math>P</math>'' over an open subset ''<math>U</math>'' of ''<math>M</math>''. Then the [[pullback (differential geometry)|pullback]] ''s''<sup>*</sup>''ω'' of a principal connection is a 1-form on ''<math>U</math>'' 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 <math>(sg)^* \omega = \operatorname{Ad}(g)^{-1}s^* \omega + g^{-1} dg</math>. 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 ''<math>G</math>'' acts on the [[tangent bundle]] ''<math>TP</math>'' by right translation.  The [[Quotient space (topology)|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> We see again the link to the horizontal bundle and thus Ehresmann connection.

===Affine property===
If ''ω'' and ''ω''′ are principal connections on a principal bundle ''P'', then the difference {{nowrap|''ω''′ − ''ω''}} is a <math>\mathfrak g</math>-valued 1-form on ''P'' that 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.