Connection (principal bundle)

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 G-bundle P over a smooth manifold M is a particular type of connection which is compatible with the action of the group G. 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 G-bundle P over a smooth manifold M is a particular type of connection which is compatible with the 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 sections of that bundle along 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. Let π:P→M be a smooth principal G-bundle over a smooth manifold M. Then a principal G-connection on P is a differential 1-form on P with values in the Lie algebra g {displaystyle {mathfrak {g}}} 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 Ω 1 ( P , g ) ≅ C ∞ ( P , T ∗ P ⊗ g ) {displaystyle Omega ^{1}(P,{mathfrak {g}})cong C^{infty }(P,T^{*}Potimes {mathfrak {g}})} such that 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. Most known non-trivial computations of principal G-connections are done with homogeneous spaces because of the triviality of the (co)tangent bundle. (For example, let G → H → H / G {displaystyle G o H o H/G} , be a principal G-bundle over H / G {displaystyle H/G} ) This means that 1-forms on the total space are canonically isomorphic to C ∞ ( H , g ∗ ) {displaystyle C^{infty }(H,{mathfrak {g}}^{*})} , where g ∗ {displaystyle {mathfrak {g}}^{*}} is the dual lie algebra, hence G-connections are in bijection with C ∞ ( H , g ∗ ⊗ g ) G {displaystyle C^{infty }(H,{mathfrak {g}}^{*}otimes {mathfrak {g}})^{G}} . 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 bundle VP to P × g {displaystyle P imes {mathfrak {g}}} , where VP = ker(dπ) is the kernel of the tangent mapping d π : T P → T M {displaystyle {mathrm {d} }pi colon TP o TM} 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 H p g = d ( R g ) p ( H p ) {displaystyle H_{pg}=mathrm {d} (R_{g})_{p}(H_{p})} . A trivializing section of a principal bundle P is given by a section s of P over an open subset U of M. Then the pullback s*ω of a principal connection is a 1-form on U with values in g {displaystyle {mathfrak {g}}} .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)*ω = Ad(g)−1 s*ω+g−1dg. The principal connection is uniquely determined by this family of g {displaystyle {mathfrak {g}}} -valued 1-forms, and these 1-forms are also called connection forms or connection 1-forms, particularly in older or more physics-oriented literature.