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The bundle TP/G is called the bundle of principal connections (Kobayashi 1957). A 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.
Using principal bundles, a connection can be realized as a Lie algebra-valued differential form. See connection (principal bundle). An approach to connections which makes direct use of the notion of transport of "data" (whatever that may be) is the Ehresmann connection.
This theory is most succinctly captured by passing to the principal bundle connection on the frame bundle of and using the theory of principal bundles. Each of the above examples can be seen as special cases of this construction: the dual bundle corresponds to the inverse transpose (or inverse adjoint) representation, the tensor product to the ...
A principal -bundle, where denotes any topological group, is a fiber bundle: together with a continuous right action such that preserves the fibers of (i.e. if then for all ) and acts freely and transitively (meaning each fiber is a G-torsor) on them in such a way that for each and , the map sending to is a homeomorphism.
A principal bundle with structure group , or a principal -bundle, consists of a quintuple (,,,,) where : is a smooth fibre bundle with fibre space isomorphic to a Lie group, and represents a free and transitive right group action of on which preserves the fibres, in the sense that for all , () = for all .
A principal connection on the principal bundle Q induces a connection on any associated vector bundle: in particular on the tangent bundle. A linear connection ∇ on TM arising in this way is said to be compatible with Q. Connections compatible with Q are also called adapted connections.
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If there is a principal bundle P whose base space is space or spacetime and structure group is a Lie group, then the sections of P form a principal homogeneous space of the group of gauge transformations. Connections (gauge connection) define this principal bundle, yielding a covariant derivative ∇ in each associated vector bundle.