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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.
Given a principal connection on P, one obtains a G-connection on the associated fiber bundle E = P × G F via pullback. Conversely, given a G-connection on E it is possible to recover the principal connection on the associated principal bundle P. To recover this principal connection, one introduces the notion of a frame on the typical fiber F.
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.
Choose any connection form ω in P, and let Ω be the associated curvature form; i.e., =, the exterior covariant derivative of ω. If [] is a homogeneous polynomial function of degree k; i.e., () = for any complex number a and x in , then, viewing f as a symmetric multilinear functional on (see the ring of polynomial functions), let
If the bundle is endowed with a bundle metric, an inner product on its vector space fibers, a metric connection is defined as a connection that is compatible with the bundle metric. A Yang-Mills connection is a special metric connection which satisfies the Yang-Mills equations of motion. A Riemannian connection is a metric connection on the ...
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The definition may be phrased for a connection on a vector bundle or principal bundle, with the two perspectives being essentially interchangeable. Here the definition of principal bundles is presented, which is the form that appears in Hitchin's work. [1] [5] [6]
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