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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.
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.
Let π : P → M be a principal bundle with a structure Lie group G. A principal connection on P usually is described by a Lie algebra-valued connection one-form on P. At the same time, a principal connection on P is a global section of the jet bundle J 1 P → P which is equivariant with respect to the canonical right action of G in P.
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. Concretely speaking, adapted connections can be ...
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 ...
A principal bundle connection form may be thought of as a projection operator on the tangent bundle of the principal bundle . The kernel of the connection form is given by the horizontal subspaces for the associated Ehresmann connection. Suppose that E is a smooth principal G-bundle over M.
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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