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A principal G-connection on a principal G-bundle over a smooth manifold is a particular type of connection that is compatible with the action of the group . 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 .
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.
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 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.
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.
where is a connection on a principal bundle , is a section of an associated spinor bundle and / is the induced Dirac operator of the induced covariant derivative on this associated bundle. The first term is an interacting term in the Lagrangian between the spinor field (the field representing the electron-positron) and the gauge field ...
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.
In particular, on a principal bundle, a principal connection is a natural reinterpretation of the connection form as a tensorial object. On the other hand, the connection form has the advantage that it is a differential form defined on the differentiable manifold , rather than on an abstract principal bundle over it.