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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 genera
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 of G which is G-equivariant and reproduces the Lie algebra generators of the fundamental vector fields on P .