Connection on a principal bundle

Summary

A connection on a principal bundle with structure group $G$ is an Ehresmann connection that is invariant under the action of $G$. It is usually given by a connection form, a $1$-form on the total space with values in the Lie algebra of $G$. A principal connection induces connections on all associated vector bundles, so it is the universal form of a connection. In physics these connections are the gauge fields, and their curvature is the field strength.