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Any principal bundle over a smooth manifold admits at least one principal connection.
Let be a paracompact smooth manifold and let be a principal -bundle, where is a Lie group. Then admits a principal connection.
In particular, the affine space is nonempty.
Let be a topological space.
An open cover of is locally finite if every point has a neighborhood that meets only finitely many sets .
The space is paracompact if every open cover of admits a locally finite open refinement (i.e., there is a locally finite open cover such that each for some ).
An -dimensional smooth manifold is an -dimensional topological manifold equipped with a smooth structure.
Let be a Lie group and let be a smooth manifold. A principal -bundle over is a quadruple consisting of a smooth manifold , a surjective submersion , and a smooth right action
such that:
A morphism of principal -bundles over the same base is a smooth -equivariant map commuting with projections to .
Principal bundles are the natural setting for principal connections; once a connection is chosen, one can define parallel transport along paths and the associated holonomy group, and the curvature measures the failure of horizontal distributions to be integrable (see curvature).
Let be a principal G-bundle with structure Lie group . For each , define the vertical subspace
Here is a tangent space and is the differential of the projection.
A principal connection on is a smooth assignment of a subspace (the horizontal subspace) for every such that: