Ehresmann connection

Summary

An Ehresmann connection on a fibre bundle is a choice of horizontal directions at every point of the total space. These directions are complementary to the directions along the fibres. It works for any smooth fibre bundle, not only for vector bundles, and needs no linear structure. It lets one lift curves in the base to horizontal curves in the total space, which gives parallel transport between fibres. Connections on vector bundles and on principal bundles are Ehresmann connections with extra symmetry.