Relation To Principal and Ehresmann Connections
Let E → M be a vector bundle of rank k and let F(E) be the principal frame bundle of E. Then a (principal) connection on F(E) induces a connection on E. First note that sections of E are in one-to-one correspondence with right-equivariant maps F(E) → Rk. (This can be seen by considering the pullback of E over F(E) → M, which is isomorphic to the trivial bundle F(E) × Rk.) Given a section σ of E let the corresponding equivariant map be ψ(σ). The covariant derivative on E is then given by
where XH is the horizontal lift of X (recall that the horizontal lift is determined by the connection on F(E)).
Conversely, a connection on E determines a connection on F(E), and these two constructions are mutually inverse.
A connection on E is also determined equivalently by a linear Ehresmann connection on E. This provides one method to construct the associated principal connection.
Read more about this topic: Connection (vector Bundle)
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