Pullback of connection on a vector bundle

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Definition

Suppose f:N \to M is a smooth map between differential manifolds N and M. Let E be a vector bundle over M, and f^*E denote the pullback of E via f (hence, f^*E is a vector bundle over N).

Given a connection \nabla for the vector bundle E, we can define a connection f^*(\nabla) for the vector bundle f^*(E), called the pullback of \nabla, as the unique connection satisfying the following:

(f^*\nabla)_X(f^*s) = f^*(\nabla_{(Df)(X)}(s))

This is to be understood as follows. Start with a section s \in \Gamma(E). Take the pullback of s to get a section f^*s \in \Gamma(f^*E). Then, given a vector field X on N, (f^*\nabla)_X should send f^*s to the pullback via f of \nabla_{Df(x)}(s).

Related facts

Induced connection on submanifold

Further information: induced connection on submanifold

if M is a Riemannian manifold and N is a submanifold, then we can use a linear connection on M to induce a linear connection on N. This involves two steps:

  • Pull back the connection on TM, to the connection on the pullback bundle on N namely TM|_N
  • Project this to the connection on TN, using the inner product structure on TM|_N

Connection along a curve

Further information: connection along a curve

A connection along a curve can be viewed as a special case of a pullback connection, where the pullback is to the interval (0,1).