Pullback of connection on a vector bundle
Given a connection for the vector bundle , we can define a connection for the vector bundle , called the pullback of , as the unique connection satisfying the following:
This is to be understood as follows. Start with a section . Take the pullback of to get a section . Then, given a vector field on , should send to the pullback via of .
Induced connection on submanifold
Further information: induced connection on submanifold
if is a Riemannian manifold and is a submanifold, then we can use a linear connection on to induce a linear connection on . This involves two steps:
- Pull back the connection on , to the connection on the pullback bundle on namely
- Project this to the connection on , using the inner product structure on
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 .