# Pullback of connection on a vector bundle

## Contents

## Definition

Suppose is a smooth map between differential manifolds and . Let be a vector bundle over , and denote the pullback of via (hence, is a vector bundle over ).

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 .

## Related facts

### 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 .