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 .