Pullback of connection on a vector bundle

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 $(0,1)$.