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 $TM|_{N}$