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 $\mathrm{\nabla }$ for the vector bundle $E$, we can define a connection $f^{*}(\mathrm{\nabla })$ for the vector bundle $f^{*}(E)$, called the pullback of $\mathrm{\nabla }$, as the unique connection satisfying the following:

$(f^{*}\mathrm{\nabla }{)}_X(f^{*}s)=f^{*}({\mathrm{\nabla }}_{(Df)(X)}(s))$

This is to be understood as follows. Start with a section $s\in \mathrm{\Gamma }(E)$. Take the pullback of $s$ to get a section $f^{*}s\in \mathrm{\Gamma }(f^{*}E)$. Then, given a vector field $X$ on $N$, $(f^{*}\mathrm{\nabla }{)}_X$ should send $f^{*}s$ to the pullback via $f$ of ${\mathrm{\nabla }}_{Df(x)}(s)$.

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 $(0,1)$.