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 follows:

$(f^{*}\mathrm{\nabla }{)}_X(s)=p\mapsto f^{-1}({\mathrm{\nabla }}_{(Df)(X(p))}(f(s)))$

This is to be understood as follows. Starting with a point $p\in N$, a tangent vector $X(p)$ at $p$, and a section $s$ of $f^{*}(E)$ we proceed as follows:

• Differentiate