I have a question related to the definition of the pull-back connection, more specifically about its uniqueness or the canonical way to induce it.

The definition that one finds in general goes along the following lines: let $F:P\rightarrow B$ be map between differentiable manifolds, let $\pi:E\rightarrow B$ be a vector bundle and $\nabla$ a connection on $E$. Then the connection $F^*\nabla$ is uniquely determined by the following relation

$(F^*\nabla)_X(F^* s):=F^*(\nabla_{df(X)} s)$.

This should uniquely determine the connection right?

Let us start with the most trivial case, when $B$ is a point. Then $E$ is a vector space and the pull-back of $E$ is a trivial vector space over $P$. A connection on $E\rightarrow pt$ is an endomorphism of $E$ (which trivially satisfies the Leibniz relation). Could anybody explain how does that canonically induce a connection on $P\times E$, presumably the trivial connection $d$ if one starts with the zero endomorphism of $E$? Leibniz relation does not suffice. One could say let us make a convention here. But the more general question is how does one define $(F^*\nabla)_X$ when $X\in\ker{dF}$, in general?

varying opensin $B$. Local sections of $F^{\ast}(E)$ are function-linear combinations of $F$-pullbacks of local sections of $E$, so the pullback rule (using pullback of 1-forms and of local sections) and Leibniz yield uniqueness. Construction with local coords gives local existence (& yields d when $B$ is pt and $E$ trivial), so by uniqueness get global existence. $\endgroup$6more comments