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I've tried to prove the naturality of the pullback of a connection. I've reduced it to the following question:

Is the pullback a surjective mapping on the space of sections of a vector bundle? i.e., suppose I have a smooth vector bundle $~E\to M$, and a smooth map $~f:N\to M$. Then is $$f^*:\Gamma(E)\to \Gamma(f^*E)$$ surjective, where $f^*E$ is the pullback bundle over $N$?

It seems intuitive, but I'm struggling to prove it.

David Roberts
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Generally no. For example, if $M$ is a one-point space and $E=\mathbb R^n\to M$ is the trivial rank-$n$ vector bundle over $M$, then $f^*E$ is the trivial bundle $N\times \mathbb R^n\to N$. In this case, $\Gamma(E)$ is finite-dimensional (just a copy of $\mathbb R^n$), while $\Gamma(f^*E)$ is infinite-dimensional (essentially the set of all smooth functions from $N$ to $\mathbb R^n$).

Jack Lee
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    Haha I'm a fan of your books on manifolds. Kinda wish there was a section on pullback bundles in one of them :) – David Roberts Nov 15 '14 at 23:12
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    @David, thank you. I might someday write a book about bundles (I have a few draft chapters hanging around somewhere), which will definitely contain a section on pullback bundles. But don't hold your breath. – Jack Lee Nov 15 '14 at 23:21