Let $E \to M$ be a smooth vector bundle with a connection $\nabla$. Let $f: N \to M$ be a smooth map between manifolds $N$ and $M$. We can define the pullback of the connection $f^*\nabla$ on the pullback bundle $f^*E \to N$ by the following argument.
Any section $s$ of $f^*E \to N$ can be written as $s = \sum_{i=1}^N g_if^*s_i$ where $s_i \in \Gamma(M, E)$. Then define $(f^*\nabla)(f^*s_i) := f^*(\nabla(s_i))$. Finally, we can linearly extend it to $\Gamma(N, f^*E)$. Checking that it's indeed a connection is not hard. The thing I found nontrivial to check is the well-definedness. Say, if $s = \sum_{i=1}^N g_if^*s_i = \sum_{i=1}^M h_if^*s^\prime_i$, then is $(f^*\nabla)(\sum_{i=1}^N g_if^*s_i) = (f^*\nabla)(\sum_{i=1}^M h_if^*s^\prime_i)$? Or we can reduce this question to the following one:
If $f^*s(x_0) = 0$ then do we have $f^*(\nabla(s))(x_0) = 0$? I cannot solve this one...maybe can someone give a hint or a solution?