Let $\mathcal{V}\stackrel{\pi_N}{\longrightarrow}N$ be a $V$-vector bundle on a smooth manifold $N$, let $D$ a connection on this bundle and let $f:M\rightarrow N$ be a smooth function. Then we can define the pullback bundle $f^*\mathcal{V}\stackrel{\pi_M}{\longrightarrow}M$ as the bundle given by all the elements of type $\{(p,v)|p\in M,\ \pi_N(v)=f(p)\}$ (with the obvious projection $\pi_M$). Now I have been told that $D$ induces a connection ${}^fD$ on the pullback bundle which is completely determined by the fact that if $\eta$ is a section of $\mathcal{V}\stackrel{\pi_N}{\longrightarrow}N$ and $v\in T_pM$, then
$${}^fD_vf^*\eta=D_{f^*v}\eta$$
I have a feeling that it is not true, because there could be sections of $f^*\mathcal{V}\stackrel{\pi_M}{\longrightarrow}M$ that are not the pullback of any section of $\mathcal{V}\stackrel{\pi_N}{\longrightarrow}N$. Am I wrong or there is really something missing to describe completely ${}^fD$?
Best Answer
The correct formula should be $\ {}^fD_v(f^*\eta) = D_{f_*v}\eta$. The connection is determined by what it does to a basis of sections on open sets over which the bundle is trivial. Of course, having a trivialization of $\mathcal V$ over $U\subset N$ gives a trivialization of $f^*\mathcal V$ over $f^{-1}(U)\subset M$.