I've some problems with the definition of the pullback vector bundle. Say $F : M \rightarrow N$ be a $C^{\infty}$ function between differential varieties, and $\pi : E \rightarrow N$ a vector bundle over $N$. We define $F^*E$, the pullback vector bundle of $E$, by saying:
- $(F^*E)_p = E_{F(p)}$ be the fiber over $p \in M$
- $F^*E = \bigsqcup_p \, (F^*E)_p$
- $\tilde{\pi} : F^*E \rightarrow M$, with $\tilde{\pi} = F^*(\pi) = \pi \circ F$
With this definitions $\tilde{\pi}: F^*E \rightarrow M$ is a vector bundle
Now, the problem is in the definition of $\tilde{\pi}$. I expect that:
$$\tilde{\pi}(F^*E)_p = p$$
but
$$\tilde{\pi}(F^*E)_p = \tilde{\pi} E_{F(p)} = \pi \circ F(E_{F(p)}) \neq p = F^{-1} \circ \pi (E_{F(p)})$$
Where i go wrong?
Thanks