My question is quite straightforward : given a $C^r$-manifold $X$ and a vector bundle $\pi : E \to X$ on $X$, is it true that $E \simeq \bigsqcup_{x \in} E_x$, where $E_x = \pi^{-1}(x)$? That is, can we always think of a vector bundle as the disjoint union of the fibers $E_x$, not just intuitively but also rigourously ? A priori, it seems there is not canonical diffeomorphism between the two.
Thanks a lot for any help!
As sets, there is a bijection between the two, yes. When you ask for a "diffeomorphism," you first need to give a smoothness structure on the disjoint union of the fibers. If your smoothness structure is inherited from the structure on each fiber in the original vector bundle $E$, then there's generally no such diffeomorphism, for if $X$ is connected, then so is $E$, but the disjoint union will typically have infinitely many connected components (if $X$ is anything other than a single point). Hence the two sets are not even homoemorphic, let alone diffeomorphic.