Sufficient conditions for triviality of pullback vector bundle to imply triviality of original vector bundle Let $E$ be a vector bundle over a smooth manifold $N$ and $f\colon M \to N$ a smooth surjective map. Is it possible that $f^*E$ is trivial while $E$ is non-trivial? If the previous question has a negative answer, can one give conditions on $f$ (not so strong, i.e. a diffeomorphism) such that the claim "$f^*E$ is trivial if and only if $E$ is trivial" hold? Maybe for line bundles?

The answer to the first question is yes. Take $M=\Bbb{R}$, $N=S^1$, $E$ the Möbius bundle. The pullback of $E$ is trivial on $\Bbb{R}$.

In general, if $M$ is contractible, then the answer to this question tells us that all vector bundles on $M$ are trivial. Thus, it seems unlikely that there would be a good condition such that $f^*E$ being trivial implies that $E$ is trivial as well.

That said, the characterization of vector bundles in the answer to the linked question suggests that if $f$ is a homotopy equivalence, then $f^*E$ being trivial should imply $E$ is also trivial. While still a strong condition, it is much weaker than $f$ being a diffeomorphism.