Suppose that $M$ is a non-compact manifold with one end which is the universal cover of some closed manifold $N$.

Is $M $ necessarily homeomorphic to the total space of some vector bundle?

In fact the only examples I can think up are much more limited, just of the form $M = \Sigma \times \mathbb{R}^n$ where $\Sigma$ is a closed simply connected manifold.

Cross posted on stack exchange https://math.stackexchange.com/questions/4417368/universal-covers-with-one-end.