A surjective submersion (surmersion) between smooth manifolds $ \pi : M \rightarrow B $ is a fiber bundle if it is proper (Ehresmann theorem).
In particular, there always exists local sections of $\pi$. One can try to glue this local section in a global one, but there are obstructions.
The questions is:
What are the obstruction for the existence of an immersion $\iota :B \rightarrow M$ such that $\pi \circ \iota$ is a covering of $B$ ? In particular the case where the total space and the base space are compact.
Any reference is welcomed too.
A bit more general: What can we get by gluing local section ? When is it possible to get a possibly immersed compact manifold by gluing multi-valued section (in the case where the base space is compact).