Proper Immersion between smooth manifolds

Question

I want to verify if the following proposition holds. If yes, I would like to find a textbook having its proof. It is the following : Let M and N be smooth manifolds and F: M-->N be a proper immersion, such that the fibers of each point in M have constant and finite cardinality. Then M is a covering space of N. I am not sure if the statement is correct.

Answer 1

As stated, it's not correct. For example, let $N=\mathbb R^2$, $M=\mathbb R$, and $F(x) = (x,0)$. Then $F$ is a proper immersion and its fibers are all singletons, but $F$ is far from being a covering map.

If you add the hypotheses that $M$ and $N$ are connected and have the same dimension, then the conclusion is true, even without assuming that the fibers have the same cardinality. In that case, $F$ is a local diffeomorphism, and every proper local diffeomorphism between connected smooth manifolds is a covering map. For a proof, see Proposition 4.46 in my Introduction to Smooth Manifolds (2nd ed.).