In the answer to this question it is proved that
Any local diffeomorphism from a compact connected $n$-dimensional manifold ($n>1$) to the sphere is a diffeomorphism.
This result seems to contradict the proof of Theorem 1 in a classical paper by Chern and Lashof, as explained below.
Let $M$ be a compact $n$-dimensional submanifold of $\mathbb{R}^{n+N}$. Let $B_{\nu}$ denote the unit normal bundle of $M$, whose fiber at $p \in M$ is the $(N-1)$-sphere inside the normal space of $M$ at $p$. The authors define a map $\overline{\nu}$ which simply translates any unit normal vector of $M$ to the unit sphere centered at the origin in $\mathbb{R}^{n+N}$ : $$\begin{align} B_{\nu} &\to S^{n+N-1}_{0}\\ (p,\nu) &\mapsto \nu \,. \end{align} $$
In the proof of Theorem 1 (p. 312, first paragraph) they show that
Every point of $S^{n+N-1}_{0}$ is covered at least twice by $\overline{\nu}$.
Now, if $\overline{\nu}$ is a local diffeomorphism, $N>1$, and $M$ is connected, then the map $\overline{\nu}$ satisfies the conditions of the statement above (as the unit normal bundle is compact and connected in such case). Being a diffeomorphism, it cannot cover more than once the sphere.
What am I missing?
EDIT (in view of the comments): I am not claiming that the map $\overline{\nu}$ is always a local diffeomorphism, only that there are cases where it is.