Let $M$ and $N$ be smooth compact $3$-dimensional manifolds. Assume that $f \colon M \to N$ is a $C^2$-map, such that
- $\operatorname{rk} \ Df = 3$ in at least one point;
- if $\operatorname{rk} Df \neq 3$, then $\operatorname{rk} Df = 1$.
Does it follow that $f$ is surjective?
More generally, one can ask about maps of $n$-dimensional manifolds, such that the rank of differential equals $n$ in at least one point, and if it is not $n$, then it is strictly less then $n-1$. My feeling is that in this case the map has to be surjective. The euristic explanation is: since $\operatorname{rk} Df$ is maximal in at least one point, the map is generically open; the image $f(M)$ contains an interiour, and its closure is a $C^1$-manifold with boundary. The boundary is of dimension $n-1$ and for a generic point on it the rank of the differential should equal $n-1$.
However I am not able to find any discussion in the literature, although the topic seems to be pretty classic.
N.B.: The $C^2$-restriction on the map is nut essential; I would be interested in any results for smooth maps as well.