Let $M$ be a compact, connected, orientable surface in $\mathbb{R}^3$. Let $N$ be the cyllinder in $\mathbb{R}^3$ defined by $x^2+y^2=1$. Suppose $f:M\to N$ is $C^{\infty}$. Show that $f_*:TM\to TN$ is singular at 2 or more points.
So far I've shown $f_*$ is singular at at least one point since otherwise f is surjective. The argument here is that by the inverse function theorem, $f$ is a local diffeomorphism. A local homeomorphism $f:M\to N$ is an open map and since $M$ is compact, $f(M)$ is therefore both closed and open. Since $N$ is connected, $f(M)=N$.
To find another point I tried to remove the singular point from $M$ and again use the inverse function theorem, but there is a lot less to work with since $M-\{\text{pt}\}$ is not compact.
Let's denote the components of $f$ by $f_1,f_2,f_3$.
In particular, $f_3 \colon M \to \mathbb{R}$ is continuous, and since $M$ is compact, $f_3$ attains its maximum and minimum at some points of $M$.
What happens at these points?