I was considering following problem.
Let us have a torus $T^2$ and real projective plane $\mathbb{R}P^2$. Let $f: T^2 \to \mathbb{R}$ and $g: \mathbb{R}P^2 \to \mathbb{R}$ be a Morse functions. Proof that $f$ and $g$ have at least one saddle point. Find minimal number of saddle points of $f$ and $g$.
So I know that $f$ and $g$ have at least two critical points, because torus and projective plane are compact. Moreover those points are non-degenerate. I should probably use two dimentional square model of those manifolds, but to be honest, I don't know what to do next. I would be grateful for any hints.