Which compact (necessarily orientable) smooth $2$-manifolds are parallelizable?
I'm aware that the sphere $\mathbb{S}^2$ is not parallelizable, whereas the torus $\mathbb{T}^2 = \mathbb{S}^1 \times \mathbb{S}^1$ is. This leaves the case of connected sums of tori $\Sigma_g = \mathbb{T}^2 \# \cdots \# \mathbb{T}^2$.
Note: This is not a homework question. I'm asking purely out of curiosity, especially because I have no idea how one would approach this problem (aside from the techniques mentioned in this problem, which are slightly above my level).
If $S$ is a parallelizable surface, then it has a flat Riemannian metric -- just choose a smooth global frame and declare it to be orthonormal. If in addition $S$ is compact, Gauss-Bonnet implies that it has Euler characteristic zero. The only compact orientable surface with $\chi(S)=0$ is the torus.