Consider the family $\mathbb{T}$ of compact oriented surfaces homeomorphic to the torus $\mathcal{T} = S^1 \times S^1$. Consider arbitrary continuous mappings $k: \mathcal{T} \rightarrow \mathbb{R}$ which obey the condition $\int_\mathcal{T} k = 0$.

For which curvature-like mappings $k$ does exist a surface $T \in > \mathbb{T}$ which $k$ is the Gaussian curvature of?

For the topological sphere the answer seems to be: for all. Can the proof of the latter - which I do not know - be generalized for arbitrary compact oriented surfaces? Or is it already a general proof for arbitrary surfaces?

topologicalmanifolds is what's throwing me now. – Ryan Budney Feb 25 '13 at 8:54