# Possible curvatures of the topological torus

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?

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How are you integrating a function that takes values in $\mathbb T$? $\mathbb T$ isn't a vector space, and what measure on $\mathcal T$ are you integrating with respect to? – Ryan Budney Feb 25 '13 at 8:25
It sounds like you're asking the question, which functions $S^1 \times S^1 \to \mathbb R$ can be realized as Gauss curvatures for some Levi-Cevita connection on $S^1 \times S^1$ ? – Ryan Budney Feb 25 '13 at 8:28
@Ryan: Sorry, this was a typo, it should have been $\mathbb{R}$. – Hans Stricker Feb 25 '13 at 8:42
@Ryan: Is "Levi-Civita connection on $S^1\times S^1$" the same as "homeomorphic to $S^1\times S^1$"? If so: yes, that's what I am asking. – Hans Stricker Feb 25 '13 at 8:44
If you don't demand some regularity of your manifold your curvature could be a distribution rather than a continuous function. That's why I'm asking what you mean by curvature -- curvature of what? If it's the curvature of a Levi-Cevita connection presumably we're talking about smooth manifolds and fairly standard notions of curvature. Your insistence on talking about topological manifolds is what's throwing me now. – Ryan Budney Feb 25 '13 at 8:54