Flatness of torus and surfaces of higher genus

Question

For the very first sight it may be surprise that the ordinary torus $S^1 \times S^1$ is flat: one argument to see this is the following. One can imagine a torus as a square with opposite sides identified and the square is obviously flat. However there are surfaces with higher genus (higher numbers of "holes") which also can be represented as polygons (4$g$ polygon for a surface of genus $g$) with sides indetified. However higher genus surfaces are negatively curved. So my question is

Why higher genus surfaces are negatively curved while they can be represented as flat polygons with sides identified?

Answer 1

If you try to turn those polygon identifications into flat metrics you run into a problem at the vertices: the angles don't add up. This issue can only be addressed for the square.

Answer 2

Theorem Benzecri. A closed surface is flat if and only it is euler number is zero.

Goldman, W. Two papers which change my life: Milnor seminal work on flat manifolds and bundles.

http://arxiv.org/pdf/1108.0216.pdf

p.4 which has a good outline of the proof.

To be more precise, a surface of genus $g>1$ than cannot be endowed with a differentiable metric whose curvature vanishes identically, since it cannot be endowed with a Koszul derivative which is flat, here flat means that the curvature and the torsion form vanish identically, since we know that the torsion form of the Levi-Civita connection vanishes, a differentiable manifold endowed with a flat differentiable metric has a flat structure; i.e it is an affine manifold. It is a well-known theorem that a Riemannian closed flat manifolds are finitely covered by the torus $T^n$, their fundamental groups are crystallographic groups, so again this theorem also implies that the only oriented surface endowed with a flat metric is the torus.

Remark that in the case of dimension 2, Milnor in his paper On the existence of a connection with curvature zero, Comm. Math. Helv. 32 (1958), 215-223 Milnor has generalized the work of Benzecri, by computing an equality which represents an obstruction for a bundle defined on a surface to be flat.