The Gauss-Bonnet Formula of Polytopal Manifolds and the Characterization of Embedded Graphs with Nonnegative Curvature

Let M be a connected d-manifold without boundary obtained from a (possibly infinite) collection P of polytopes of R by identifying them along isometric facets. Let V (M) be the set of vertices of M . For each v ∈ V (M), define the discrete Gaussian curvature κM (v) as the normal angle-sum with sign, extended over all polytopes having v as a vertex. Our main result is as follows: If the absolute total curvature ∑ v∈V (M) |κM (v)| is finite, then the limiting curvature κM (p) for every end p of M can be well-defined and holds the Gauss-Bonnet formula: ∑ v∈V (M)∪EndM κM (v) = χ(M). In particular, if G is a (possibly infinite) graph embedded in a 2-manifold M without boundary such that every face has at least 3 sides, and if the combinatorial curvature ΦG(v) ≥ 0 for all v ∈ V (G), then the number of vertices with non-vanishing curvature is finite. Furthermore, if G is finite, then M has four choices: sphere, torus, projective plane, and Klein bottle. If G is infinite, then M has three choices: cylinder without boundary, plane, and projective plane minus one point.