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^d by identifying them along isometric facets. Let V(M) be the set of vertices of M. For each υ ∈ V(M), define the discrete Gaussian curvature κ_M(υ) as the normal angle-sum with sign, extended over all polytopes having υ as a vertex. Our main result is as follows: If the absolute total curvature Σ_υ∈V(M) |κ_M(υ)| is finite, then the limiting curvature κ_M(p) for every end p ∈ End M can be well-defined and the Gauss-Bonnet formula holds: Σ υ∈V(M) ∪End M κ_M(υ) = 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 φ_G(υ) ≥ 0 for all υ ∈ V(G), then the number of vertices with nonvanishing 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.