The Gauss-Bonnet formula of polytopal manifolds and the characterization of embedded graphs with nonnegative curvature

Abstract

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