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 $\mathcal P$ of polytopes of ${\mathbb{R}}^d$ by identifying them along isometric facets. Let $V(M)$ be the set of vertices of $M$. For each $v\in V(M)$, define the discrete Gaussian curvature $\kappa_M(v)$ 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 $\sum_{v\in V(M)}\vert\kappa_M(v)\vert$ is finite, then the limiting curvature $\kappa_M(p)$ for every end $p\in\operatorname{End} M$ can be well-defined and the Gauss-Bonnet formula holds:

$$\sum_{v\in V(M)\cup\operatorname{End} M}\kappa_M(v)=\chi(M).$$

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 $\Phi_G(v)\geq 0$ for all $v\in V(G)$, 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.