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 v is an element of V(M), define tire 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 Sigma(v is an element of V(M))vertical bar kappa(M)(v)vertical bar is finite, then the limiting curvature kappa(M)(p) for every end p is an element of End M can be well-defined and the Gauss-Bonnet, formula holds: Sigma(v is an element of V(M)boolean OR 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) >= 0 for all v is an element of V(G), then the number of vertices with nonvanishing curvature is finite. Furthermore, if G is finite, their 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.