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

Authors Chen, Beifang View this author's profile
Issue Date 2009
Source Proceedings of the American Mathematical Society , v. 137, (5), 2009, p. 1601-1611
Summary 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.
ISSN 0002-9939
Language English
Format Article
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