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

Author: Beifang Chen
Journal: Proc. Amer. Math. Soc. 137 (2009), 1601-1611
MSC (2000): Primary 05C10, 52B70; Secondary 05C75, 57M15, 57N05, 57P99
Published electronically: November 20, 2008
MathSciNet review: 2470818
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Abstract: 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:

$\displaystyle \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.

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Additional Information

Beifang Chen
Affiliation: Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

Keywords: Discrete curvature, combinatorial curvature, Gauss-Bonnet formula, Euler relation, infinite graph, embedded graph, nonnegative curvature, finiteness theorem
Received by editor(s): March 2, 2007
Received by editor(s) in revised form: February 15, 2008, and July 6, 2008
Published electronically: November 20, 2008
Additional Notes: The author was supported in part by the RGC Competitive Earmarked Research Grants 600703 and 600506.
Communicated by: Jon G. Wolfson
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.