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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The Gauss-Bonnet formula of polytopal manifolds and the characterization of embedded graphs with nonnegative curvature
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by Beifang Chen PDF
Proc. Amer. Math. Soc. 137 (2009), 1601-1611 Request permission


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 $\sum _{v\in V(M)}|\kappa _M(v)|$ 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.
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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
  • Email:
  • 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
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 137 (2009), 1601-1611
  • MSC (2000): Primary 05C10, 52B70; Secondary 05C75, 57M15, 57N05, 57P99
  • DOI:
  • MathSciNet review: 2470818