The GaussBonnet formula of polytopal manifolds and the characterization of embedded graphs with nonnegative curvature
Author:
Beifang Chen
Journal:
Proc. Amer. Math. Soc. 137 (2009), 16011611
MSC (2000):
Primary 05C10, 52B70; Secondary 05C75, 57M15, 57N05, 57P99
Published electronically:
November 20, 2008
MathSciNet review:
2470818
Fulltext PDF Free Access
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Abstract: Let be a connected manifold without boundary obtained from a (possibly infinite) collection of polytopes of by identifying them along isometric facets. Let be the set of vertices of . For each , define the discrete Gaussian curvature as the normal anglesum with sign, extended over all polytopes having as a vertex. Our main result is as follows: If the absolute total curvature is finite, then the limiting curvature for every end can be welldefined and the GaussBonnet formula holds: In particular, if is a (possibly infinite) graph embedded in a manifold without boundary such that every face has at least sides, and if the combinatorial curvature for all , then the number of vertices with nonvanishing curvature is finite. Furthermore, if is finite, then has four choices: sphere, torus, projective plane, and Klein bottle. If is infinite, then 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:
mabfchen@ust.hk
DOI:
http://dx.doi.org/10.1090/S0002993908097396
PII:
S 00029939(08)097396
Keywords:
Discrete curvature,
combinatorial curvature,
GaussBonnet 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.
