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 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 angle-sum 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 well-defined and the Gauss-Bonnet formula holds:

**1.**A. D. Aleksandrov and V. A. Zalgaller,*Intrinsic geometry of surfaces*, Translated from the Russian by J. M. Danskin. Translations of Mathematical Monographs, Vol. 15, American Mathematical Society, Providence, R.I., 1967. MR**0216434****2.**Carl B. Allendoerfer and André Weil,*The Gauss-Bonnet theorem for Riemannian polyhedra*, Trans. Amer. Math. Soc.**53**(1943), 101–129. MR**0007627**, 10.1090/S0002-9947-1943-0007627-9**3.**Thomas Banchoff,*Critical points and curvature for embedded polyhedra*, J. Differential Geometry**1**(1967), 245–256. MR**0225327****4.**Jeff Cheeger and David G. Ebin,*Comparison theorems in Riemannian geometry*, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. North-Holland Mathematical Library, Vol. 9. MR**0458335****5.**Jeff Cheeger, Werner Müller, and Robert Schrader,*On the curvature of piecewise flat spaces*, Comm. Math. Phys.**92**(1984), no. 3, 405–454. MR**734226****6.**Beifang Chen,*The Gram-Sommerville and Gauss-Bonnet theorems and combinatorial geometric measures for noncompact polyhedra*, Adv. Math.**91**(1992), no. 2, 269–291. MR**1149626**, 10.1016/0001-8708(92)90019-H**7.**Beifang Chen and Guantao Chen,*Gauss-Bonnet formula, finiteness condition, and characterizations of graphs embedded in surfaces*, Graphs Combin.**24**(2008), no. 3, 159–183. MR**2410938**, 10.1007/s00373-008-0782-z**8.**Matt DeVos and Bojan Mohar,*An analogue of the Descartes-Euler formula for infinite graphs and Higuchi’s conjecture*, Trans. Amer. Math. Soc.**359**(2007), no. 7, 3287–3300 (electronic). MR**2299456**, 10.1090/S0002-9947-07-04125-6**9.**H. Groemer,*On the extension of additive functionals on classes of convex sets*, Pacific J. Math.**75**(1978), no. 2, 397–410. MR**0513905****10.**M. Gromov,*Hyperbolic groups*, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR**919829**, 10.1007/978-1-4613-9586-7_3**11.**Yusuke Higuchi,*Combinatorial curvature for planar graphs*, J. Graph Theory**38**(2001), no. 4, 220–229. MR**1864922**, 10.1002/jgt.10004**12.**M. Ishida, Pseudo-curvature of a graph,*Lecture at `Workshop on topological graph theory'*, Yokohama National University, 1990.**13.**P. McMullen,*Non-linear angle-sum relations for polyhedral cones and polytopes*, Math. Proc. Cambridge Philos. Soc.**78**(1975), no. 2, 247–261. MR**0394436****14.**S. B. Myers,*Riemannian manifolds with positive mean curvature*, Duke Math. J.**8**(1941), 401–404. MR**0004518****15.**David A. Stone,*A combinatorial analogue of a theorem of Myers*, Illinois J. Math.**20**(1976), no. 1, 12–21. MR**0410602****16.**Wolfgang Woess,*A note on tilings and strong isoperimetric inequality*, Math. Proc. Cambridge Philos. Soc.**124**(1998), no. 3, 385–393. MR**1636552**, 10.1017/S0305004197002429

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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/S0002-9939-08-09739-6

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.