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
Abstract 
References 
Similar Articles 
Additional Information
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.
 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
(35 #7267)
 2.
Carl
B. Allendoerfer and André
Weil, The GaussBonnet theorem for
Riemannian polyhedra, Trans. Amer. Math.
Soc. 53 (1943),
101–129. MR 0007627
(4,169e), http://dx.doi.org/10.1090/S00029947194300076279
 3.
Thomas
Banchoff, Critical points and curvature for embedded
polyhedra, J. Differential Geometry 1 (1967),
245–256. MR 0225327
(37 #921)
 4.
Jeff
Cheeger and David
G. Ebin, Comparison theorems in Riemannian geometry,
NorthHolland Publishing Co., Amsterdam, 1975. NorthHolland Mathematical
Library, Vol. 9. MR 0458335
(56 #16538)
 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
(85m:53037)
 6.
Beifang
Chen, The GramSommerville and GaussBonnet theorems and
combinatorial geometric measures for noncompact polyhedra, Adv. Math.
91 (1992), no. 2, 269–291. MR 1149626
(92m:52021), http://dx.doi.org/10.1016/00018708(92)90019H
 7.
Beifang
Chen and Guantao
Chen, GaussBonnet formula, finiteness condition, and
characterizations of graphs embedded in surfaces, Graphs Combin.
24 (2008), no. 3, 159–183. MR 2410938
(2009g:05037), http://dx.doi.org/10.1007/s003730080782z
 8.
Matt
DeVos and Bojan
Mohar, An analogue of the DescartesEuler
formula for infinite graphs and Higuchi’s conjecture, Trans. Amer. Math. Soc. 359 (2007), no. 7, 3287–3300 (electronic).
MR
2299456 (2008e:05041), http://dx.doi.org/10.1090/S0002994707041256
 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
(58 #24003)
 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
(89e:20070), http://dx.doi.org/10.1007/9781461395867_3
 11.
Yusuke
Higuchi, Combinatorial curvature for planar graphs, J. Graph
Theory 38 (2001), no. 4, 220–229. MR 1864922
(2002i:05109), http://dx.doi.org/10.1002/jgt.10004
 12.
M. Ishida, Pseudocurvature of a graph, Lecture at `Workshop on topological graph theory', Yokohama National University, 1990.
 13.
P.
McMullen, Nonlinear anglesum relations for polyhedral cones and
polytopes, Math. Proc. Cambridge Philos. Soc. 78
(1975), no. 2, 247–261. MR 0394436
(52 #15238)
 14.
S.
B. Myers, Riemannian manifolds with positive mean curvature,
Duke Math. J. 8 (1941), 401–404. MR 0004518
(3,18f)
 15.
David
A. Stone, A combinatorial analogue of a theorem of Myers,
Illinois J. Math. 20 (1976), no. 1, 12–21. MR 0410602
(53 #14350a)
 16.
Wolfgang
Woess, A note on tilings and strong isoperimetric inequality,
Math. Proc. Cambridge Philos. Soc. 124 (1998), no. 3,
385–393. MR 1636552
(99f:52026), http://dx.doi.org/10.1017/S0305004197002429
 1.
 A.D. Aleksandrov and V.A. Zalgaller, Intrinsic Geometry of Surfaces, Translations of Mathematical Monographs, vol. 15, Amer. Math. Soc., Providence, RI, 1967. MR 0216434 (35:7267)
 2.
 C.B. Allendoerfer and A. Weil, The GaussBonnet theorem for Riemannian polyhedra, Trans. Amer. Math. Soc. 53 (1943), 101129. MR 0007627 (4:169e)
 3.
 T. Banchoff, Critical points and curvature for embedded polyhedra, J. Diff. Geom. 1 (1967), 245256. MR 0225327 (37:921)
 4.
 J. Cheeger and D.G. Ebin, Comparison Theorems in Riemannian Geometry, NorthHolland Publishing Co., 1975. MR 0458335 (56:16538)
 5.
 J. Cheeger, W. Müller, and R. Schrader, On the curvature of piecewise flat spaces. Comm. Math. Phys. 92 (1984), 405454. MR 734226 (85m:53037)
 6.
 B. Chen, The GramSommerville and GaussBonnet theorems and combinatorial geometric measures for noncompact polyhedra, Advances in Math. 91 (1992), 269291. MR 1149626 (92m:52021)
 7.
 B. Chen and G. Chen, GaussBonnet formula, finiteness condition, and characterizations of graphs embedded in surfaces, Graphs and Combin. 24 (2008), 159183. MR 2410938
 8.
 M. DeVos and B. Mohar, An analogue of the DescartesEuler formula for infinite graphs and Higuchi's conjecture, Trans. Amer. Math. Soc. 359 (2007), 32873300. MR 2299456 (2008e:05041)
 9.
 H. Groemer, On the extension of additive functionals on classes of convex sets, Pacific J. Math. 75 (1978), 397410. MR 0513905 (58:24003)
 10.
 M. Gromov, Hyperbolic groups, Essays in group theory, S. M. Gersten (Editor), M.S.R.I. Publ. 8, Springer, 1987, pp. 75263. MR 919829 (89e:20070)
 11.
 Y. Higuchi, Combinatorial curvature for planar graphs, J. Graph Theory 38 (2001), 220229. MR 1864922 (2002i:05109)
 12.
 M. Ishida, Pseudocurvature of a graph, Lecture at `Workshop on topological graph theory', Yokohama National University, 1990.
 13.
 P. McMullen, Nonlinear anglesum relations for polyhedral cones and polytopes, Math. Proc. Cambridge Philos. Soc. 78 (1975), 247261. MR 0394436 (52:15238)
 14.
 S.B. Myers, Riemannian manifolds with positive mean curvature, Duke Math. J. 8 (1941), 401404. MR 0004518 (3:18f)
 15.
 D. Stone, A combinatorial analogue of a theorem of Myers, Illinois J. Math. 20 (1976), 1221. MR 0410602 (53:14350a)
 16.
 W. Woess, A note on tilings and strong isoperimetric inequality, Math. Proc. Camb. Phil. Soc. 124 (1998), 385393. MR 1636552 (99f:52026)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
05C10,
52B70,
05C75,
57M15,
57N05,
57P99
Retrieve articles in all journals
with MSC (2000):
05C10,
52B70,
05C75,
57M15,
57N05,
57P99
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.
