On combinatorial Gauss-Bonnet Theorem for general Euclidean simplicial complexes

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• 作者：Stephan Klaus
• 关键词：Curvature ; dihedral angle ; Euclidean simplex ; triangulation ; Euler characteristic ; Euler manifold ; combinatorial manifold ; pseudo manifold
• 刊名：Frontiers of Mathematics in China
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：11
• 期：5
• 页码：1345-1362
• 全文大小：199 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Chinese Library of Science
  
• 出版者：Higher Education Press, co-published with Springer-Verlag GmbH
• ISSN：1673-3576
• 卷排序：11

文摘

For a finitely triangulated closed surface M2, let α_x be the sum of angles at a vertex x. By the well-known combinatorial version of the 2- dimensional Gauss-Bonnet Theorem, it holds Σ_x(2π - α_x) = 2πχ(M2), where χ denotes the Euler characteristic of M2, α_x denotes the sum of angles at the vertex x, and the sum is over all vertices of the triangulation. We give here an elementary proof of a straightforward higher-dimensional generalization to Euclidean simplicial complexes K without assuming any combinatorial manifold condition. First, we recall some facts on simplicial complexes, the Euler characteristics and its local version at a vertex. Then we define δ(τ) as the normed dihedral angle defect around a simplex τ. Our main result is Σ_τ (-1)dim(τ)δ(τ) = χ(K), where the sum is over all simplices τ of the triangulation. Then we give a definition of curvature κ(x) at a vertex and we prove the vertex-version Σ_x∈K0 κ(x) = χ(K) of this result. It also possible to prove Morse-type inequalities. Moreover, we can apply this result to combinatorial (n + 1)-manifolds W with boundary B, where we prove that the difference of Euler characteristics is given by the sum of curvatures over the interior of W plus a contribution from the normal curvature along the boundary B: $$\chi \left( W \right) - \frac{1}{2}\chi \left( B \right) = \sum {_{\tau \in W - B}} {\left( { - 1} \right)^{\dim \left( \tau \right)}} + \sum {_{\tau \in B}} {\left( { - 1} \right)^{\dim \left( \tau \right)}}\rho \left( \tau \right)$$.KeywordsCurvaturedihedral angleEuclidean simplextriangulationEuler characteristicEuler manifoldcombinatorial manifoldpseudo manifoldMSC51M2052B7053A5553C2053C2355U0555U1057Q0557Q1557R0557R20References1.Allendoerfer C B, Weil A. The Gauss-Bonnet Theorem for Riemannian polyhedra. Trans Amer Math Soc, 1943, 53: 101–129MathSciNetCrossRefMATHGoogle Scholar2.Banchoff T F. Critical points and curvature for embedded polyhedra. 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