On combinatorial Gauss-Bonnet Theorem for general Euclidean simplicial complexes

Abstract

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 ∑_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 simplexτ. Our main result is ∑_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 {\sum }_{x\in K_{\mathrm{0}}}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:χ(W) −{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}χ(B) = ∑_τ∈W−B(−1)^dim(τ)δ(τ) +∑_τ∈B(−1)^dim(τ)ρ(τ).

Key words： Curvature; dihedral angle; Euclidean simplex; triangulation; Euler characteristic; Euler manifold; combinatorial manifold; pseudo manifold

References

1 Allendoerfer C B, Weil A. The Gauss-Bonnet Theorem for Riemannian polyhedra. Trans Amer Math Soc, 1943, 53: 101–129

2 Banchoff T F. Critical points and curvature for embedded polyhedra. J Differential Geom, 1967, 1: 245–256

3 Banchoff T F. Critical points and curvature for embedded polyhedral surfaces. Amer Math Monthly, 1970, 77: 475–485

4 Banchoff T F. Critical points and curvature for embedded polyhedra. II. In: Differential Geometry (College Park, Md, 1981/1982). Progr Math, Vol 32. Boston: Birkhäuser, 1983, 34–55

5 Berger M. Geometry I. Berlin: Springer, 1987

6 Bloch E D. The angle defect for arbitrary polyhedra. Beiträge Algebra Geom, 1998, 39: 379–393

7 Bloch E D. Critical points and the angle defect. Geom Dedicata, 2004, 109: 121–137

8 Bloch E D. The angle defect for odd-dimensional simplicial manifolds. Discrete Comput Geom, 2006, 35(2): 311–328

9 Chern S S. A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds, Ann of Math, 1944, 45: 747–752

10 Conway J H, Guy R K. The Book of Numbers. Berlin: Springer, 1996, 107–109

11 Hopf H. Differential Geometry in the Large. Lecture Notes in Math, Vol 1000. Berlin: Springer, 1989

12 Klee V. A combinatorial analogue of Poincaré’s duality theorem. Canad J Math, 1964, 16: 517–531

13 Levitt N. The Euler characteristic is the unique locally determined numerical homotopy invariant of finite complexes. Discrete Comput Geom, 1992, 7(1): 59–67

14 MacLaurin C, Robertson G. Euler characteristic in odd dimensions. Austral Math Soc Gaz, 2003, 30(4): 195–199

15 Milnor J W, Stasheff J D. Characteristic Classes. Ann of Math Stud, Vol 76. Princeton: Princeton Univ Press, 1974

16 Murakami J. Volume formulas for a spherical tetrahedron. Proc Amer Math Soc, 2012, 140(9): 3289–3295

17 Rourke C, Sanderson B. Introduction to Piecewise-Linear Topology. Berlin: Springer, 1982

18 Spanier E H. Algebraic Topology. Berlin: Springer, 1966

19 Wall C T C. Arithmetic invariants of subdivision of complexes. Canad J Math, 1966, 18: 92–96

20 Wu H-H. Historical development of the Gauss-Bonnet Theorem. Sci China Ser A, 2008, 51(4): 777–784