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).

Keywords

Curvature / dihedral angle / Euclidean simplex / triangulation / Euler characteristic / Euler manifold / combinatorial manifold / pseudo manifold

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