Abstract
For a finitely triangulated closed surface M 2, let αx be the sum of angles at a vertex x. By the well-known combinatorial version of the 2- dimensional Gauss-Bonnet Theorem, it holds Σx(2π - αx) = 2πχ(M 2), where χ denotes the Euler characteristic of M 2, αx 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 Στ (-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:
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Klaus, S. On combinatorial Gauss-Bonnet Theorem for general Euclidean simplicial complexes. Front. Math. China 11, 1345–1362 (2016). https://doi.org/10.1007/s11464-016-0575-2
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DOI: https://doi.org/10.1007/s11464-016-0575-2
Keywords
- Curvature
- dihedral angle
- Euclidean simplex
- triangulation
- Euler characteristic
- Euler manifold
- combinatorial manifold
- pseudo manifold