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We prove a Gauss-Bonnet formula X(G) = sum_x K(x), where K(x)=(-1)^dim(x) (1-X(S(x))) is a curvature of a vertex x with unit sphere S(x) in the Barycentric refinement G1 of a simplicial complex G. K(x) is dual to (-1)^dim(x) for which Gauss-Bonnet is the definition of Euler characteristic X. Because the connection Laplacian L'=1+A' of G is unimodular, where A' is the adjacency matrix of of the connection graph G', the Green function values g(x,y) = (1+A')^-1_xy are integers and 1-X(S(x))=g(x,x). Gauss-Bonnet for K^+ reads therefore as str(g)=X(G), where str is the super trace. As g is a time-discrete heat kernel, this is a cousin to McKean-Singer str(exp(-Lt)) = X(G) for the Hodge Laplacian L=dd^* +d^*d which lives on the same Hilbert space than L'. Both formulas hold for an arbitrary finite abstract simplicial complex G. Writing V_x(y)= g(x,y) for the Newtonian potential of the connection Laplacian, we prove sum_y V_x(y) = K(x), so that by the new Gauss-Bonnet formula, the Euler characteristic of G agrees with the total potential theoretic energy sum_x,y g(x,y)=X(G) of G. The curvature K now relates to the probability measure p minimizing the internal energy U(p)=sum_x,y g(x,y) p(

Author

Knill, Oliver

Published

2017

Language

EN