The Angle Defect for Odd-Dimensional Simplicial Manifolds

In a 1967 paper, Banchoff stated that a certain type of polyhedral curvature, that applies to all finite polyhedra, was zero at all vertices of an odd-dimensional polyhedral manifold; one then obtains an elementary proof that odd-dimensional manifolds have zero Euler characteristic. In a previous paper, the author defined a different approach to curvature for arbitrary simplicial complexes, based upon a direct generalization of the angle defect. The generalized angle defect is not zero at the simplices of every odd-dimensional manifold. In this paper we use a sequence based upon the Bernoulli numbers to define a variant of the angle defect for finite simplicial complexes that still satisfies a Gauss-Bonnet type theorem, but is also zero at any simplex of an odddimensional simplicial complex K (of dimension at least 3), such that (link(i , K)) = 2 for all i-simplices i of K, where i is an even integer

• Type: Journal
• Year: 2006
• Where: DCG
• Authors: Ethan D. Bloch
• Keywords: Angle Defect, DCG 2006, Odd-dimensional Manifold, Simplicial Complexes