  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
odd-dimensional simplicial complex K (of dimension at least 3), such that the
Euler characteristic of the link of each i-simplex equals 2, where i is a
non-negative even integer that is less than n. As a corollary, an elementary
proof is given that any such simplicial complex has Euler characteristic zero.
