Constant Scalar Curvature Metrics on Boundary Complexes of Cyclic Polytopes

In this paper we give examples of constant scalar curvature metrics on piecewise-flat triangulated 3-manifolds. These types of metrics are possible candidates for “best” metrics on triangulated 3-manifolds. In the pentachoron, the triangulation formed by the simplicial boundary of the 4-simplex, we find that its stucture is completely deterimed with a vertex transitive metric. Further this metric is a constant scalar curvature metric. Looking at a type of triangulated 3-manifolds, known as boundaries complexes of cyclic polytopes in 4-dimensions, with a metric called a cyclic length metric, we find this entire class of metrics on these manifolds are constant scalar curvature metrics. 1. Notions of Triangulations and Constant Scalar Curvature It is a classical problem in smooth Reimannian geometry to find constant scalar curvature metric on smooth manifolds. Here we will conisder the same problem in the discrete case. The piecewise-flat triangulated 3-manifold is a discrete version of the compact smooth 3-manifold. These manifolds are formed from 3-dimensional Euclidean tetrahedra, “glued” together in R, with various specifications, so that the manifold is without boundary. It is on these manifolds that we will try to find constant scalar curvature metrics. We derive many of these discrete notions of curvature and metrics from [CGY10] and [Gli09]. Since curvature is a geometric concept, before studying curvature one must know the most basic geometric concept of length on these simpilicies. This geometric assignment of lengths is known as a metric on these manifolds. Definition 1.1. Let (M, T) be a triangulation for a 3-manifold and let CM represent the Cayley-Menger determinent. A metric ` of triangulation (M, T) is a complete set of edge lengths for the triangulation such that for each tetrahedron, t∈T, CM(t) > 0. Date: July 30, 2010.