Piecewise constant curvature manifolds are discrete analogues of Riemannian manifolds in which edge lengths play the role of the metric tensor. These triangulated manifolds are specified by two types of data: 1. Each edge in the triangulation is assigned a real-valued length. 2. For each simplex in the triangulation, there exists an isometric embedding of that simplex into one of three background geometries (Euclidean, hyperbolic, or spherical). In particular, this isometry respects the edge length data. By making the edge lengths functions of scalars, called conformal parameters, that are assigned to the vertices of the triangulation we obtain a conformal structure - that is, a parameterization of a discrete conformal class. We discuss how our definition of conformal structure places several existing notions of a discrete conformal class in a common framework. We then describe discrete analogues of scalar curvature for 2-and 3-manifolds and study how these curvatures depend on the conformal parameters. This leads us to some local rigidity theorems - we identify circumstances in which the mapping from conformal parameters to scalar curvatures is a local diffeomorphism. In three dimensions, we focus on the case of hyperbolic background geometry. We study a discrete analogue of the Einstein-Hilbert (or total scalar curvature) functional and investigate when this functional is locally convex.
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