Piecewise linear manifolds: Einstein metrics and Ricci flows

Abstract

This article provides an attempt to extend concepts from the theory of Riemannian manifolds to piecewise linear spaces. In particular we propose an analogue of the Ricci tensor, which we give the name of an Einstein vector field. On a given set of piecewise linear spaces we define and discuss (normalized) Einstein flows. Piecewise linear Einstein metrics are defined and examples are provided. Criteria for flows to approach Einstein metrics are formulated. Second variations of the total scalar curvature at a specific Einstein space are calculated.Date: April 25, 2018File: PLEEv2.tex.

A Note on Discrete Einstein Metrics

“…Remark 1. We have recently learned that R. Schrader got the same result (Theorem 3.1, [12]) as our Theorem 1.1 in this paper. R. Schrader [12] also provided some interesting analogues of Ricci tensor and Ricci flow in the theory of piecewise linear spaces.…”

Piecewise flat curvature and Ricci flow in three dimensions

“…While the simplicial complex gives the topological structure of a piecewise flat manifold S n , the set of edge-lengths completely fixes the geometry of each simplex, and therefore that of S n . More detail about the edge-lengths as the piecewise flat metric can be found in [23].…”

Reflection positivity in simplicial gravity

“…Further analogies between quantities in the p.l. and the continuum context will be provided below, and more are given in the first appendix of [34]. Given z, we denote by |σ k |(z) the volume of this euclidean k−simplex σ k (z), see also (2.5) below.…”

“…In [34] we introduced and discussed the vector field given as the gradient of the Regge curvature. So the components of the vector field ∇ z + R(K + , z + ) also define observables and expectations like…”

“…By the discussion in [34] the vector field ∇ z + R(K + , z + )) can only become singular, when z + tends to the boundary of M(K + ) in such a way, that the volume of one (or more) of the (n − 2)−simplexes tends to zero. Analogously the gradient of the volume can become singular, when the volume of one (or more) of the n−simplexes becomes small.…”