Piecewise linear manifolds: Einstein metrics and Ricci flows

TL;DR

This paper extends Riemannian geometry concepts to piecewise linear spaces, introducing an Einstein vector field, defining Ricci flows, and analyzing conditions for convergence to Einstein metrics.

Contribution

It proposes a novel analogue of the Ricci tensor for piecewise linear spaces and develops Ricci flow methods to find Einstein metrics in this setting.

Findings

Defined piecewise linear Einstein metrics

Formulated criteria for Ricci flows to approach Einstein metrics

Calculated second variations of total scalar curvature

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) Ricci 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.