Piecewise Flat Curvature and Ricci Flow in Three Dimensions

TL;DR

This paper develops discrete scalar, sectional, and Ricci curvatures on simplicial triangulations of manifolds, demonstrating their convergence to smooth curvature values and defining a Ricci flow in the piecewise flat setting.

Contribution

It introduces a novel method for defining and computing discrete curvatures and Ricci flow on triangulated manifolds, bridging discrete and smooth geometric analysis.

Findings

Piecewise flat curvatures converge to smooth values.

Ricci flow defined via edge-length changes converges to smooth Ricci flow.

Method applicable to diverse manifold triangulations.

Abstract

Discrete forms of the scalar, sectional and Ricci curvatures are constructed on simplicial piecewise flat triangulations of smooth manifolds, depending directly on the simplicial structure and a choice of dual tessellation. This is done by integrating over volumes which include appropriate samplings of hinges for each type of curvature, with the integrals based on the parallel transport of vectors around hinges. Computations for triangulations of a diverse set of manifolds show these piecewise flat curvatures to converge to their smooth values. The Ricci curvature also gives a piecewise flat Ricci flow as a fractional rate of change of edge-lengths, again converging to the smooth Ricci flow for the manifolds tested.