Fat Triangulations and Differential Geometry

TL;DR

This paper explores the implications of fat triangulations in differential geometry, linking them to curvature convergence, quasiconformal mappings, and metric approaches to Regge calculus, extending previous results to broader manifold classes.

Contribution

It advances the understanding of fat triangulations' geometric consequences, extends their applicability to almost Riemannian manifolds, and connects them to metric curvature and Regge calculus.

Findings

Established convergence of Lipschitz-Killing curvatures in piecewise-flat approximations.

Extended triangulation results to almost Riemannian manifolds.

Connected fatness of triangulations to metric curvature and excess.

Abstract

We study the differential geometric consequences of our previous result on the existence of fat triangulations, in conjunction with a result of Cheeger, M\"{u}ller and Schrader, regarding the convergence of Lipschitz-Killing curvatures of piecewise-flat approximations of smooth Riemannian manifolds. A further application to the existence of quasiconformal mappings between manifolds, as well as an extension of the triangulation result to the case of almost Riemannian manifolds, are also given. In addition, the notion of fatness of triangulations and its relation to metric curvature and to excess is explored. Moreover, applications of the main results, and in particular a purely metric approach to Regge calculus, are also investigated.