Approximating Riemannian manifolds by polyhedra

TL;DR

This paper investigates how to approximate Riemannian manifolds with polyhedra, providing a proof of Regge's theorem that relates polyhedral curvature to the scalar curvature integral, advancing geometric approximation methods.

Contribution

It offers a proof of Regge's theorem within the Riemannian framework, connecting polyhedral curvature to scalar curvature approximation.

Findings

Proof of Regge's theorem close to original intuition

Polyhedral approximation of scalar curvature

Bridging Riemannian geometry and polyhedral models

Abstract

This is a study on approximating a Riemannian manifold by polyhedra. Our scope is understanding Tullio Regge's [52] article in the restricted Riemannian frame. We give a proof of the Regge theorem along lines close to its original intuition: one can approximate a compact domain of a Riemannian manifold by polyhedra in such a way that the integral of the scalar curvature is approximated by a corresponding polyhedral curvature.