Riemannian Geometry of a Discretized Circle and Torus

TL;DR

This paper explores the Riemannian geometry of discretized circles and tori within noncommutative geometry, classifying connections, analyzing curvature, and connecting discrete models to continuous limits.

Contribution

It provides a full classification of linear connections on finite cyclic groups and analyzes their geometric properties in the noncommutative setting.

Findings

Multiple classes of metric-compatible connections identified

Curvature and scalar curvature computed for these metrics

Continuous limits of discrete geometries derived

Abstract

We extend the results of Riemannian geometry over finite groups and provide a full classification of all linear connections for the minimal noncommutative differential calculus over a finite cyclic group. We solve the torsion-free and metric compatibility condition in general and show that there are several classes of solutions, out of which only special ones are compatible with a metric that gives a Hilbert $C^\ast$-module structure on the space of the one-forms. We compute curvature and scalar curvature for these metrics and find their continuous limits.