Riemannian Geometry of a Discretized Circle and Torus

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.

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Cited by in corpus (3)

• Fuzzy and discrete black hole models
• On multimatrix models motivated by random noncommutative geometry II: A Yang-Mills-Higgs matrix model
• Quantum Riemannian geometry of the discrete interval and q-deformation