THE RIEMANNIAN GEOMETRY OF A DISCRETIZED CIRCLE AND TORUS

Since the inception of noncommutative geometry, the generalization of Riemannian geometry to the noncommutative setup was a challenge. In this talk, we propose techniques that allow us to provide a complete classification of all linear connections for the minimal noncommutative differential calculus over a finite cyclic group. We find general solutions for the torsion-free and metric-compatibility conditions, and show that there are several classes of such solutions containing special ones that are compatible with a metric that gives a Hilbert C*-module structure on the space of one-forms. We compute curvature and scalar curvature for these metrics, and find their continuous limits. Based on the joint paper with A. Bochniak and P. Zalecki: <SIGMA 16 (2020) 143>.