Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 16 (2020), 143, 28 pages      arXiv:2007.01241      https://doi.org/10.3842/SIGMA.2020.143
Contribution to the Special Issue on Noncommutative Manifolds and their Symmetries in honour of Giovanni Landi

Riemannian Geometry of a Discretized Circle and Torus

Arkadiusz Bochniak, Andrzej Sitarz and Paweł Zalecki
Institute of Theoretical Physics, Jagiellonian University, prof. Stanisława Łojasiewicza 11, 30-348 Kraków, Poland

Received July 03, 2020, in final form December 15, 2020; Published online December 23, 2020

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.

Key words: noncommutative Riemannian geometry; linear connections; curvature.

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