Riemannian Geometry of a Discretized Circle and Torus

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...

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Published inSymmetry, integrability and geometry, methods and applications
Main Authors Bochniak, Arkadiusz, Sitarz, Andrzej, Zalecki, Pawel
Format Journal Article
LanguageEnglish
Published 01.01.2020
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Summary: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∗-module structure on the space of the one-forms. We compute curvature and scalar curvature for these metrics and find their continuous limits.
ISSN:1815-0659
1815-0659
DOI:10.3842/SIGMA.2020.143