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 in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2020 |
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2020
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/211076 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Riemannian Geometry of a Discretized Circle and Torus. Arkadiusz Bochniak, Andrzej Sitarz and Paweł Zalecki. SIGMA 16 (2020), 143, 28 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862569677343948800 |
|---|---|
| author | Bochniak, Arkadiusz Sitarz, Andrzej Zalecki, Paweł |
| author_facet | Bochniak, Arkadiusz Sitarz, Andrzej Zalecki, Paweł |
| citation_txt | Riemannian Geometry of a Discretized Circle and Torus. Arkadiusz Bochniak, Andrzej Sitarz and Paweł Zalecki. SIGMA 16 (2020), 143, 28 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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 *-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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| first_indexed | 2026-03-13T11:43:16Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-211076 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-13T11:43:16Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Bochniak, Arkadiusz Sitarz, Andrzej Zalecki, Paweł 2025-12-23T13:11:03Z 2020 Riemannian Geometry of a Discretized Circle and Torus. Arkadiusz Bochniak, Andrzej Sitarz and Paweł Zalecki. SIGMA 16 (2020), 143, 28 pages 1815-0659 2020 Mathematics Subject Classification: 46L87; 83C65 arXiv:2007.01241 https://nasplib.isofts.kiev.ua/handle/123456789/211076 https://doi.org/10.3842/SIGMA.2020.143 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 *-module structure on the space of the one-forms. We compute curvature and scalar curvature for these metrics and find their continuous limits. We would like to thank the referees for their valuable comments on the content of our manuscript and their suggestions for improving the document. PZ was supported by the Faculty of Physics, Astronomy and Applied Computer Science of the Jagiellonian University under the DSC scheme: U1U/P05/NW/03.27. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Riemannian Geometry of a Discretized Circle and Torus Article published earlier |
| spellingShingle | Riemannian Geometry of a Discretized Circle and Torus Bochniak, Arkadiusz Sitarz, Andrzej Zalecki, Paweł |
| title | Riemannian Geometry of a Discretized Circle and Torus |
| title_full | Riemannian Geometry of a Discretized Circle and Torus |
| title_fullStr | Riemannian Geometry of a Discretized Circle and Torus |
| title_full_unstemmed | Riemannian Geometry of a Discretized Circle and Torus |
| title_short | Riemannian Geometry of a Discretized Circle and Torus |
| title_sort | riemannian geometry of a discretized circle and torus |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211076 |
| work_keys_str_mv | AT bochniakarkadiusz riemanniangeometryofadiscretizedcircleandtorus AT sitarzandrzej riemanniangeometryofadiscretizedcircleandtorus AT zaleckipaweł riemanniangeometryofadiscretizedcircleandtorus |