Levi-Civita's Theorem for Noncommutative Tori
We show how to define Riemannian metrics and connections on a noncommutative torus in such a way that an analogue of Levi-Civita's theorem on the existence and uniqueness of a Riemannian connection holds. The major novelty is that we need to use two different notions of noncommutative vector fi...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2013 |
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2013
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/149363 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Levi-Civita's Theorem for Noncommutative Tori / L. Rosenberg // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 11 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862721715660914688 |
|---|---|
| author | Rosenberg, J. |
| author_facet | Rosenberg, J. |
| citation_txt | Levi-Civita's Theorem for Noncommutative Tori / L. Rosenberg // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 11 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We show how to define Riemannian metrics and connections on a noncommutative torus in such a way that an analogue of Levi-Civita's theorem on the existence and uniqueness of a Riemannian connection holds. The major novelty is that we need to use two different notions of noncommutative vector field. Levi-Civita's theorem makes it possible to define Riemannian curvature using the usual formulas.
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| first_indexed | 2025-12-07T18:32:48Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-149363 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T18:32:48Z |
| publishDate | 2013 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Rosenberg, J. 2019-02-21T07:21:13Z 2019-02-21T07:21:13Z 2013 Levi-Civita's Theorem for Noncommutative Tori / L. Rosenberg // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 11 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 46L87; 58B34; 46L08; 46L08 DOI: http://dx.doi.org/10.3842/SIGMA.2013.071 https://nasplib.isofts.kiev.ua/handle/123456789/149363 We show how to define Riemannian metrics and connections on a noncommutative torus in such a way that an analogue of Levi-Civita's theorem on the existence and uniqueness of a Riemannian connection holds. The major novelty is that we need to use two different notions of noncommutative vector field. Levi-Civita's theorem makes it possible to define Riemannian curvature using the usual formulas. This paper is a contribution to the Special Issue on Noncommutative Geometry and Quantum Groups in
 honor of Marc A. Rief fel. The full collection is available at http://www.emis.de/journals/SIGMA/Rieffel.html.
 This research was supported by NSF grant DMS-1206159. The author thanks the referees and
 the participants at the Fields Institute Focus Program for several interesting comments and
 discussions. I would like to thank Joakim Arnlind for pointing out a mistake in the original
 formulation of Proposition 3.4 en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Levi-Civita's Theorem for Noncommutative Tori Article published earlier |
| spellingShingle | Levi-Civita's Theorem for Noncommutative Tori Rosenberg, J. |
| title | Levi-Civita's Theorem for Noncommutative Tori |
| title_full | Levi-Civita's Theorem for Noncommutative Tori |
| title_fullStr | Levi-Civita's Theorem for Noncommutative Tori |
| title_full_unstemmed | Levi-Civita's Theorem for Noncommutative Tori |
| title_short | Levi-Civita's Theorem for Noncommutative Tori |
| title_sort | levi-civita's theorem for noncommutative tori |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/149363 |
| work_keys_str_mv | AT rosenbergj levicivitastheoremfornoncommutativetori |