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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| Format: | Article |
| Language: | English |
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National Academy of Science of Ukraine
2013-11-01
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| Series: | Symmetry, Integrability and Geometry: Methods and Applications |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.3842/SIGMA.2013.071 |
| _version_ | 1818025080301551616 |
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| author | Jonathan Rosenberg |
| author_facet | Jonathan Rosenberg |
| author_sort | Jonathan Rosenberg |
| collection | DOAJ |
| 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. |
| first_indexed | 2024-12-10T04:10:25Z |
| format | Article |
| id | doaj.art-44c7b828270d446ea8f30a1447051fb0 |
| institution | Directory Open Access Journal |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2024-12-10T04:10:25Z |
| publishDate | 2013-11-01 |
| publisher | National Academy of Science of Ukraine |
| record_format | Article |
| series | Symmetry, Integrability and Geometry: Methods and Applications |
| spelling | doaj.art-44c7b828270d446ea8f30a1447051fb02022-12-22T02:02:45ZengNational Academy of Science of UkraineSymmetry, Integrability and Geometry: Methods and Applications1815-06592013-11-01907110.3842/SIGMA.2013.071Levi-Civita's Theorem for Noncommutative ToriJonathan RosenbergWe 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.http://dx.doi.org/10.3842/SIGMA.2013.071noncommutative torusnoncommutative vector fieldRiemannian metricLevi-Civita connectionRiemannian curvatureGauss-Bonnet theorem |
| spellingShingle | Jonathan Rosenberg Levi-Civita's Theorem for Noncommutative Tori Symmetry, Integrability and Geometry: Methods and Applications noncommutative torus noncommutative vector field Riemannian metric Levi-Civita connection Riemannian curvature Gauss-Bonnet theorem |
| 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 |
| topic | noncommutative torus noncommutative vector field Riemannian metric Levi-Civita connection Riemannian curvature Gauss-Bonnet theorem |
| url | http://dx.doi.org/10.3842/SIGMA.2013.071 |
| work_keys_str_mv | AT jonathanrosenberg levicivitastheoremfornoncommutativetori |