Existence and uniqueness of the Levi-Civita connection on noncommutative differential forms
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909457311596544 |
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| author | Mesland, Bram Rennie, Adam |
| author_facet | Mesland, Bram Rennie, Adam |
| contents | We combine Hilbert module and algebraic techniques to give necessary and sufficient conditions for the existence of an Hermitian torsion-free connection on the bimodule of differential one-forms of a first order differential calculus. In the presence of the extra structure of a bimodule connection, we give sufficient conditions for uniqueness. We prove that any $θ$-deformation of a compact Riemannian manifold admits a unique Hermitian torsion-free bimodule connection and provide an explicit construction of it. Specialising to classical Riemannian manifolds yields a novel construction of the Levi-Civita connection on the cotangent bundle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_13735 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Existence and uniqueness of the Levi-Civita connection on noncommutative differential forms Mesland, Bram Rennie, Adam Quantum Algebra Mathematical Physics Differential Geometry Operator Algebras 58B34 We combine Hilbert module and algebraic techniques to give necessary and sufficient conditions for the existence of an Hermitian torsion-free connection on the bimodule of differential one-forms of a first order differential calculus. In the presence of the extra structure of a bimodule connection, we give sufficient conditions for uniqueness. We prove that any $θ$-deformation of a compact Riemannian manifold admits a unique Hermitian torsion-free bimodule connection and provide an explicit construction of it. Specialising to classical Riemannian manifolds yields a novel construction of the Levi-Civita connection on the cotangent bundle. |
| title | Existence and uniqueness of the Levi-Civita connection on noncommutative differential forms |
| topic | Quantum Algebra Mathematical Physics Differential Geometry Operator Algebras 58B34 |
| url | https://arxiv.org/abs/2403.13735 |