Levi-Civita's Theorem for Noncommutative Tori

Abstract

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.

Proposition 3.4 corrected

References in corpus (1)

• The Gauss-Bonnet Theorem for the noncommutative two torus

Cited by in corpus (27)

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• Laplace-Beltrami Operators on Noncommutative Tori
• Cartan structure equations and Levi-Civita connection in noncommutative geometry
• Noncommutative Ricci flow in a matrix geometry
• Pseudodifferential calculus on noncommutative tori, I. Oscillating integrals
• Scalar Curvature of a Levi-Civita Connection on Cuntz algebra with three generators
• Gauss-Bonnet for matrix conformally rescaled Dirac
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• Hypergeometric function and Modular Curvature II. Connes-Moscovici functional relation after Lesch's work
• Local invariants of non-commutative tori
• Connes Trace Theorem for Curved Noncommutative Tori. Application to Scalar Curvature
• Pseudodifferential calculus on noncommutative tori, II. Main properties
• Dixmier Trace Formulas and Negative Eigenvalues of Schroedinger Operators on Curved Noncommutative Tori
• Curvature and geometric modules of noncommutative spheres and tori
• On q-deformed Levi-Civita connections
• Heat equation and Poisson equation in matrix geometry
• Einstein-Hilbert Action and on the Gauss-Bonnet Theorem for Riemannian Noncommutative Tori
• L2 norm preserving flow in matrix geometry
• Projective real calculi over matrix algebras
• Metrized Quantum Vector Bundles over Quantum Tori Built from Riemannian Metrics and Rosenberg's Levi-Civita Connections
• Levi-Civita connections from toral actions
• Gauge theory models on $\kappa$-Minkowski space: Results and prospects