Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 071, 9 pages      arXiv:1307.3775
Contribution to the Special Issue on Noncommutative Geometry and Quantum Groups in honor of Marc A. Rieffel

Levi-Civita's Theorem for Noncommutative Tori

Jonathan Rosenberg
Department of Mathematics, University of Maryland, College Park, MD 20742, USA

Received July 26, 2013, in final form November 19, 2013; Published online November 21, 2013

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.

Key words: noncommutative torus; noncommutative vector field; Riemannian metric; Levi-Civita connection; Riemannian curvature; Gauss-Bonnet theorem.

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