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.

Keywords

• noncommutative torus
• noncommutative vector field
• Riemannian metric
• Levi-Civita connection
• Riemannian curvature
• Gauss-Bonnet theorem