Levi-Civita's Theorem for Noncommutative Tori

TL;DR

This paper extends Levi-Civita's theorem to noncommutative tori by defining suitable Riemannian metrics and connections, using two notions of noncommutative vector fields, enabling curvature computation similar to classical geometry.

Contribution

It introduces a framework for Riemannian geometry on noncommutative tori, establishing existence and uniqueness of Levi-Civita connections in this setting.

Findings

Established Levi-Civita's theorem analogue for noncommutative tori

Defined Riemannian metrics and connections using two notions of vector fields

Enabled curvature calculations with classical formulas

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.