On Noncommutative Levi-Civita Connections

TL;DR

This paper investigates Levi-Civita connections on noncommutative tori, highlighting their non-uniqueness, curvature properties, and the validity of the Gauss-Bonnet theorem under certain deformations.

Contribution

It provides new insights into the structure of Levi-Civita connections in noncommutative geometry, especially regarding their non-uniqueness and curvature characteristics.

Findings

Non-uniqueness of torsion-free metric-compatible connections without fixed inner derivation operator.

Nontrivial curvature form of inner *-derivations.

Gauss-Bonnet theorem holds for specific non-conformal deformations of noncommutative tori.

Abstract

We make some observations about Rosenberg's Levi-Civita connections on noncommutative tori, noting the non-uniqueness of torsion-free metric-compatible connections without prescribed connection operator for the inner *-derivations, the nontrivial curvature form of the inner *-derivations, and the validity of the Gauss-Bonnet theorem for two classes of non-conformal deformations of the flat metric on the noncommutative two-tori, including the case of non-commuting scalings along the principal directions of a two-torus.