Riemannian curvature of the noncommutative 3-sphere

TL;DR

This paper extends classical Riemannian geometry concepts to noncommutative spaces by developing a pseudo-Riemannian calculus, proving an analogue of Levi-Civita's theorem, and explicitly analyzing the noncommutative 3-sphere's curvature.

Contribution

It introduces a framework for noncommutative pseudo-Riemannian calculus, establishing existence and uniqueness of Levi-Civita connections and computing curvature for the noncommutative 3-sphere.

Findings

Existence of a unique torsion-free, metric connection in the noncommutative setting

Curvature operator retains classical symmetry properties

Explicit calculation of scalar curvature for the noncommutative 3-sphere

Abstract

In order to investigate to what extent the calculus of classical (pseudo-)Riemannian manifolds can be extended to a noncommutative setting, we introduce pseudo-Riemannian calculi of modules over noncommutative algebras. In this framework, it is possible to prove an analogue of Levi-Civita's theorem, stating that there exists at most one torsion-free and metric connection for a given (metric) module, satisfying the requirements of a real metric calculus. Furthermore, the corresponding curvature operator has the same symmetry properties as the classical Riemannian curvature. As our main motivating example, we consider a pseudo-Riemannian calculus over the noncommutative 3-sphere and explicitly determine the torsion-free and metric connection, as well as the curvature operator together with its scalar curvature.