On the Gauss-Chern-Bonnet theorem for the noncommutative 4-sphere

TL;DR

This paper develops a differential calculus on the noncommutative 4-sphere, establishing a Gauss-Chern-Bonnet type theorem by constructing metrics, connections, and localizations within this noncommutative geometric framework.

Contribution

It introduces a differential calculus and formulates a Gauss-Chern-Bonnet theorem for the noncommutative 4-sphere, extending classical geometric results to noncommutative geometry.

Findings

Existence of unique torsion-free connections for conformal metrics

Localization of the vector field module enables the theorem formulation

Formulation of a Gauss-Chern-Bonnet type theorem in noncommutative setting

Abstract

We construct a differential calculus over the noncommutative 4-sphere in the framework of pseudo-Riemannian calculi, and show that for every metric in a conformal class of perturbations of the round metric, there exists a unique metric and torsion-free connection. Furthermore, we find a localization of the projective module corresponding to the space of vector fields, which allows us to formulate a Gauss-Chern-Bonnet type theorem for the noncommutative 4-sphere.