A new look at Levi-Civita connection in noncommutative geometry

TL;DR

This paper establishes the existence and uniqueness of Levi-Civita connections for a broad class of pseudo-Riemannian metrics in noncommutative geometry, extending previous results and discussing star-compatibility.

Contribution

It introduces a general framework for Levi-Civita connections in noncommutative settings, encompassing bilinear metrics and conformal deformations, with new existence and uniqueness proofs.

Findings

Proves existence and uniqueness of Levi-Civita connections for strongly sigma-compatible metrics.

Extends previous results to include conformal deformations of metrics.

Discusses star-compatibility conditions for Levi-Civita connections.

Abstract

We prove the existence and uniqueness of Levi-Civita connections for strongly sigma-compatible pseudo-Riemannian metrics on tame differential calculi. Such pseudo-Riemannian metrics properly contain the classes of bilinear metrics as well as their conformal deformations. This extends the previous results in references 9 and 10. Star-compatibility of Levi-Civita connections for bilinear pseudo-Riemannian metrics are also discussed.