A new look at Levi-Civita connection in noncommutative geometry

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.

The original article has been split into two parts. The other part is arXiv 2101.07221. This version includes new results on star-compatibility of Levi-Civita connections. The main result holds for strongly σ-compatible metrics. This version is to be viewed as a follow up to references 9, 10, 11 and 13