Comparison of Levi-Civita connections in noncommutative geometry

TL;DR

This paper compares different constructions of Levi-Civita connections in noncommutative geometry, providing translations between them and extending existence results under certain assumptions.

Contribution

It offers a unified framework for various Levi-Civita connection constructions and extends their existence results in noncommutative geometry.

Findings

Provides direct translations between different Levi-Civita connection constructions.

Extends existence results for Hermitian torsion-free connections.

Clarifies conditions under which these constructions are equivalent.

Abstract

We compare the constructions of Levi-Civita connections for noncommutative algebras developed in arXiv:1505.07330, arXiv:1809.06721, arXiv:2403.13735. The assumptions in these various constructions differ, but when they are all defined, we provide direct translations between them. An essential assumption is that the (indefinite) Hermitian inner product on differential forms/vector fields provides an isomorphism with the module dual. By exploiting our translations and clarifying the simplifications that occur for centred bimodules, we extend the existence results for Hermitian torsion-free connections in arXiv:1505.07330, arXiv:1809.06721.