Existence and uniqueness of the Levi-Civita connection on noncommutative differential forms

TL;DR

This paper establishes conditions for the existence and uniqueness of the Levi-Civita connection in noncommutative geometry, providing explicit constructions and extending classical results to quantum deformations.

Contribution

It offers necessary and sufficient conditions for Hermitian torsion-free connections on noncommutative differential forms, including explicit construction for $ heta$-deformations.

Findings

Existence of Hermitian torsion-free connections under certain conditions

Uniqueness of the Levi-Civita connection on $ heta$-deformations

Explicit construction of the Levi-Civita connection in noncommutative setting

Abstract

We combine Hilbert module and algebraic techniques to give necessary and sufficient conditions for the existence of an Hermitian torsion-free connection on the bimodule of differential one-forms of a first order differential calculus. In the presence of the extra structure of a bimodule connection, we give sufficient conditions for uniqueness. We prove that any $\theta$-deformation of a compact Riemannian manifold admits a unique Hermitian torsion-free bimodule connection and provide an explicit construction of it. Specialising to classical Riemannian manifolds yields a novel construction of the Levi-Civita connection on the cotangent bundle.