Levi-Civita connections for a class of spectral triples

TL;DR

This paper introduces a new definition of Levi-Civita connection for noncommutative pseudo-Riemannian metrics within spectral triples, proving existence and uniqueness for various noncommutative geometries including fuzzy spheres and quantum manifolds.

Contribution

It provides a novel, consistent framework for Levi-Civita connections in noncommutative geometry, addressing previous limitations and extending to new classes of noncommutative manifolds.

Findings

Existence and uniqueness of Levi-Civita connections are established for several noncommutative geometries.

The new definition resolves issues in prior approaches, such as the quantum Heisenberg manifold.

In matrix geometries, the connection coincides with previously known torsion-less unitary connections.

Abstract

We give a new definition of Levi-Civita connection for a noncommutative pseudo-Riemannian metric on a noncommutative manifold given by a spectral triple. We prove the existence-uniqueness result for a class of modules of one forms over a large class of noncommutative manifolds, including the matrix geometry of the fuzzy 3-sphere, the quantum Heisenberg manifolds and Connes-Landi deformations of spectral triples on the Connes-Dubois Violette-Rieffel-deformation of a compact manifold equipped with a free toral action. It is interesting to note that in the example of the quantum Heisenberg manifold, the definition of metric compatibility given in the paper by Frolich et al failed to ensure the existence of a unique Levi-Civita connection. In the case of the matrix geometry, the Levi-Civita connection that we get coincides with the unique real torsion-less unitary connection obtained by Frolich et al.