On the existence of noncommutative Levi-Civita connections in derivation based calculi

TL;DR

This paper investigates the conditions for the existence of Levi-Civita connections in noncommutative geometry, providing algebraic criteria and explicit examples, extending classical metric symmetry concepts.

Contribution

It establishes necessary and sufficient algebraic conditions for Levi-Civita connections in derivation-based noncommutative calculi, including symmetry conditions and explicit computations.

Findings

Derived a criterion based on the image of an operator from the hermitian form.

Identified a symmetry condition extending classical metric symmetry.

Explicitly computed examples for noncommutative 3-tori.

Abstract

We study the existence of Levi-Civita connections, i.e torsion free connections compatible with a hermitian form, in the setting of derivation based noncommutative differential calculi over $\ast$-algebras. We prove a necessary and sufficient condition for the existence of Levi-Civita connections in terms of the image of an operator derived from the hermitian form. Moreover, we identify a necessary symmetry condition on the hermitian form that extends the classical notion of metric symmetry in Riemannian geometry. The theory is illustrated with explicit computations for free modules of rank three, including noncommutative 3-tori. We note that our approach is algebraic and does not rely on analytic tools such as $C^\ast$-algebra norms.