Levi-Civita connections on quantum spheres

TL;DR

This paper develops and characterizes q-deformed Levi-Civita connections on quantum spheres, providing explicit formulas and conditions for metric compatibility and torsion freeness, extending to general Hopf algebra settings.

Contribution

It introduces q-deformed connections on quantum spheres, establishes their existence on projective modules, and derives explicit formulas for Levi-Civita connections, generalizing to Hopf algebras.

Findings

Explicit formulas for q-deformed Levi-Civita connections.

Conditions for metric compatibility and torsion freeness.

Existence of connections on projective modules over quantum spheres.

Abstract

We introduce $q$-deformed connections on the quantum 2-sphere and 3-sphere, satisfying a twisted Leibniz rule in analogy with $q$-deformed derivations. We show that such connections always exist on projective modules. Furthermore, a condition for metric compatibility is introduced, and an explicit formula is given, parametrizing all metric connections on a free module. On the quantum 3-sphere, a q-deformed torsion freeness condition is introduced and we derive explicit expressions for the Christoffel symbols of a Levi-Civita connection for a general class of metrics. We also give metric connections on a class of projective modules over the quantum 2-sphere. Finally, we outline a generalization to any Hopf algebra with a (left) covariant calculus and associated quantum tangent space.