Integrable geodesic motion on 3D curved spaces from non-standard quantum deformations

TL;DR

This paper explores how non-standard quantum deformations of sl(2) Poisson coalgebras generate integrable and superintegrable geodesic motions on 3D curved spaces, linking curvature to deformation parameters.

Contribution

It introduces a method to derive integrable geodesic Hamiltonians on 3D manifolds with variable or constant curvature from quantum algebra deformations.

Findings

Deformation parameter z determines the curvature of the 3D space.

Constructed Hamiltonians exhibit integrability and superintegrability.

Method generalizes to higher-dimensional spaces.

Abstract

The link between 3D spaces with (in general, non-constant) curvature and quantum deformations is presented. It is shown how the non-standard deformation of a sl(2) Poisson coalgebra generates a family of integrable Hamiltonians that represent geodesic motions on 3D manifolds with a non-constant curvature that turns out to be a function of the deformation parameter z. A different Hamiltonian defined on the same deformed coalgebra is also shown to generate a maximally superintegrable geodesic motion on 3D Riemannian and (2+1)D relativistic spaces whose sectional curvatures are all constant and equal to z. This approach can be generalized to arbitrary dimension.