Second-order superintegrable systems and Weylian geometry

TL;DR

This paper explores the geometric structure of second-order superintegrable systems, revealing their connection to Weyl geometry and extending conformal superintegrability concepts within this framework.

Contribution

It uncovers the Weyl geometric nature of superintegrable systems and extends conformal superintegrability to Weyl structures, providing new geometric insights.

Findings

Revealed Weyl structure underlying superintegrable systems

Extended conformal superintegrability to Weyl geometry

Connected superintegrable systems to semi-Weyl structures

Abstract

Abundant second-order maximally conformally superintegrable Hamiltonian systems are re-examined, revealing their underlying natural Weyl structure and offering a clearer geometric context for the study of St\"ackel transformations (also known as coupling constant metamorphosis). This also allows us to naturally extend the concept of conformal superintegrability from the realm of conformal geometries to that of Weyl structures. It enables us to interpret superintegrable systems of the above type as semi-Weyl structures, a concept related to statistical manifolds and affine hypersurface theory.