Abundant Superintegrable Systems and Hessian Structures

TL;DR

This paper demonstrates that many non-degenerate second-order superintegrable systems naturally induce Hessian structures with adapted coordinates, especially on constant curvature Riemannian manifolds, and provides explicit coordinate computations in low dimensions.

Contribution

It establishes a link between superintegrable systems and Hessian structures, introducing natural coordinates and explicit examples in 2D and 3D.

Findings

Superintegrable systems induce Hessian structures.

Explicit Hessian coordinates are computed for specific systems.

Applicable to systems on constant curvature manifolds.

Abstract

We show that a large class of non-degenerate second-order (maximally) superintegrable systems gives rise to Hessian structures, which admit natural (Hessian) coordinates adapted to the superintegrable system. In particular, abundant superintegrable systems on Riemannian manifolds of constant sectional curvature fall into this class. We explicitly compute the natural Hessian coordinates for examples of non-degenerate second-order superintegrable systems in dimensions two and three.