# Curvature as an integrable deformation

## TL;DR

This paper reviews how classical Hamiltonian systems can be generalized from Euclidean space to curved spaces like the sphere and hyperbolic plane by using constant Gaussian curvature as a deformation parameter, preserving integrability.

## Contribution

It introduces a unified geometric framework for extending integrable systems to curved spaces and demonstrates superintegrability of specific Hamiltonians like anisotropic oscillators and Hénon-Heiles systems.

## Key findings

- Construction of integrable Hamiltonians on curved spaces
- Superintegrability of anisotropic oscillator with commensurate frequencies
- Extension of Hénon-Heiles system to spherical and hyperbolic geometries

## Abstract

The generalization of (super)integrable Euclidean classical Hamiltonian systems to the two-dimensional sphere and the hyperbolic space by preserving their (super)integrability properties is reviewed. The constant Gaussian curvature of the underlying spaces is introduced as an explicit deformation parameter, thus allowing the construction of new integrable Hamiltonians in a unified geometric setting in which the Euclidean systems are obtained in the vanishing curvature limit. In particular, the constant curvature analogue of the generic anisotropic oscillator Hamiltonian is presented, and its superintegrability for commensurate frequencies is shown. As a second example, an integrable version of the H\'enon-Heiles system on the sphere and the hyperbolic plane is introduced. Projective Beltrami coordinates are shown to be helpful in this construction, and further applications of this approach are sketched