Curvature computations for a two-component Camassa-Holm equation with vorticity

TL;DR

This paper explores the geometric structure of a two-component Camassa-Holm system with vorticity, focusing on the computation of sectional curvature and its implications for the system's dynamics.

Contribution

It provides a detailed geometric formalism and computes the sectional curvature of the configuration manifold for the 2CH system with vorticity, revealing directions with positive curvature.

Findings

Sectional curvature $K$ can be strictly positive.

Existence of directions with curvature bounded away from zero.

Advances understanding of the geometric properties of the 2CH system.

Abstract

In the present paper, a two-component Camassa-Holm (2CH) system with vorticity is studied as a geodesic flow on a suitable Lie group. The paper aims at presenting various details of the geometric formalism and a major result is the computation of the sectional curvature $K$ of the underlying configuration manifold. As a further result, we show that there are directions for which $K$ is strictly positive and bounded away from zero.