The curvature of semidirect product groups associated with two-component Hunter-Saxton systems

TL;DR

This paper investigates the geometric properties of two-component Hunter-Saxton equations, showing they are well-posed as geodesic flows and analyzing their sectional curvature, revealing constant positive curvature for 2HS and large positive curvature subspaces for 2μHS.

Contribution

It demonstrates the local well-posedness of two-component Hunter-Saxton equations as geodesic flows and characterizes their sectional curvature properties, extending geometric understanding of these systems.

Findings

Sectional curvature of 2HS is constant and positive.

2μHS admits a large subspace with positive sectional curvature.

Both equations are locally well-posed as geodesic flows.

Abstract

In this paper, we study two-component versions of the periodic Hunter-Saxton equation and its $\mu$-variant. Considering both equations as a geodesic flow on the semidirect product of the circle diffeomorphism group $\Diff(\S)$ with a space of scalar functions on $\S$ we show that both equations are locally well-posed. The main result of the paper is that the sectional curvature associated with the 2HS is constant and positive and that 2$\mu$HS allows for a large subspace of positive sectional curvature. The issues of this paper are related to some of the results for 2CH and 2DP presented in [J. Escher, M. Kohlmann, and J. Lenells, J. Geom. Phys. 61 (2011), 436-452].