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

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

2010-10-12

arxiv.org/abs/1010.2363v2

J. Phys. A: Math. Theor. 44 (2011) 225203 (17pp)

Physics

Mathematical Physics

19 pages

Scientific paper

10.1088/1751-8113/44/22/225203

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].

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