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In this paper, we study two-component versions of the periodic Hunter-Saxtonequation and its $\mu$-variant. Considering both equations as a geodesic flowon the semidirect product of the circle diffeomorphism group $\Diff(\S)$ with aspace of scalar functions on $\S$ we show that both equations are locallywell-posed. The main result of the paper is that the sectional curvatureassociated with the 2HS is constant and positive and that 2$\mu$HS allows for alarge subspace of positive sectional curvature. The issues of this paper arerelated 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].

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