  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].
