  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.
