  We use geometric methods to study two natural two-component generalizations
of the periodic Camassa-Holm and Degasperis-Procesi equations. We show that
these generalizations can be regarded as geodesic equations on the semidirect
product of the diffeomorphism group of the circle $\Diff(S^1)$ with some space
of sufficiently smooth functions on the circle. Our goals are to understand the
geometric properties of these two-component systems and to prove local
well-posedness in various function spaces. Furthermore, we perform some
explicit curvature calculations for the two-component Camassa-Holm equation,
giving explicit examples of large subspaces of positive curvature.
