  In this paper, we provide the geometric formulation to the two-component
Camassa-Holm equation (2-mCHE). We also study the relation between the 2-mCHE
and the M-CV equation. We have shown that these equations arise from the
invariant space curve flows in three-dimensional Euclidean geometry. Using this
approach we have established the geometrical equivalence between the 2-mCHE and
the M-CV equation. The gauge equivalence between these equations is also
considered.
