  In this paper, multi-component generalizations to the Camassa-Holm equation,
the modified Camassa-Holm equation with cubic nonlinearity are introduced.
Geometric formulations to the dual version of the Schr\"odinger equation, the
complex Camassa-Holm equation and the multi-component modified Camassa-Holm
equation are provided. It is shown that these equations arise from
non-streching invariant curve flows respectively in the three-dimensional
Euclidean geometry, the two-dimensional M\"obius sphere and $n$-dimensional
sphere ${\mathbb S}^n(1)$. Integrability to these systems is also studied.
