  The Miura transformation plays a crucial role in the study of integrable
systems. There have been various extensions of the Miura transformation, which
have been used to relate different kinds of integrable equations and to
classify the bi-Hamiltonian structures. In this paper, we are mainly concerned
with the geometric aspects of the Miura transformation. The generalized Miura
transformations from the mKdV-type hierarchies to the KdV-type hierarchies are
constructed under both algebraic and geometric settings. It is shown that the
Miura transformations not only relate integrable curve flows in different
geometries but also induce the transition between different moving frames.
Other geometric formulations are also investigated.
