  In this paper we show that if one writes down the structure equations for the
evolution of a curve embedded in an (n)-dimensional Riemannian manifold with
constant curvature this leads to a symplectic, a Hamiltonian and an hereditary
operator. This gives us a natural connection between finite dimensional
geometry, infinite dimensional geometry and integrable systems. Moreover one
finds a Lax pair in (\orth{n+1}) with the vector modified Korteweg-De Vries
equation (vmKDV) \[ \vk{t}=
  \vk{xxx}+\fr32 ||\vk{}||^2 \vk{x} \] as integrability condition. We indicate
that other integrable vector evolution equations can be found by using a
different Ansatz on the form of the Lax pair. We obtain these results by using
the {\em natural} or {\em parallel} frame and we show how this can be gauged by
a generalized Hasimoto transformation to the (usual) {\em Fren{\^e}t} frame. If
one chooses the curvature to be zero, as is usual in the context of integrable
systems, then one loses information unless one works in the natural frame.
