Integrable systems in $n$-dimensional Riemannian geometry

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 a 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 $\mathfrak o_{n+1}$ with the vector modified Korteweg–De Vries equation (vmKDV)
$$ u_t=u_{xxx}+\frac{3}{2}\|u\|^2 u_x $$
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 natural or parallel frame and we show how this can be gauged by a generalized Hasimoto transformation to the (usual) Frenêt 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.