The submanifold geometries associated to Grassmannian systems

Abstract

There is a hierarchy of commuting soliton equations associated to each symmetric space U/K. When U/K has rank n, the first n flows in the hierarchy give rise to a natural first order non-linear system of partial diffferential equations in n variables, the so called U/K-system. Let G_{m,n} denote the Grassmannian of n-dimensional linear subspaces in R^{m+n}, and G_{m,n}^1 the Grassmannian of space like m-dimensional linear subspaces in the Lorentzian space R^{m+n,1}. In this paper, we use techniques from soliton theory to study submanifolds in space forms whose Gauss-Codazzi equations are gauge equivalent to the G_{m,n}-system or the G_{m,n}^1-system. These include submanifolds with constant sectional curvatures, isothermic surfaces, and submanifolds admitting principal curvature coordinates. The dressing actions of simple elements on the space of solutions of the G_{m,n} and G_{m,n}^1 systems correspond to B\"acklund, Darboux and Ribaucour transformations for submanifolds.