The submanifold geometries associated to Grassmannian systems

Original · EN

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 Gₘ,ₙ denote the Grassmannian of n-dimensional linear subspaces in Rᵐ⁺ⁿ, and Gₘ,ₙ¹ the Grassmannian of space like m-dimensional linear subspaces in the Lorentzian space Rᵐ⁺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 Gₘ,ₙ-system or the Gₘ,ₙ¹-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 Gₘ,ₙ and Gₘ,ₙ¹ systems correspond to Bäcklund, Darboux and Ribaucour transformations for submanifolds.