2000/06/28 by Martina Brück, Brück, Martina, Xi Du +5
Mathematics · Physics and Astronomy · #35 #53 #Differential Geometry (math.DG) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #math.DG #msc:35 #msc:53 #nlin.SI
paper · pdf · doi:10.48550/arxiv.math/0006216
104 pages, 7 figures
arxiv created 2000/06/28 · arxiv updated 2009/11/30
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.