The submanifold geometries associated to Grassmannian systems (Q2784249)

The Gauss and Codazzi equation of the surfaces with special geometric properties (called ``soliton'' equations) admit ``Lax pairs'', i.e. they can be written as the condition for a family of connections to be flat. The existence of a Lax pair is one of the characteristic properties of soliton equations and it often gives rise to the dressing action [\textit{A. I. Bobenko}, Aspects Math. E23, 83-127 (1994; Zbl 0841.53003); \textit{C.-L. Terng} and \textit{K. Uhlenbeck}, Notices Am. Math. Soc. 47, No. 1, 17-25 (2000; Zbl 0987.37072)].NEWLINENEWLINENEWLINEThe main goal of this monograph is to study submanifold geometries associated for\break \(O(m+n)/ O(m)\times O(n)\) and \(O(m+n,1)/ O(m)\times O(n,1)\)-systems, which include submanifolds with constant sectional curvature in space forms, isothermic surfaces and submanifolds admitting special principal curvature coordinates.NEWLINENEWLINENEWLINEA systematic approach for associating submanifold geometries to given soliton equations is developed. The material is organized as follows: After some general facts about \(U/K\)-systems, some submanifolds associated to \(G_{m,n}\)- and \(G^1_{m,n}\)-systems are described and relations between isothermic surfaces and \(G^1_{m,1}\)-systems are studied. The dressing action of a rational map with two simple poles on solutions of the \(G_{m,n}\)- and \(G_{m,n}^1\)-systems are written down explicitly and the corresponding geometric transformations are given.NEWLINENEWLINENEWLINEThe dressing action of a rational map with two poles on the space of solutions of the \(G^1_{n,1}\)-system gives rise to Darboux transformations [Burstall-preprint, math-dg/0003096]. A relation between the dressing action of loops with one simple pole and Bäcklund transformations is given, a permutability formula for dressing actions and finite type solutions of the \(U/K\)-system are given at the end.