Grassmann Geometries in Infinite Dimensional Homogeneous Spaces and an Application to Reflective Submanifolds

Abstract

Let U be a real form of a complex semisimple Lie group, and (�, �) a pair of commuting involutions on U. This data corresponds to a reflective submanifold of a symmetric space, U/K. We define an associated integrable system, and describe how to produce solutions from curved flats. This gives many new examples of submanifolds as integrable systems. The solutions are shown to correspond to various special submani- folds, depending on which homogeneous space, U/L, one projects to. We apply the construction to a question which generalizes, to the context of reflective submanifolds of arbitrary symmetric spaces, the problem of isometric immersions of space forms with negative extrinsic curva- ture and flat normal bundle. For this problem, we prove that the only cases where local solutions exist are the previously known cases of space forms, in addition to our new example of constant curvature Lagrangian immersions into complex projective and complex hyperbolic spaces. We also prove non-existence of global solutions in the compact case. For other reflective submanifolds, lower dimensional solutions exist, and can be described in terms of Grassmann geometries. We consider one example in detail, associated to the group G2, obtaining a special class of surfaces in S 6 .