ABSTRACT

This chapter focuses on the Grassmannian manifold associated to a bounded symmetric domain. It provides information on bounded symmetric domains, compact type symmetric manifolds, and quasi-inverses. The chapter shows how the compact Hermitian symmetric space may be generated from its non-compact dual. Infinite dimensional bounded symmetric domains exist- the unit ball of a Hubert space for example. However, infinite dimensional compact manifolds cannot exist so if the duality theory of finite dimensions is to extend, some loosening of the requirements on the dual symmetric manifold of a bounded symmetric domain will have to be made. There are several ways to construct a compact Hermitian space from its non-compact dual. The method developed by W. Kaup works for infinite dimensions. The chapter also provides information on one finite dimensional construction, to see how far this can be extended in infinite dimensions.