Holomorphic maps between closed SU(l, m)-orbits in Grassmannian

In this paper, we study germs of smooth CR mappings sending a closed orbit of SU (l,m) into a closed orbit of SU (l',m') in Grassmannian manifolds. We show that if the signature difference of the Levi forms of two orbits is not too large, then the mapping can be factored into a simple form and one of the factors extends to a totally geodesic embedding of the ambient Grassmannian with respect to the standard metric. As an application, we give a sufficient condition for a proper holomorphic mapping between type I bounded symmeric domains to be the product of trivial embedding and a holomorphic mapping into a subdomain.11Nsciescopu