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

Orbits of $SU(\ell, m)$ in a Grassmannian manifold have homogeneous CR structures. In this paper, we study germs of smooth CR mappings sending a closed orbit of $SU(\ell,m)$ into a closed orbit of $SU(\ell',m')$ 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 into another Grassmannian with respect to the standard metric. As an application, we give a sufficient condition for a smooth CR mapping sending a closed orbit of $SU(\ell,m)$ into a closed orbit of $SU(\ell',m')$ in Grassmannian manifolds to extend as a totally geodesic embedding of the Grassmannian into another Grassmannian.