HOMOGENEOUS SUBMANIFOLDS OF HIGHER RANK AND PARALLEL MEAN-CURVATURE

Let M(n), n greater-than-or-equal-to 2, be an orbit of a representatio n of a compact Lie group which is irreducible and full as a submanifol d of the ambient space. We prove that if M admits a nontrivial (i.e., not a multiple of the position vector) locally defined parallel normal vector field, then M is (also) an orbit of the isotropy representatio n of a simple symmetric space. So, in particular, compact homogeneous irreducible submanifolds of the Eucildean space with parallel mean cur vature (not minimal in a sphere) are characterized (and classified). T he proof is geometric and related to the normal holonomy groups and th e theorem of Thorbergsson.