Parallel submanifolds of the real 2-Grassmannian

Abstract

A submanifold of a Riemannian symmetric space is called parallel if its second fundamental form is a parallel section of the appropriate tensor bundle. We classify parallel submanifolds of the Grassmannian G + (IR n+2 ) which parameterizes the oriented 2-planes of the Euclidean space IR n+2 . Our main result states that every complete parallel submanifold of G + (IR n+2 ), which is not a curve, is contained in some totally geodesic submanifold as a symmetric submanifold. This result holds also if the ambient space is the non-compact dual of G + (IR n+2 ).