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 \(\rmG^+_2(\R^{n+2})\) which parameterizes the oriented 2-planes of the Euclidean space \(\R^{n+2}\)\,. Our main result states that every complete parallel submanifold of \(\rmG^+_2(\R^{n+2})\)\,, 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 \(\rmG^+_2(\R^{n+2})\)\,.