Parallel submanifolds of the real 2-Grassmannian

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