Parallel submanifolds of the real 2-Grassmannian

Published 28 Jul 2011 in math.DG | (1107.5761v3)

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 $\rmG^{+_2(\R^{n+2})$\,.}