Submanifolds with homothetic Gauss map in codimension two

Let f : M-n -> Rn+p be an isometric immersion of an n-dimensional Riemannian manifold M-n into the (n + p)-dimensional Euclidean space. Its Gauss map phi: M-n -> G(n)(Rn+p) into the Grassmannian G(n)(Rn+p) is defined by assigning to every point of M-n its tangent space, considered as a vector subspace of Rn+p. The third fundamental form III of f is the pullback of the canonical Riemannian metric on G(p)(Rn+p) via f. In this article we derive a complete classification of all those f with codimension two for which the Gauss map phi is homothetic; i.e., III is a constant multiple of the Riemannian metric on M-n. We furthermore study and classify codimension two submanifolds with homothetic Gaussmap in real space forms of nonzero curvature. To conclude, based on a strong connection established between homothetic Gauss map and minimal Einstein submanifolds, we pose a conjecture suggesting a possible complete classification of the submanifolds with the former property in arbitrary codimension.