Submanifolds with totally umbilical Gauss image

The submanifolds whose Gauss images are totally umbilical submanifolds of the Grassmann manifold are under consideration. The main result is the following classification theorem: if the Gauss image of a submanifold F in a Euclidean space is totally umbilical then either the Gauss image is totally geodesic, or F is the surface in E ⁴ of the special structure. Submanifolds in a Euclidean space with totally geodesic Gauss image were classified earlier.