Gauss map harmonicity and mean curvature of a hypersurface in a homogeneous manifold

We define a Gauss map of an orientable hypersurface in a homogeneous manifold with an invariant Riemannian metric. Our main objective is to extend to this setting some results on the Gauss map of a constant mean curvature hypersurface of an Euclidean space, namely the Ruh - Vilm theorem relating the harmonicity of the Gauss map and the constancy of the mean curvature, and the Hoffman - Osserman - Schoen theorem characterizing the plane and the circular cylinder as the only complete constant mean curvature surfaces whose Gauss image is contained in a closed hemisphere of the sphere.