Hypersurfaces with null higher order mean curvature

A hypersurface Mn immersed in a space form is r-minimal if its (r + 1)th-curvature (the (r + 1)th elementary symmetric function of its principal curvatures) vanishes identically. Let W be the set of points which are omitted by the totally geodesic hypersurfaces tangent to M. We will prove that if an orientable hypersurface Mn is r-minimal and its rth-curvature is nonzero everywhere, and the set W is nonempty and open, then Mn has relative nullity n − r. Also we will prove that if an orientable hypersurface Mn is r-minimal and its rth-curvature is nonzero everywhere, and the ambient space is euclidean or hyperbolic and W is nonempty, then Mn is r-stable.