On the quasi Gauss map for a compact sub-manifold in Euclidean space

Let  σ be the quasi Gauss map of a compact and oriented  n-dimensional isometric immersion sub-manifold  Mn in the  (n+p)-dimensional Euclid space  Rn+p. Denote by ξ the unit mean curvature vector field to  Mn and denote by  Hi the  i-mean curvature along the direction  ξ.Assume that  Hi>0,  i=1,2,…,r for some integer  r  (1≤r≤n-1) and  Hr is a constant. By applying an integral formula recently given by themselves, it is proven that if the image σ(Mn) lies within an open  n-dimension unit semi sphere  Sn+ then  Mn must be totally quasi umbilical. This result generalizes a relevant theorem on hypersurfaces in Euclid space.