On constant curvature submanifolds of space forms

We prove a converse to well-known results by E. Cartan and J. D. Moore. Let f:Mcn?Qc˜n+p be an isometric immersion of Riemannian manifold with constant sectional curvature c into space form c˜, free weak-umbilic points if c>c˜. show that the substantial codimension f is p=n?1 if, as shown Moore, first normal bundle possesses lowest possible rank n?1. These submanifolds are class has been extensively studied due their many properties. For instance, they holonomic admit Bäcklund Ribaucour transformations.