Pseudo-spherical and pseudo-hyperbolic submanifolds via the quadric representation, I

Abstract.

We consider the quadric representation of a submanifold into non flat pseudo-Riemannian space forms. Then we classify submanifolds such that the mean curvature vector field of its quadric representation is proper for the Laplacian. We have got a complete characterization of hypersurfaces whose quadric representation satisfies ${\it \tilde \Delta} {\tilde H} = {\it\lambda} {\tilde H} + {\it\mu} (\varphi -\varphi _0)$. As for surfaces into De Sitter and anti De Sitter worlds we have also found nice characterizations for minimal B-scrolls and complex circles.