Pseudo-Spherical Submanifolds with 1-Type Pseudo-Spherical Gauss Map

In this work, we study pseudo-Riemannian submanifolds of a pseudo-sphere with 1-type pseudo-spherical Gauss map. First, we classify Lorentzian surfaces in a 4-dimensional pseudo-sphere Ss4(1) with index s, s= 1 , 2 , and having harmonic pseudo-spherical Gauss map. Then we give a characterization theorem for pseudo-Riemannian submanifolds of a pseudo-sphere Ssm-1(1)⊂Esm with 1-type pseudo-spherical Gauss map, and we classify spacelike surfaces and Lorentzian surfaces in the de Sitter space S14(1)⊂E15 with 1-type pseudo-spherical Gauss map. Finally, according to the causal character of the mean curvature vector we obtain the classification of submanifolds of a pseudo-sphere having 1-type pseudo-spherical Gauss map with nonzero constant component in its spectral decomposition.