Classification of surfaces in a pseudo-sphere with 2-type pseudo-spherical Gauss map

In this article, we study submanifolds in a pseudo-sphere with 2-type pseudo-spherical Gauss map. We give a characterization theorem for Lorentzian surfaces in the pseudo-sphere S⁴ ₂ ⊂ E⁵ ₂ with zero mean curvature vector in S⁴ ₂ and 2-type pseudo-spherical Gauss map. We also prove that non-totally umbilical proper pseudo-Riemannian hypersurfaces in a pseudo-sphere Sⁿ⁺¹ _s ⊂ Eⁿ⁺² _s with non-zero constant mean curvature has 2-type pseudo-spherical Gauss map if and only if it has constant scalar curvature. Then, for n=2 we obtain the classification of surfaces in S³ ₁ ⊂ E⁴ ₁ with 2-type pseudo-spherical Gauss map. Finally, we give an example of surface with null 2-type pseudo-spherical Gauss map which does not appear in Riemannian case, and we give a characterization theorem for Lorentzian surfaces in S³ ₁ ⊂ E⁴ ₁ with null 2-type pseudo-spherical Gauss map.