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‐sphereS 2 4 ⊂ E 2 5with zero mean curvature vector in S 2 4 and 2‐type pseudo‐spherical Gauss map. We also prove that non‐totally umbilical proper pseudo‐Riemannian hypersurfaces in a pseudo‐sphereS s n + 1 ⊂ E s n + 2with 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 inS 1 3 ⊂ E 1 4with 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 inS 1 3 ⊂ E 1 4with null 2‐type pseudo‐spherical Gauss map.