Pseudo-spherical submanifolds with 1-type pseudo-spherical Gauss map

Abstract

In this work, we study the pseudo-Riemannian submanifolds of a pseudo-sphere with 1-type pseudo-spherical Gauss map. First, we classify the Lorentzian surfaces in a 4-dimensional pseudo-sphere ${\mathbb{S}}_s^4(1)$ 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 ${\mathbb{S}}_s^{m-1}(1)\subset {\mathbb{E}}_s^m$ with 1-type pseudo-spherical Gauss map, and we classify spacelike surfaces and Lorentzian surfaces in the de Sitter space ${\mathbb{S}}_1^4(1)\subset {\mathbb{E}}_1^5$ 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.