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

Abstract

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 \({\mathbb{S}^4_s(1)}\) 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 \({\mathbb{S}^{m-1}_s(1)\subset\mathbb{E}^m_s}\) with 1-type pseudo-spherical Gauss map, and we classify spacelike surfaces and Lorentzian surfaces in the de Sitter space \({\mathbb{S}^4_1(1)\subset\mathbb{E}^5_1}\) 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.