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

TL;DR

This paper classifies and characterizes pseudo-Riemannian submanifolds of pseudo-spheres with 1-type pseudo-spherical Gauss map, focusing on Lorentzian and spacelike surfaces, and explores their geometric properties and classifications.

Contribution

It provides a comprehensive classification and characterization of pseudo-Riemannian submanifolds with 1-type pseudo-spherical Gauss map, including Lorentzian and spacelike surfaces in pseudo-spheres and de Sitter space.

Findings

Classified Lorentzian surfaces in 4D pseudo-spheres with harmonic pseudo-spherical Gauss map.

Characterized submanifolds with 1-type pseudo-spherical Gauss map in pseudo-spheres.

Classified submanifolds based on the causal character of the mean curvature vector.

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 $\mathbb{S}^4_s(1)$ 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.