Pseudo-hyperbolic Gauss maps of Lorentzian surfaces in anti-de Sitter space

TL;DR

This paper classifies the type numbers of pseudo-hyperbolic Gauss maps for Lorentzian surfaces in anti-de Sitter space, exploring their behavior across families and higher-dimensional hypersurfaces.

Contribution

It provides a comprehensive analysis of the type numbers of pseudo-hyperbolic Gauss maps for various Lorentzian surfaces in anti-de Sitter space, including constant curvature and B-scroll type hypersurfaces.

Findings

Type numbers determined for all oriented Lorentzian surfaces with specific curvature conditions.

Behavior of type numbers along parallel families analyzed.

Type number of Lorentzian B-scroll hypersurfaces in higher-dimensional anti-de Sitter space studied.

Abstract

In this paper, we determine the type numbers of the pseudo-hyperbolic Gauss maps of all oriented Lorentzian surfaces of constant mean and Gaussian curvatures and non-diagonalizable shape operator in the $3$-dimensional anti-de Sitter space. Also, we investigate the behavior of type numbers of the pseudo-hyperbolic Gauss map along the parallel family of such oriented Lorentzian surfaces in the $3$-dimensional anti-de Sitter space. Furthermore, we investigate the type number of the pseudo-hyperbolic Gauss map of one of Lorentzian hypersurfaces of B-scroll type in a general dimensional anti-de Sitter space.