$HS_{r}$-valued Gauss maps and umbilic spacelike surfaces of codimension two

TL;DR

This paper introduces a new pair of Gauss maps for spacelike surfaces of codimension two in Lorentz-Minkowski space, aiding in the analysis of flat and umbilic surfaces.

Contribution

It constructs and validates the extbf n_r^{ ext{±}}-Gauss maps, providing a practical tool for studying specific geometric properties of spacelike surfaces.

Findings

The extbf n_r^{ ext{±}}-Gauss maps are well-defined.

These maps help analyze flat spacelike surfaces.

They are useful for studying umbilic spacelike surfaces.

Abstract

To study spacelike surfaces of codimension two in the Lorentz-Minkowski space $\Bbb R^{n+1}_1,$ we construct a pair of maps whose values are in $HS_r:=H_+^n(\textbf v,1)\cap \{x_{n+1}=r\},$ called $\textbf n_r^{\pm}$-Gauss maps. It is showed that they are well-defined and useful to study practically flat as well as umbilic spacelike surfaces of codimension two in $\Bbb R^{n+1}_1.$