$LS_{\lowercase {r}}$-valued Gauss maps and spacelike surfaces of revolution in $\mathbb R_1^4$

TL;DR

This paper introduces $ ext{LS}_r$-valued Gauss maps for spacelike surfaces in Lorentz-Minkowski space, enabling the analysis of umbilical surfaces and parametrizations of revolution surfaces of hyperbolic and elliptic types.

Contribution

It constructs new lightcone-valued Gauss maps for spacelike surfaces in $ ext{R}_1^4$, providing tools for classifying umbilical surfaces and deriving parametrizations of revolution surfaces.

Findings

Defined $ ext{LS}_r$-valued Gauss maps for spacelike surfaces.

Characterized umbilical spacelike surfaces using these maps.

Obtained parametrizations of hyperbolic and elliptic revolution surfaces.

Abstract

To study spacelike surfaces in the Lorentz-Minkowski space $\Bbb R^{4}_1,$ we construct a pair of maps whose values are in the lightcone, called $\mathfrak l_r^{\pm} $-Gauss maps. We can use these maps to study umbilical spacelike surfaces and find parametrizations of spacelike surfaces of revolution of hyperbolic and eliptic types in some particular cases.