Some Classifications for Gauss Map of Tubular Hypersurfaces in $\mathbb{E}^{4}_{1}$ Concerning Linearized Operators $\mathcal{L}_{k}$

TL;DR

This paper classifies tubular hypersurfaces in 4D Lorentz-Minkowski space based on their Gauss map properties related to linearized operators, extending understanding of their geometric and harmonic characteristics.

Contribution

It introduces classifications for tubular hypersurfaces with specific Gauss map types concerning the linearized operators $ ext{L}_1$ and $ ext{L}_2$ in Lorentz-Minkowski space.

Findings

Classified hypersurfaces with generalized $ ext{L}_k$ 1-type Gauss map.

Identified hypersurfaces with $ ext{L}_k$-harmonic Gauss map.

Analyzed the $ ext{L}_1$ operator for envelopes of pseudo hyperspheres.

Abstract

In this study, we deal with the Gauss map of tubular hypersurfaces in 4-dimensional Lorentz-Minkowski space concerning the linearized operators $\mathcal{L}_{1}$ (Cheng-Yau) and $\mathcal{L}_{2}$. We obtain the $\mathcal{L}_{1}$ (Cheng-Yau) operator of the Gauss map of tubular hypersurfaces that are formed as the envelope of a family of pseudo hyperspheres {or pseudo hyperbolic hyperspheres} whose centers lie on timelike or spacelike curves with non-null Frenet vectors in $\mathbb{E}^{4}_{1}$ and give some classifications for these hypersurfaces which have generalized $\mathcal{L}_{k}$ 1-type Gauss map, first kind $\mathcal{L}_{k}$-pointwise 1-type Gauss map, second kind $\mathcal{L}_{k}$-pointwise 1-type Gauss map and $\mathcal{L}_{k}$-harmonic Gauss map, $k\in\{1,2\}$.