Some Classifications for Gauss Map of Tubular Hypersurfaces in ${\mathbb{E}}_1^4$ Concerning Linearized Operators ${\mathcal{L}}_k$

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 ${\mathbb{E}}_1^4$ 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, $k\in \{1,2\}$.