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Rotational hypersufaces in $E^4_1$ with generalized $L_k$ 1-type Gauss map

Ahmet Kazan, Mustafa Altin, Nurettin Cenk Turgay

Abstract

In this paper, we study the Gauss map of rotational hypersurfaces in 4-dimensional Lorentz-Minkowski space concerning the linear second order differential operators $L_1$ and $L_2$, where $L_1$ is usually called as the Cheng-Yau operator. We obtain some classifications of rotational hypersurfaces which have $L_k$-harmonic Gauss map, $L_k$-pointwise 1-type Gauss map and generalized $L_k$ 1-type Gauss map, where $k = 1,2$.

Rotational hypersufaces in $E^4_1$ with generalized $L_k$ 1-type Gauss map

Abstract

In this paper, we study the Gauss map of rotational hypersurfaces in 4-dimensional Lorentz-Minkowski space concerning the linear second order differential operators and , where is usually called as the Cheng-Yau operator. We obtain some classifications of rotational hypersurfaces which have -harmonic Gauss map, -pointwise 1-type Gauss map and generalized 1-type Gauss map, where .
Paper Structure (12 sections, 27 theorems, 101 equations)

This paper contains 12 sections, 27 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Linearized Operators for the Gauss Map of Hypersurfaces in $\mathbb E_{1}^{n+1}$
  4. Rotational hypersurfaces in $\mathbb E_{1}^{4}$
  5. On Hypersurfaces with Generalized $L_1$ 1-Type Gauss Map
  6. Rotational Hypersurfaces about Spacelike Axis
  7. Rotational Hypersurfaces about Timelike Axis
  8. Rotational Hypersurfaces about Lightlike Axis
  9. On Hypersurfaces with Generalized $L_2$ 1-Type Gauss Map
  10. Rotational Hypersurfaces about Spacelike Axis
  11. Rotational Hypersurfaces about Timelike Axis
  12. Rotational Hypersurfaces about Lightlike Axis

Key Result

Proposition 1

If the rotational hypersurface (surfspace) in $\mathbb E_{1}^{4}$ is flat (resp. minimal), then we obtain where

Theorems & Definitions (32)

  • Definition 1
  • Remark 1
  • Proposition 1
  • Remark 2
  • Corollary 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Proposition 2
  • ...and 22 more