Rotational hypersurfaces family satisfying $\mathbb{L}_{n-3}\mathcal{G}=\mathcal{A}\mathcal{G}$ in ${\mathbb{E}}^{n}$

TL;DR

This paper classifies a family of rotational hypersurfaces in Euclidean space based on a differential operator acting on their Gauss map, revealing a specific relationship between the Gauss map and a matrix through a classification theorem.

Contribution

It provides a classification theorem for rotational hypersurfaces satisfying a specific Gauss map equation involving the operator 1L_{n-3}1 and a matrix 1A1, extending understanding of their geometric properties.

Findings

Established a classification theorem for the hypersurfaces.

Connected the Gauss map with a matrix via the operator 1L_{n-3}1.

Analyzed the properties of the Gauss map under the specified operator.

Abstract

In this paper, we investigate rotational hypersurfaces family in $n$ -dimensional Euclidean space ${\mathbb{E}}^{n}$. Our focus is on studying the Gauss map $\mathcal{G}$ of this family with respect to the operator $\mathbb{L}_{k}$, which acts on functions defined on the hypersurfaces. The operator $\mathbb{L}_{k}$ can be viewed as a modified Laplacian and is known by various names, including the Cheng--Yau operator in certain cases. Specifically, we focus on the scenario where $k=n-3$ and $n\geq 3$. By applying the operator $\mathbb{L}_{n-3}$ to the Gauss map $\mathcal{G}$, we establish a classification theorem. This theorem establishes a connection between the $n\times n$ matrix $\mathcal{A}$, and the Gauss map $\mathcal{G}$ through the equation $\mathbb{L}_{n-3}\mathcal{G}=\mathcal{A}\mathcal{G}$.