On rotational surfaces in 3 dimensional de Sitter space with Weingarten condition

TL;DR

This paper classifies spacelike and timelike rotational surfaces in 3D de Sitter space that satisfy specific Weingarten conditions, providing explicit profile curves and curvature relations.

Contribution

It introduces a classification of Weingarten rotational surfaces in de Sitter space with explicit curvature relations and profile curves.

Findings

Classification of Weingarten rotational surfaces in de Sitter space.

Explicit formulas for profile curves based on principal curvatures.

Identification of special cases with linear and power relations between curvatures.

Abstract

In this article, we study spacelike and timelike rotational surfaces in a 3--dimensional de Sitter space $\mathbb{S}^3_1$ which are the orbit of a regular curve under the action of the orthogonal transformation of 4--dimensional Minkowski space $\mathbb{E}^4_1$ leaving a spacelike, a timelike or a degenerate plane pointwise fixed. We determine the profile curve of such Weingarten rotational surfaces parameterized by the principal curvature. Then, we classify spacelike and timelike Weingarten rotational surface in $\mathbb{S}^3_1$ with the principal curvatures $\kappa$ and $\lambda$ satisfying $\kappa=a\lambda+b$ or $\kappa=a\lambda^m$ for special cases of constants $a, b$ and $m$.