On Rotational Surfaces in 3-Dimensional de Sitter Space with Weingarten Condition

Abstract

In this article, we study spacelike and timelike rotational surfaces in a 3-dimensional de Sitter space S31 which are the orbit of a regular curve under the action of the orthogonal transformation of a 4-dimensional Minkowski space E41 leaving a Riemannian plane, a Lorentzian plane or a degenerate plane pointwise fixed. First, we determine the profile curve of the rotational surfaces in S31 whose the principal curvatures say κ and λ satisfy a certain relation κ = f(λ) for a continuous function f, which is called a Weingarten surface. Also, the profile curve of such surfaces are parametrized with respect to the principal curvature, not arc-length parameter. Then, we find the parametrization of the profile curve of spacelike and timelike Weingarten rotational surfaces in S31 by choosing the certain relation as κ = aλ + b or κ = aλm where a, b and m are constants. Under these circumstances, we classify the spacelike and timelike rotational surfaces in S31 with constant mean curvature, namely minimal or maximal surfaces, or constant Gaussian curvature parametrized in terms of principal curvature.