Timelike rotational surfaces of elliptic, hyperbolic and parabolic types in Minkowski space E41 with pointwise 1-type Gauss map

Abstract

Abstract.In this work, we focus on a class of timelike rotational surfaces in Minkowski space E 4 1 with 2-dimensional axis. There are three types of rotational surfaces with 2-dimensional axis, called rotational surfaces of elliptic, hyperbolic or parabolic type. We obtain all flat timelike rotational surface of elliptic and hyperbolic types with pointwise 1-type Gauss map of the first and second kind. We also prove that there exists no flat timelike rotational surface of parabolic type in E 4 1 with pointwise 1-type Gauss map.

Bour's Theorem of Spacelike Surfaces in Minkowski 4--Space

“…For λ = 0, the three kinds of helicoidal surfaces in E 4 1 given by ( 11)-( 13) become the rotational surfaces of hyperbolic, elliptic and parabolic type in E 4 1 , respectively. For the details, see [13] and [6]. Throughout this article, the corresponding rotational surfaces are denoted by R 1 , R 2 and R 3 .…”

Rotational Surfaces with Pointwise 1-Type Gauss Map in Pseudo Euclidean Space $\mathbb{E}_{2}^{4}$

“…Theorem 1. Let M 1 be rotation surface of elliptic type given by the parametrization (5). If M 1 has harmonic Gauss map then it has constant Gaussian curvature.…”

“…for some smooth function f on M and some constant vector C. A submanifold of a Euclidean space or pseudo-Euclidean space is said to have pointwise 1-type Gauss map, if its Gauss map satisfies (1) for some smooth function f on M and some constant vector C. If the vector C in (1) is zero, a submanifold with pointwise 1-type Gauss map is said to be of the first kind, otherwise it is said to be of the second kind. A lot of papers were recently published about rotational surfaces with pointwise 1-type Gauss map in four dimensional Euclidean and pseudo Euclidean space in [1], [3], [4], [8], [9] [11].Timelike and spacelike rotational surfaces of elliptic, hyperbolic and parabolic types in Minkowski space E 4 1 with pointwise 1-type Gauss map were studied in [5,7]. Aksoyak and Yaylı in [2] studied boost invariant surfaces (rotational surfaces of hyperbolic type) with pointwise 1-type Gauss map in Minkowski space E 4 1 .…”

“…If rotational surfaces of elliptic type M 1 given by(5) is minimal then it has pointwise 1-type Gauss map of the first kind.…”

On rotational surfaces in pseudo-Euclidean space $\mathbb{E}^4_t$ with pointwise 1-type Gauss map

“…Recently, a classification of flat spacelike and timelike rotational surfaces in E 4 1 with pointwise 1-type Gauss map were given [1,7].…”