MERIDIAN SURFACES IN 𝔼⁴WITH POINTWISE 1-TYPE GAUSS MAP

Abstract

Abstract. In the present article we study a special class of surfaces in the four-dimensional Euclidean space, which are one-parameter systems of meridians of the standard rotational hypersurface. They are called meridian surfaces. We show that a meridian surface has a harmonic Gauss map if and only if it is part of a plane. Further, we give necessary and sufficient conditions for a meridian surface to have pointwise 1-type Gauss map and find all meridian surfaces with pointwise 1-type Gauss map.

On Generalized Rotational Surfaces in Euclidean Spaces

“…For n = 1 and m = 3, the radius vector (4.1) satisfying the indicated properties describes the rotational surface M in E 4 given with the parametrization Actually, the surfaces given with the parametrization (4.21) are know the meridian surfaces in E 4 (see [4], [15]). The tangent space is spanned by the vector fields…”

“…They also gave some special classes of generalized rotational surfaces as examples. See also [6], [12] and [20] for the rotational surfaces with constant Gaussian curvature in Euclidean 4-space E 4 . For higher dimensional case N. H. Kuiper defined rotational embedded submanifolds in Euclidean spaces [17].…”

Meridian Surfaces on Rotational Hypersurfaces with Lightlike Axis in ${\mathbb E}^4_2$

“…with the property that the restriction of the metric onto the tangent space T p M is of signature (1,1), and the restriction of the metric onto the normal space N p M is of signature (1, 1).…”

“…The classification of meridian surfaces with constant Gauss curvature, with constant mean curvature, Chen meridian surfaces and meridian surfaces with parallel normal bundle is given in [15] and [17]. The meridian surfaces in E 4 with pointwise 1-type Gauss map are classified in [1]. The idea from the Euclidean space is used in [16], [18], and [19] for the construction of meridian spacelike surfaces lying on rotational hypersurfaces in E 4 1 with timelike, spacelike, or lightlike axis.…”

Surfaces of Revolution With Pointwise 1-Type Gauss Map in Pseudo-Galilean Space

“…A helicoid, a catenoid and a right cone are the typical examples of surfaces with pointwise 1-type Gauss map. Many results of submanifolds with pointwise 1-type Gauss map were obtained in [1], [3], [5], [6], [7], [9], [12], etc, when the ambient spaces are the Euclidean space, Minkowski space and Galilean space.…”