FLAT ROTATIONAL SURFACES WITH POINTWISE 1-TYPE GAUSS MAP IN E⁴

Abstract

Abstract. In this paper we study general rotational surfaces in the 4-dimensional Euclidean space E 4 and give a characterization of flat general rotational surface with pointwise 1-type Gauss map. Also, we show that a flat general rotational surface with pointwise 1-type Gauss map is a Lie group if and only if it is a Clifford torus.

Special proper pointwise slant surfaces of a locally product Riemannian manifold

“…Thus, M is called a locally product manifold, which admits a locally product structure defined by the existence of a separating coordinate system. A locally product manifold always admits a natural tensor field F of type (1,1) given by…”

“…X uu (u, v) = (− cosh v cos u, − sinh v cos u, − cosh v sin u, − sinh v sin u), X uv (u, v) = (− sinh v cos u, − cosh v sin u, sinh v cos u, cosh v cos u), X vv (u, v) = (cosh v cos u, sinh v cos u, cosh v sin u, sinh v sin u). The surface given in Example 5.7 is a tensor product surface of E 4 , which was defined in [25] and studied in [1,2,6,4,5,26], etc. The surface is a proper pointwise slant surface with the Wirtinger function θ = |u − v| , but it is not special.…”

“…A slant surface M of a Kaehlerian manifold is called special slant if, with respect to some suitable adapted orthonormal frame {e 1 , e 2 , e 3 , e 4 } , the shape operator of the surface takes the following forms:…”

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

“…for some smooth function f on M and some constant vector C. A submanifold M of a pseudo-Euclidean space E m s 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. A submanifold with pointwise 1-type Gauss map is said to be of the first kind if the vector C in (1) is zero vector. Otherwise, the pointwise 1-type Gauss map is said to be of the second kind.…”

“…Kim and Yoon in [16] obtained the complete classification theorems for the flat rotation surfaces with finite type Gauss map and pointwise 1-type Gauss map. The authors in [1] studied flat general rotational surfaces in the 4-dimensional Euclidean space E 4 with pointwise 1-type Gauss map and they showed that a non-planar flat general rotational surfaces with pointwise 1-type Gauss map is a Lie group if and only if it is a Clifford Torus.…”