Betul Bulca

MERIDIAN SURFACES IN 4 WITH POINTWISE 1-TYPE GAUSS MAP

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.

TENSOR PRODUCT SURFACES WITH POINTWISE 1-TYPE GAUSS MAP

Tensor product immersions of a given Riemannian manifold was initiated by B.-Y. Chen. In the present article we study the tensor product surfaces of two Euclidean plane curves. We show that a tensor product surface M of a plane circle c1 centered at origin with an Euclidean planar curve c2 has harmonic Gauss map if and only if M is a part of a plane. Further, we give necessary and sufficient conditions for a tensor product surface M of a plane circle c1 centered at origin with an Euclidean planar curve c2 to have pointwise 1-type Gauss map.

Spherical Product Surfaces in E4

Abstract In the present study we calculate the coefficients of the second fundamental form and curvature ellipse of spherical product surfaces in E4. Otsuki rotational surfaces and Ganchev-Milousheva rotational surfaces are the special type of spherical product surfaces in E4. Further, we give necessary and sufficient condition for the origin of NpM to lie on the curvature ellipse of such surfaces. Finally we get the necessary condition for Ganchev-Milousheva rotational surfaces in E4 to become flat or Chen type. We also give some examples of the projections of these surfaces in E3

On Spherical Product Surfaces in E3

In the present study we consider spherical product surfaces X = axb of two 2D curves in E3. We prove that if a spherical product surface patch X = axb has vanishing Gaussian curvature K (i.e. a flat surface) then either a or b is a straight line. Further, we prove that if a(u) is a straight line and b(v) is a 2D curve then the spherical product is a non-minimal and flat surface. We also prove that if b(v) is a straight line passing through origin and a(u) is any 2D curve (which is not a line) then the spherical product is both minimal and flat. We also give some examples of spherical product surface patches with potential applications to visual cyberworlds.

Meridian Surfaces with Constant Mean Curvature in Pseudo-Euclidean 4-Space with Neutral Metric

In the present paper we consider a special class of Lorentz surfaces in the four-dimensional pseudo-Euclidean space with neutral metric which are one-parameter systems of meridians of rotational hypersurfaces with timelike, spacelike, or lightlike axis and call them meridian surfaces. We give the complete classification of minimal and quasi-minimal meridian surfaces. We also classify the meridian surfaces with non-zero constant mean curvature.

Focal representation of k-slant Helices in Em+1

Abstract The focal representation of a generic regular curve γ in Em+1 consists of the centers of the osculating hyperplanes. A k-slant helix γ in Em+1 is a (generic) regular curve whose unit normal vector Vk makes a constant angle with a fixed direction in Em+1. In the present paper we proved that if γ is a k-slant helix in Em+1, then the focal representation Cγ of γ in Em+1 is an (m− k + 2)-slant helix in Em+1.

On Total Shear Curvature of Surfaces in E^{n+2}

In this study we consider the surfaces in 2  n E . First, we give preleminaries of second fundamental form and curvature properties of the surfaces. Further, we obtained some results related with the total shear curvature of the surfaces. Finally, we give an example of a surface in Euclidean 4-space 4 E with vanishing shear curvature.

Surface pencils in Euclidean 4-space 𝔼4

In this paper, we study the problem of constructing a family of surfaces (surface pencils) from a given curve in 4-dimensional Euclidean space 𝔼4. We have shown that the generalized rotation surfaces in 𝔼4 are the special type of surface pencils. Further, the curvature properties of these surfaces are investigated. Finally, we give some examples of flat surface pencils in 𝔼4.

On translation surfaces in 4-dimensional Euclidean space

We consider translation surfaces in Euclidean spaces. Firstly, we give some results of translation surfaces in the 3-dimensional Euclidean space E 3 . Further, we consider translation surfaces in the 4-dimensional Euclidean space E 4 . We prove that a translation surface is flat in E 4 if and only if it is either a hyperplane or a hypercylinder. Finally we give necessary and sufficient condition for a quadratic triangular Bezier surface in E 4 to become a translation surface.