Dae Won Yoon

Ruled surfaces with pointwise 1-type Gauss map

Abstract In this paper, we study ruled surfaces in a three-dimensional Minkowski space with pointwise 1-type Gauss map and obtain the complete classification theorems for those. We also obtain a new characterization of minimal ruled surfaces in a three-dimensional Minkowski space.

Classification of ruled surfaces in Minkowski 3-spaces

Abstract In this paper, we study some properties about the second Gaussian curvature of ruled surfaces in a three-dimensional Minkowski space. Furthermore, we classify ruled surfaces in a three-dimensional Minkowski space in terms of the second Gaussian curvature, the mean curvature and the Gaussian curvature.

CLASSIFICATIONS OF ROTATION SURFACES IN PSEUDO-EUCLIDEAN SPACE

In this article, we study rotation surfaces in the 4- dimensional pseudo-Euclidean space E 4. Also, we obtain the com- plete classification theorems for the flat rotation surfaces with finite type Gauss map, pointwise 1-type Gauss map and an equation in terms of the mean curvature vector. In fact, we characterize the flat rotation surfaces of finite type immersion with the Gauss map and the mean curvature vector field, namely the Gauss map of finite type, pointwise 1-type Gauss map and some algebraic equations in terms of the Gauss map and the mean curvature vector field related to the Laplacian of the surfaces with respect to the induced metric.

Optimal inequalities for the Casorati curvatures of submanifolds of real space forms endowed with semi-symmetric metric connections

In this paper, we prove two optimal inequalities involving the intrinsic scalar curvature and extrinsic Casorati curvature of submanifolds of real space forms endowed with a semi-symmetric metric connection. Moreover, we show that in both cases, the equality at all points characterizes the invariantly quasi-umbilical submanifolds.MSC:53C40, 53B05.

HELICOIDAL SURFACES AND THEIR GAUSS MAP IN MINKOWSKI 3-SPACE II

The helicoidal surface is a generalization of rotation surface in a Minkowski space. We study helicoidal surfaces in a Minkowski 3-space in terms of their Gauss map and provide some examples of new classes of helicoidal surfaces with constant mean curvature in a Minkowski 3-space.

ON THE GAUSS MAP OF SURFACES OF REVOLUTION WITHOUT PARABOLIC POINTS

In this article, we study surfaces of revolution without par- abolic points in a Euclidean 3-space whose Gauss map G satisfies the condition ¢ h G = AG,A 2 Mat(3,R), where ¢ h denotes the Laplace op- erator of the second fundamental form h of the surface and Mat(3,R) the set of 3◊3-real matrices, and also obtain the complete classification the- orem for those. In particular, we have a characterization of an ordinary sphere in terms of it.

HELICOIDAL SURFACES WITH PRESCRIBED CURVATURES IN Nil3

In the present paper, we study helicoidal surfaces in the 3-dimensional Heisenberg group Nil3. Also, we construct helicoidal surfaces in Nil3 with prescribed Gaussian curvature or mean curvature given by smooth functions. As the results, we classify helicoidal surfaces with constant Gaussian curvature or constant mean curvature.

Ruled surfaces of non-degenerate third fundamental forms in Minkowski 3-spaces

In this paper, we study ruled surfaces in a Minkowski 3-space satisfying some equation in terms of a position vector field and Laplacian operator with respect to non-degenerate third fundamental form. Furthermore, we give a new example of null scroll in a Minkowski 3-space.

SOME TRANSLATION SURFACES IN THE 3-DIMENSIONAL HEISENBERG GROUP

Abstract. In this paper, we define translation surfaces in the 3-dimen-sional Heisenberg group H 3 obtained as a product of two planar curveslying in planes, which are not orthogonal, and study minimal translationsurfaces in H 3 . 1. IntroductionMinimal surfaces are one of main objects which have drawn geometers’ in-terest for a very long time. In 1744, L. Euler found that the only minimalsurfaces of revolution are the planes and the catenoids, and in 1842 E. Catalanproved that the planes and the helicoids are the only minimal ruled surfacesin the 3-dimensional Euclidean space E 3 . Also, H. F. Scherk in 1835 studiedtranslation surfaces in E 3 defined as graph of the function z(x,y) = f(x)+g(y)and he proved that, besides the planes, the only minimal translation surfacesare the surfaces given by(1.1) z =1alog cos(ax)cos(ay) =1alog|cos(ax)| −1alog|cos(ay)|,where f(x) and g(y) are smooth functions on some interval of R and a is anon-zero constant. These surfaces are now referred as Scherk’s minimal sur-faces. The study of minimal surfaces of revolution, ruled surfaces and trans-lation surfaces in the Euclidean space was extended to the Lorentz version byO. Kobayashi [4] and I. V. de Woestijne [8]. R. Lo´pez [5] studied transla-tion surfaces in the 3-dimensional hyperbolic space H

HELICOIDAL SURFACES OF THE THIRD FUNDAMENTAL FORM IN MINKOWSKI 3-SPACE

Abstract. We study helicoidal surfaces with the non-degenerate thirdfundamental form in Minkowski 3-space. In particular, we mainly focuson the study of helicoidal surfaces with light-like axis in Minkowski 3-space. As a result, we classify helicoidal surfaces satisfying an equationin terms of the position vector field and the Laplace operator with respectto the third fundamental form on the surface. 1. IntroductionWe study a (pseudo-)Riemannian manifold as a submanifold of a (pseudo-)Euclidean space via an isometric immersion by Nash’s Theorem. Let x :M → E 3 be an isometric immersion of a connected surface M in a Euclidean3-space E 3 . Denote by ∆ the Laplacian with respect to the induced metric onM. Takahashi ([11]) proved that minimal surfaces and spheres are the onlysurfaces in E 3 satisfying the condition ∆x = λx, λ ∈ R. As a generalization ofTakahashi’s Theorem, Garay ([4]) classified the hypersurfaces whose coordinatefunctions in E m are eigenfunctions of their Laplacian, that is, the hypersurfacesin E