• Journal of the Korean Mathematical Society
• /
• v.46 no.1
• /
• pp.215-223
• /
• 2009

The helicoidal surfaces with pointwise 1-type or harmonic gauss map in Euclidean 3-space are studied. The notion of pointwise 1-type Gauss map is a generalization of usual sense of 1-type Gauss map. In particular, we prove that an ordinary helicoid is the only genuine helicoidal surface of polynomial kind with pointwise 1-type Gauss map of the first kind and a right cone is the only rational helicoidal surface with pointwise 1-type Gauss map of the second kind. Also, we give a characterization of rational helicoidal surface with harmonic or pointwise 1-type Gauss map.

• Journal of the Korean Mathematical Society
• /
• v.44 no.2
• /
• pp.407-442
• /
• 2007

The study of Euclidean submanifolds with finite type "classical" Gauss map was initiated by B.-Y. Chen and P. Piccinni in [11]. On the other hand, it was believed that for spherical sub manifolds the concept of spherical Gauss map is more relevant than the classical one (see [20]). Thus the purpose of this article is to initiate the study of spherical submanifolds with finite type spherical Gauss map. We obtain several fundamental results in this respect. In particular, spherical submanifolds with 1-type spherical Gauss map are classified. From which we conclude that all isoparametric hypersurfaces of $S^{n+1}$ have 1-type spherical Gauss map. Among others, we also prove that Veronese surface and equilateral minimal torus are the only minimal spherical surfaces with 2-type spherical Gauss map.

• Bulletin of the Korean Mathematical Society
• /
• v.52 no.1
• /
• pp.301-312
• /
• 2015

In this paper, we study a class of spacelike rotational surfaces in the Minkowski 4-space $\mathbb{E}^4_1$ with meridian curves lying in 2-dimensional spacelike planes and having pointwise 1-type Gauss map. We obtain all such surfaces with pointwise 1-type Gauss map of the first kind. Then we prove that the spacelike rotational surface with flat normal bundle and pointwise 1-type Gauss map of the second kind is an open part of a spacelike 2-plane in $\mathbb{E}^4_1$.

• Bulletin of the Korean Mathematical Society
• /
• v.42 no.1
• /
• pp.23-27
• /
• 2005

In this paper, we study rotation surfaces in a Euclidean space with pointwise 1-type Gauss map and obtain by the use of the concept of pointwise finite type Gauss map, a characterization theorem for rotation surfaces of constant mean curvature.

• The Pure and Applied Mathematics
• /
• v.18 no.4
• /
• pp.369-377
• /
• 2011

In this article, we study generalized slant cylindrical surfaces (GSCS's) with pointwise 1-type Gauss map of the first and second kinds. Our main results state that GSCS's with pointwise 1-type Gauss map of the first kind coincide with surfaces of revolution with constant mean curvature; and the right cones are the only polynomial kind GSCS's with pointwise 1-type Gauss map of the second kind.

• Journal of the Korean Mathematical Society
• /
• v.41 no.6
• /
• pp.1007-1021
• /
• 2004

In this paper, we study rotation surfaces in the Minkowski 3-dimensional space with pointwise 1-type Gauss map and obtain by the use of the concept of pointwise finite type Gauss map, a characterization theorem concerning rotation surfaces and constancy of the mean curvature of certain open subsets on these surfaces.

• Bulletin of the Korean Mathematical Society
• /
• v.51 no.6
• /
• pp.1863-1874
• /
• 2014

In this paper, we study spacelike rotational surfaces which are called boost invariant surfaces in Minkowski 4-space $\mathbb{E}^4_1$. We give necessary and sufficient condition for flat spacelike rotational surface to have pointwise 1-type Gauss map. Also, we obtain a characterization for boost invariant marginally trapped surface with pointwise 1-type Gauss map.

• Honam Mathematical Journal
• /
• v.38 no.2
• /
• pp.305-316
• /
• 2016

In this paper we study general rotational surfaces in the 4-dimensional Euclidean space $\mathbb{E}^4$ 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.

• Journal of the Korean Mathematical Society
• /
• v.35 no.2
• /
• pp.315-330
• /
• 1998

Chen and Piccinni [7] have classified all compact surfaces in a Euclidean space $R^{2＋p}$ with 1-type generalized Gauss map. Being motivated by this result, the purpose of this paper is to consider the Lorentz version of the classification theorem and to obtain a complete classification of space-like surfaces in indefinite Euclidean space $R_{p}$ $^{2＋p}$ with 1-type generalized Gauss map.p.

• Journal of the Korean Mathematical Society
• /
• v.55 no.4
• /
• pp.923-938
• /
• 2018

In this paper, we handle the Gauss map of a tubular surface which is constructed according to the parallel transport frame of its spine curve. We show that there is no tubular surface having harmonic Gauss map. Moreover, we give a complete classification of this kind of tubular surface having pointwise 1-type Gauss map in Euclidean 4-space ${\mathbb{E}}^4$.