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

• Bulletin of the Korean Mathematical Society
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• v.52 no.1
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• pp.301-312
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• 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 $\mathbb{E}^4_1$.

• The Pure and Applied Mathematics
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• v.18 no.4
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• pp.369-377
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• 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
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• v.41 no.6
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• pp.1007-1021
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• 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
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• v.38 no.4
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• pp.753-761
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• 2001

We introduce the notion of Gauss map of pointwise 1-type on ruled surfaces in the Euclidean 3-space for which vector valued functions is neither trivial nor it extends or coincides with the usual notion of 1-type, in general. We characterize the minimal helicoid in terms of it and give a complete classification of the ruled surfaces with pointwise 1-type Gauss map.

• Journal of the Korean Mathematical Society
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• v.52 no.6
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• pp.1337-1346
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• 2015

We examine the relationship of the shape operator of a surface of Euclidean 3-space with its Gauss map of pointwise 1-type. Surfaces with constant mean curvature and right circular cones with respect to some properties of the shape operator are characterized when their Gauss map is of pointwise 1-type.

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

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

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

• The Pure and Applied Mathematics
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• v.19 no.3
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• pp.229-237
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• 2012

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 the right circular cones are the only rational kind GSCS's with pointwise 1-type Gauss map of the second kind.