scholarly journals Spherical Ruled Surfaces in S3 Characterized by the Spherical Gauss Map

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2106
Author(s):  
Young Ho Kim ◽  
Sun Mi Jung
Keyword(s):  
Riemannian Manifold ◽  
Finite Type ◽  
Eigenvalue Problems ◽  
Gauss Map ◽  
Ruled Surfaces ◽  
Dimensional Sphere ◽  
3 Dimensional ◽  
Smooth Maps ◽  
The Laplace Operator ◽  
Natural Way

The Laplace operator on a Riemannian manifold plays an important role with eigenvalue problems and the spectral theory. Extending such an eigenvalue problem of smooth maps including the Gauss map, the notion of finite-type was introduced. The simplest finite-type is of 1-type. In particular, the spherical Gauss map is defined in a very natural way on spherical submanifolds. In this paper, we study ruled surfaces of the 3-dimensional sphere with generalized 1-type spherical Gauss map which generalizes the notion of 1-type. The classification theorem of ruled surfaces of the sphere with the spherical Gauss map of generalized 1-type is completed.

scholarly journals New Characterizations of the Clifford Torus and the Great Sphere

Symmetry ◽  
2019 ◽  
Vol 11 (9) ◽  
pp. 1076 ◽  
Author(s):  
Sun Mi Jung ◽  
Young Ho Kim ◽  
Jinhua Qian
Keyword(s):  
Euclidean Space ◽  
Finite Type ◽  
Three Dimensional ◽  
Gauss Map ◽  
Ruled Surfaces ◽  
Dimensional Sphere ◽  
Round Sphere ◽  
Clifford Torus ◽  
Great Sphere ◽  
Set Up

In studying spherical submanifolds as submanifolds of a round sphere, it is more relevant to consider the spherical Gauss map rather than the Gauss map of those defined by the oriented Grassmannian manifold induced from their ambient Euclidean space. In that sense, we study ruled surfaces in a three-dimensional sphere with finite-type and pointwise 1-type spherical Gauss map. Concerning integrability and geometry, we set up new characterizations of the Clifford torus and the great sphere of 3-sphere and construct new examples of spherical ruled surfaces in a three-dimensional sphere.

scholarly journals Ruled Surfaces and Tubes with Finite Type Gauss Map

1993 ◽  
Vol 16 (2) ◽  
pp. 341-349 ◽  
Author(s):  
Christos BAIKOUSSIS ◽  
Bang-yen CHEN ◽  
Leopold VERSTRAELEN
Keyword(s):  
Finite Type ◽  
Gauss Map ◽  
Ruled Surfaces

scholarly journals Characterization of Clifford Torus in Three-Spheres

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 718
Author(s):  
Dong-Soo Kim ◽  
Young Ho Kim ◽  
Jinhua Qian
Keyword(s):  
Laplace Operator ◽  
Unit Sphere ◽  
Three Dimensional ◽  
Gauss Map ◽  
Dimensional Sphere ◽  
Clifford Torus ◽  
Closed Surfaces ◽  
Dimensional Unit ◽  
The Laplace Operator

We characterize spheres and the tori, the product of the two plane circles immersed in the three-dimensional unit sphere, which are associated with the Laplace operator and the Gauss map defined by the elliptic linear Weingarten metric defined on closed surfaces in the three-dimensional sphere.

scholarly journals Extension of Eigenvalue Problems on Gauss Map of Ruled Surfaces

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 514
Author(s):  
Miekyung Choi ◽  
Young Kim
Keyword(s):  
Circular Cylinder ◽  
Eigenvalue Problems ◽  
Gauss Map ◽  
Natural Generalization ◽  
Ruled Surfaces ◽  
Usual Sense ◽  
Infinite Type ◽  
Base Curve ◽  
Smooth Map ◽  
Nice Tool

A finite-type immersion or smooth map is a nice tool to classify submanifolds of Euclidean space, which comes from the eigenvalue problem of immersion. The notion of generalized 1-type is a natural generalization of 1-type in the usual sense and pointwise 1-type. We classify ruled surfaces with a generalized 1-type Gauss map as part of a plane, a circular cylinder, a cylinder over a base curve of an infinite type, a helicoid, a right cone and a conical surface of G-type.

scholarly journals On the Gauss map of ruled surfaces

1992 ◽  
Vol 34 (3) ◽  
pp. 355-359 ◽  
Author(s):  
Christos Baikoussis ◽  
David E. Blair
Keyword(s):  
Laplace Operator ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Gauss Map ◽  
Ruled Surfaces ◽  
Connected Surface ◽  
Image Position ◽  
The Laplace Operator ◽  
Special Case

Let M2 be a (connected) surface in Euclidean 3-space E3, and let G:M2→S2(1) ⊂ E3 be its Gauss map. Then, according to a theorem of E. A. Ruh and J. Vilms [3], M2 is a surface of constant mean curvature if and only if, as a map from M2 to S2(1), G is harmonic, or equivalently, if and only ifwhere δ is the Laplace operator on M2 corresponding to the induced metric on M2 from E3 and where G is seen as a map from M2to E3. A special case of (1.1) is given byi.e., the case where the Gauss map G:M2→E3 is an eigenfunction of the Laplacian δ on M2.

scholarly journals Extension of eigenvalue problems on Gauss map of ruled surfaces

2018 ◽  
Author(s):  
Miekyung Choi ◽  
Young Ho Kim
Keyword(s):  
Circular Cylinder ◽  
Eigenvalue Problems ◽  
Gauss Map ◽  
Natural Generalization ◽  
Ruled Surfaces ◽  
Usual Sense ◽  
Infinite Type ◽  
Base Curve ◽  
Smooth Map ◽  
Nice Tool

A finite-type immersion or smooth map is a nice tool to classify submanifolds of Euclidean space, which comes from eigenvalue problem of immersion. The notion of generalized 1-type is a natural generalization of those of 1-type in the usual sense and pointwise 1-type. We classify ruled surfaces with generalized 1-type Gauss map as part of a plane, a circular cylinder, a cylinder over a base curve of an infinite type, a helicoid, a right cone and a conical surface of $G$-type.

scholarly journals Classification Theorems in Minkowski 3-space

2018 ◽  
Author(s):  
Miekyung Choi ◽  
Young Ho Kim
Keyword(s):  
Minkowski Space ◽  
Gauss Map ◽  
Ruled Surfaces ◽  
3 Dimensional ◽  
Dimensional Minkowski Space

By generalizing the notion of pointwise 1-type Gauss map, the generalized 1-type Gauss map has been recently introduced. Without any assumption, we classified all possible ruled surfaces with generalized 1-type Gauss map in a 3-dimensional Minkowski space. In particular, null scrolls do not have the proper generalized 1-type Gauss map. In fact, it is harmonic.

scholarly journals On the gauss map of ruled surfaces of type ii in 3-dimensional Pseudo-Galilean space

2011 ◽  
Vol 31 (1) ◽  
pp. 145
Author(s):  
Alper Osman Öğrenmiş ◽  
Mahmut Ergüt
Keyword(s):  
Three Dimensional ◽  
Gauss Map ◽  
Ruled Surfaces ◽  
Laplacian Operator ◽  
Type Ii ◽  
3 Dimensional ◽  
Galilean Space

In this paper, ruled surfaces of type II in a three-dimensional Pseudo-Galilean space are given. By studying its Gauss map and Laplacian operator, we obtain a classification of ruled surfaces of type II in a three-dimensional Pseudo-Galilean space.

scholarly journals Classification Theorems of Ruled Surfaces in Minkowski Three-Space

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 318
Author(s):  
Miekyung Choi ◽  
Young Kim
Keyword(s):  
Minkowski Space ◽  
Gauss Map ◽  
Ruled Surfaces ◽  
3 Dimensional ◽  
Dimensional Minkowski Space ◽  
Three Space

By generalizing the notion of the pointwise 1-type Gauss map, the generalized 1-type Gauss map has been recently introduced. Without any assumption, we classified all possible ruled surfaces with the generalized 1-type Gauss map in a 3-dimensional Minkowski space. In particular, null scrolls do not have the proper generalized 1-type Gauss map. In fact, it is harmonic.

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