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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 130495, 8 pages
http://dx.doi.org/10.1155/2013/130495
Research Article

On the Gauss Map of Surfaces of Revolution with Lightlike Axis in Minkowski 3-Space

1School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China
2Department of Mathematics, Heilongjiang Institute of Technology, Harbin 150050, China
3Rear Services Office, Chongqing Police College, Chongqing 401331, China

Received 29 March 2013; Accepted 2 July 2013

Academic Editor: Ji Gao

Copyright © 2013 Minghao Jin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

By studying the Gauss map G and Laplace operator of the second fundamental form h, we will classify surfaces of revolution with a lightlike axis in 3-dimensional Minkowski space and also obtain the surface of Enneper of the 2nd kind, the surface of Enneper of the 3rd kind, the de Sitter pseudosphere, and the hyperbolic pseudosphere that satisfy condition being a real matrix.

1. Introduction

The Gauss map is a useful tool for studying surfaces in Euclidean space and pseudo-Euclidean space.

Suppose that is a connected surface in and is the Gauss map on . According to a theorem proved by Ruh and Vilms [1], has constant mean curvature if and only if where is the Laplace operator on that corresponds to the metric induced on from . A special case of (1) is given by where the Gauss map is an eigenfunction of the Laplacian on . As a more general form of (1), Dillen et al. [2] proved that a surface of revolution in satisfies the condition if and only if is a plane, sphere, or cylinder. Baikoussis and Blair [3] proved that a ruled surface in satisfies condition (3) if and only if is a plane, helicoidal surface, or spiral surface in . Additionally, Choi and Alías et al. [46] completely classified the surfaces of revolution and ruled surfaces in 3-dimensional Minkowski space that satisfy condition (3). Kim and Yoon [7] studied ruled surfaces in such that

Recently, an interesting question was raised: what surfaces of revolution without parabolic points in Euclidean or pseudo-Euclidean space satisfy the following condition? where is the Laplace operator with respect to the second fundamental form of the surface. This operator is formally defined by for the components of the second fundamental form on , and we denote by (resp., ) the inverse matrix (resp., the determinant) of the matrix .

In [8], the authors studied surfaces of revolution without parabolic points in Euclidean 3-space and presented some classification theorems. In this paper, we will consider surfaces of revolution with lightlike axis in and present some classification results.

2. Preliminaries

Let be a 3-dimensional Minkowski space with the scalar product and Lorentz cross-product defined as for every vector and in .

A vector of is said to be spacelike if or , timelike if and lightlike or null if and . A timelike or lightlike vector in is said to be causal. Let be a smooth curve in , where is an interval in . We call spacelike, timelike, or lightlike curve if the tangent vector at any point is spacelike, timelike, or lightlike, respectively.

Let be an open interval and a plane curve lying in a plane of and a straight line in which does not intersect with the curve . A surface of revolution with axis in is defined to be invariant under the group of motions in , which fixes each point of the line [9]. Because the present paper discusses the case of lightlike axis, without loss of generality, we may assume that the axis is the line spanned by vector in the plane .

So, we choose the line spanned by the vector as axis and express the suppose curve as follows: where is a smooth positive function and is a smooth function such that . Then, the surface of revolution with such axis may be given by

Now, let us consider the Gauss map on a surface in . The map , which sends each point of to the unit normal vector to at that point, is called the Gauss map of surface . Here, denotes the sign of the vector field and is a 2-dimensional space form as follows:

A surface is called minimal if and only if its mean curvature is zero. As de Woestijne ([10]) proved, we have the following theorems.

Theorem 1 (see [10]). Every minimal, spacelike surface of revolution is congruent to a part of one of the following surfaces:(1)a spacelike plane;(2)the catenoid of the 1st kind;(3)the catenoid of the 2nd kind;(4)the surface of Enneper of the 2nd kind.

Theorem 2 (see [10]). Every minimal, timelike surface of revolution is congruent to a part of one of the following surfaces:(1)a Lorentzian plane;(2)the catenoid of the 3rd kind;(3)the catenoid of the 4th kind;(4)the catenoid of the 5th kind;(5)the surface of Enneper of the 3rd kind.

Now, we consider some examples of surfaces of revolution which are mentioned in our theorems.

Example 1 (The surface of Enneper of the 2nd kind is shown in Figure 1). The surface of Enneper of the 2nd kind is parameterized by for . Then, the components of the first and the second fundamental forms are given by So, the mean curvature on the surface is Therefore, the surface of Enneper of the 2nd kind is minimal.

130495.fig.001
Figure 1: The surface of Enneper of the 2nd kind.

Example 2 (The surface of Enneper of the 3rd kind is shown in Figure 2). The surface of Enneper of the 3rd kind is parameterized by for . Then, the components of the first and the second fundamental forms are given by So, the mean curvature on the surface is Therefore, the surface of Enneper of the 3rd kind is minimal.

130495.fig.002
Figure 2: The surface of Enneper of the 3rd kind.

Example 3 (The de Sitter pseudosphere is shown in Figure 3). The de Sitter pseudosphere with radius can be expressed as Then, its Gauss map and Laplacian are given by By a straight computation, we get which means that is, the de Sitter pseudosphere satisfies condition (1).

130495.fig.003
Figure 3: The de Sitter pseudosphere.

Example 4 (The hyperbolic pseudosphere is shown in Figure 4). The hyperbolic pseudosphere with radius is parameterized by Then, its Gauss map and Laplacian are given by By a straight computation, we get So, we have that is, the hyperbolic pseudosphere satisfies condition (1).

130495.fig.004
Figure 4: The future hyperbolic pseudosphere.

3. The Surface of Revolution with Lightlike Axis

In this section, we will classify the surfaces of revolution with lightlike axis in that satisfy condition (5).

Theorem 3. The only surfaces of revolution with lightlike axis in , whose Gauss map satisfies are locally the surface of Enneper of the 2nd kind, the surface of Enneper of the 3rd kind, the de Sitter pseudosphere, and the hyperbolic pseudosphere.

Proof. Let be a surface of revolution with lightlike axis as (9); then we may assume that the profile curve is of unit speed; thus Without lost of generality, we assume that and give a detailed proof just for the case .
Then, we may put for the smooth function . Using the natural frame of defined by we obtain the components of the first and the second fundamental forms of the surface as follows: where Gauss map is defined by , , .
So, the matrix is composed by second fundamental form as follows: Since makes Laplacian degenerate, so we can assume that for every . Then, the mean curvature on is given by By a straightforward computation, the Laplacian of the second fundamental form on with the help of (2), (27), and (29) turns out to be Accordingly, we get
By the assumption (25) and the above equation, we get the following system of differential equations: where denote the components of the matrix given by (25).
In order to prove the theorem, we have to solve the above system of ordinary differential equations. So, we get three systems of ODE, equivalently: From (35), we easily deduce that and . We put and . Therefore, the matrix satisfies Then, three systems (35) now reduce to the following equations:
By the computation (37) (38) and using , , , , , and , we easily get On the other hand, substituting and into (39) equivalently, we get the following equation: Now, we discuss five cases according to the constants and .
Case 1  . In this case, we easily get , which implies that the mean curvature vanishes identically because of (31). Therefore, the surface is minimal; from Theorem 1 it is the surface of Enneper of the 2nd kind. Furthermore, a surface of Enneper of the 2nd kind satisfies the condition (25).
Case 2 . By (40), we get Differentiating (42) with respect to , we have Substituting (42) and (43) into (41), we get from which Furthermore, (45) together with (42) becomes ; that is, On the other hand, by (27), (45), and (46), we have Then, the surface has the following expression: where , , . From this, we easily get
This equation means that the surface is contained in the hyperbolic pseudosphere centered at with radius . Also, the hyperbolic pseudosphere satisfies condition (25).
Case 3  . In this case, (40) becomes ; that is, and thus Substituting (50) and (51) into (41), we get where we put Differentiating (52) and using (50), we find where Combining (52) and (54), we show that where , .
Differentiating once again this equation and using the same algebraic techniques above, we find the following trigonometric polynomial in and satisfying where , , and are nonzero coefficients of the function . Since this polynomial is equal to zero for every , all its coefficients must be zero. Thus, we have , which is a contradiction. Consequently, there are no surfaces of revolution with lightlike axis in this case.
Case 4  . In this case, (40) becomes ; that is, and thus Substituting (58) and (59) into (41), we get where we put
Differentiating (60) and using (58), we find where Combining (60) and (62), we show that where , .
Differentiating once again this equation and using the same method above, we find the following trigonometric polynomial in and satisfying where , , and are nonzero coefficients of the function . Since this polynomial is equal to zero for every , all its coefficients must be zero. Thus, we have , which is a contradiction. Consequently, there are no surfaces of revolution with lightlike axis.
Case 5   . In this case, (40) is unchanged; that is, and thus Substituting (66) and (67) into (41), we get where we put
Differentiating (68) and using (66), we find where
Combining (68) and (70), we show that where , .
Differentiating once again this equation and using the same algebraic techniques above, we find the following trigonometric polynomial in and satisfying where where are the known polynomials in and . Since this polynomial is equal to zero for every , all its coefficients must be zero. Therefore, , which is a contradiction. Consequently, there are no surfaces of revolution with lightlike axis in this case.
When , we can assume that and . Using the same algebraic techniques as for , we easily prove from theorem (9) that the surfaces of Enneper of the 3rd kind and the de Sitter pseudosphere satisfy condition (25). This completes the proof.

Acknowledgments

The authors were supported by NSF of China (no. 11271063) and NSF of Heilongjiang Institute of Technology (no. 2012QJ19).

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