Abstract and Applied Analysis

Volume 2014 (2014), Article ID 564245, 9 pages

http://dx.doi.org/10.1155/2014/564245

Research Article

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

Minghao Jin1,2 and Donghe Pei¹

¹School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China

²Department of Mathematics, Heilongjiang Institute of Technology, Harbin 150050, China

Received 16 December 2013; Accepted 27 January 2014; Published 9 March 2014

Academic Editor: Jaume Giné

Copyright © 2014 Minghao Jin and Donghe Pei. 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.

We study surfaces of revolution with a nonlightlike axis in 3-dimensional Minkowski space and classify such surfaces in terms of the Gauss map that satisfies the condition , with Λ being a real matrix. Furthermore, this paper completes the classification problem of surfaces of revolution in Minkowski 3-space given by Jin et al. (2013).

1. Introduction

The notion of finite type immersions introduced by Chen in [1] has been widely used in studying submanifolds of Euclidean and pseudo-Euclidean spaces. Also, such a notion can be extended to smooth maps on submanifolds. Among them the Gauss map is a very useful and extensively used to deal with submanifolds [2].

Let be a connected surface in Euclidean 3-space , and let be its Gauss map. It is well known [3] that has constant mean curvature if and only if , with being the Laplace operator on corresponding to the induced metric on from . As a special case, one can consider Euclidean surfaces whose Gauss map is an eigenfuction of the Laplacian; that is,

On the other hand, Chen and Piccinni [2] proved that the only compact surface in a Euclidean 3-space satisfying (1) is a sphere. Jang [4] studied that an orientable, connected surface in a Euclidean 3-space satisfying (1) is a sphere or a circular cylinder. On the generalization of (1), Dillen et al. [5] studied surfaces of revolution in a Euclidean 3-space such that their Gauss map satisfies the condition

and proved that such surfaces are part of the planes, the spheres, and the circular cylinders.

As a Lorentz version of Dillen et al.'s result, the author proves the following [6].

Theorem 1. The only spacelike or timelike surfaces of revolution in whose Gauss map satisfies (2) are locally the following spaces:

(1) and if ,(2) and if .

Recently, we studied [7] such surface and its Gauss map satisfy the following condition in Minkowski space:

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 .

The main purpose of this note is to complete classification of surfaces of revolution in whose Gauss map satisfies the condition . Actually, we will show the de Sitter pseudosphere, the hyperbolic pseudosphere, and five kinds of catenoid satisfying the above condition.

2. Preliminaries

Let be a 3-dimensional Minkowski space with the scalar product and Lorentz cross product defined as

for any vectors 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 [8]. From this we obtain four kinds of surfaces of revolution in . If the axis is timelike (resp., spacelike), then there is a Lorentz transformation by which the axis is transformed to the axis (resp., axis or axis). Hence, without loss of generality, we may consider as the axis of revolution with the axis or axis if is not null. If the axis is null, then we may assume that this axis is the line spanned by vector on the plane .

We now introduce three different types of surfaces of revolution in .

Type 1. The surface of revolution with timelike axis.

Without loss of generality, we choose as the axis. Meanwhile suppose that has a parameter as follows:

where and are smooth functions and . Then the surface of revolution with axis may be given by

Type 2. The surface of revolution with spacelike axis.

Without loss of generality, we choose as the axis, and suppose that curve has a parameter as follows:

where and are smooth functions and . Then the surface of revolution with axis may be given by

or

Type 3. The surface of revolution with lightlike axis.

Without loss of generality, we choose a line spanned by the vector as axis, and suppose that curve has a parameter as follows:

where is smooth positive function and is smooth function such that . Then the surface of revolution with the line spanned by vector as axis may be given by

Here we only consider type 1 and type 2, as for type 3 we have already discussed in [7].

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 mean curvature . Now we consider some examples of minimal surfaces which will be mentioned in theorems.

Example 2 (the catenoid of the 1st kind is shown in Figure 1). A surface of catenoid of the 1st 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 catenoid of the 1st kind is minimal.

Figure 1: The catenoid of the 1st kind.

Example 3 (the catenoid of the 2nd kind is shown in Figure 2). A surface of catenoid 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 catenoid of the 2nd kind is minimal.

Figure 2: The catenoid of the 2nd kind.

Example 4 (the catenoid of the 3rd kind is shown in Figure 3). A surface of catenoid 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 catenoid of the 3rd kind is minimal.

Figure 3: The catenoid of the 3rd kind.

Example 5 (the catenoid of the 4th kind is shown in Figure 4). A surface of catenoid of the 4th 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 catenoid of the 4th kind is minimal.

Figure 4: The catenoid of the 4th kind.

Example 6 (the catenoid of the 5th kind is shown in Figure 5). A surface of catenoid of the 5th 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 catenoid of the 5th kind is minimal.

Figure 5: The catenoid of the 5th kind.

Example 7 (the de Sitter pseudosphere is shown in Figure 6). The de Sitter pseudosphere centered at with radius 1 is parameterized by

Then its Gauss map and Laplacian are given by

By a straight computation, we get

So we have

that is, the de Sitter pseudosphere satisfies condition (3).

Figure 6: The de Sitter pseudosphere.

3. The Surface of Revolution with Timelike Axis

In this section, we will classify the surfaces of revolution with timelike axis in that satisfy condition (3).

Theorem 8. The only surfaces of revolution with timelike axis in whose Gauss map satisfies

are locally the catenoid of the 1st kind, the catenoid of the 3rd kind, the de Sitter pseudosphere, or the hyperbolic pseudosphere.

Proof. Let be a surface of revolution with timelike axis as (7). We may assume that the profile curve is of unit speed; thus

We will give detailed proof just for the case . Then is a spacelike surface and 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 composed by the second fundamental form can be expressed as

Since the surface has no parabolic points, so 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 (4), (35), and (37) turns out to be

Accordingly, we get

By the assumption (33) and the above equation, we get the following system of differential equations:

where () denote the components of the matrix given by (33). In order to prove the theorem, we have to solve the above system of ordinary differential equations. From (42) we easily deduce that and ; that is, the matrix is diagonal. We put and . Then, the system (42) is reduced to the following equations:

By the computation (43) ×  − (44) × , we easily get

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 (39). Therefore, the surface is minimal; from theorem in [9], it is the 1st kind of catenoid. Furthermore, the 1st kind of catenoid satisfies condition (33).

Case 2  . By (45), we get

Differentiating (46) with respect to , we have

Combining (46), (47), and (43), we get

from which

Furthermore, (49) together with (46) becomes ; that is,

On the other hand, by (35) and (50), we get

Then, the surface has the following expression:

Consequently, we have

which means that the surface is contained in the hyperbolic pseudosphere centered at with radius . Also, the hyperbolic pseudosphere satisfies condition (33).

Case 3  , . In this case, (45) becomes ; that is,

and thus

Substituting (54) and (55) into (43), we get

Differentiating the above equation, we have

If we take the differentiation of the equation once again, we get

Since is a positive function and , for every . Therefore, is vanishing identically for every . Hence, we have

It implies that is a part of a Euclidean plane whose points are parabolic. Thus, there is no surface of revolution with timelike axis satisfying this case.

Case 4  , . In this case, (45) becomes ; that is,

and thus

Furthermore, by (43), (60), and (61), we get

where we put

Differentiating (62) and using (60), we find

where

Combining (62) and (64), we can show

where , .

Differentiating once again this equation and using the same algebraic techniques above, we find the following trigonometric polynomial in satisfying

where , , are coefficients as nonzero constant of the function . Since this polynomial is equal to zero for every , all its coefficients must be zero. Thus, we have . So we get a contradiction, and therefore, in this case, there are no surfaces of revolution with timelike axis.

Case 5  (let , , and ). In this case, (45) is unchanged; that is,

and thus

From which, (43) is written as

where

Differentiating (70) and using (68), we find

where

Combining (70) and (72), we have

where .

Hence, by this procedure, (74) is reduced to a linear one with respect to the function . Therefore, if we repeat this method one more time, we can find the following polynomial:

where , , are nonzero constants. Since this polynomial is equal to zero for every , all its coefficients must be zero. Therefore we conclude that ; that is, , which is a contradiction. Consequently, there are no surfaces of revolution with timelike axis in this case.

When , is a timelike surface. In this case, we can assume that and , and using the same algebraic techniques as for easily prove that the 3rd kind of catenoid and the de Sitter pseudosphere satisfy condition (33). This completes the proof.

4. The Surface of Revolution with Spacelike Axis

In this section, we will classify the surfaces of revolution with spacelike axis in that satisfy condition (3).

Theorem 9. The only surfaces of revolution with spacelike axis in whose Gauss map satisfies

are locally the 2nd kind of catenoid, the 4th kind of catenoid, the 5th kind of catenoid, the hyperbolic pseudosphere, or the de Sitter pseudosphere.

Proof. Let be a surface of revolution with spacelike axis as (9). We may assume that the profile curve is of unit speed; thus

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 .

So the matrix composed by the second fundamental form can be expressed as

Since the surface has no parabolic points, so 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 (4), (78), and (80) turns out to be

Accordingly, we get

By the assumption (76) and the above equation, we get the following system of differential equations:

where () denote the components of the matrix given by (76). In order to prove the theorem, we have to solve the above system of ordinary differential equations. From (85) we easily deduce that and ; that is, the matrix is diagonal. We put and . Then, the system (85) is reduced to the following equations:

By the computation (87) ×  − (86) × , we easily get

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 (82). Therefore, the surface is minimal; from theorem in [9], it is the 5th kind catenoid. Furthermore, the 5th kind catenoid satisfies the condition (76).

Case 2  . By (88), we get

Differentiating (89) with respect to , we have

Combining (89), (90), and (86), we get

from which

Furthermore, (92) together with (89) becomes ; that is,

On the other hand, by (78) and (93), we have

Then, the surface has the following expression:

Consequently, we have

which means that the surface is contained in the de Sitter pseudosphere centered at with radius . Also, the de Sitter pseudosphere satisfies condition (76). In the cases of 3, 4, and 5, we will use the same method of Section 3 and easily get that there are no surfaces of revolution with spacelike axis satisfies condition (76).

When the surface of revolution has the expression given by (10), we can similarly prove that the 2nd kind of catenoid, 4th kind of catenoid, the de Sitter pseudosphere, and the hyperbolic pseudosphere satisfy condition (76). This completes the proof.

Conflict of Interests

The authors declare that there is no conflict of interests in this work.

Acknowledgments

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

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