Journal of Applied Mathematics

Volume 2014 (2014), Article ID 154294, 5 pages

http://dx.doi.org/10.1155/2014/154294

Research Article

Some Surfaces with Zero Curvature in

Dae Won Yoon

Department of Mathematics Education and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea

Received 24 December 2013; Accepted 26 February 2014; Published 24 March 2014

Academic Editor: Chong Lin

Copyright © 2014 Dae Won Yoon. 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 defined as graph of the function in the product space . In particular, we completely classify flat or minimal surfaces given by , where and are smooth functions.

1. Introduction

Homogenous geometries have main roles in the modern theory of manifolds. Homogenous spaces are, in a sense, the nicest examples of Riemannian manifolds and have applications in physics [1]. To underline their importance from the mathematical point of view we roughly cite the famous Thurston conjecture. This conjecture asserts that every compact orientable 3-dimensional manifold has a canonical decomposition into pieces, each of which admits a canonical geometric structure from among the eight maximal simple connected homogenous Riemannian 3-dimensional geometries [2]. The Riemannian product space is one of the eight model spaces.

Constant mean curvature and constant Gaussian curvature surfaces are one of the main objects which have drawn geometers' interest for a very long time. Recently, the study of the geometry of surfaces in is growing very rapidly, and the interest is mainly focused on minimal and constant mean curvature surfaces [3–9].

The purpose of this paper is to study surfaces defined as graph of the function in the product space . In Sections 4 and 5 we classify minimal and flat surfaces defined as , where and are smooth functions.

2. Preliminaries

Let be the upper half plane model of the hyperbolic plane endowed with the metric, of constant Gaussian curvature , given by

The hyperbolic space , with the group structure derived by the composition of proper affine maps, is a Lie group and the metric is left invariant. Therefore, the product space is a Lie group with the left invariant product metric

On the other hand, an orthonormal basis of left invariant vector fields on is

with the only nontrivial commutator relation . It follows that the Levi-Civita connection of is expressed as

For any vectors and in the cross-product is defined by

3. Graphs in

Let us consider a surface parametrized by

where is a domain in and is a smooth function. Then is a surface defined as graph of the function defined on . In this case, we have

It follows that the coefficients of the first fundamental form of are given by

Also, the unit normal vector field to is given by

where

By a straightforward calculation, we obtain

which imply that the coefficients of the second fundamental form of are

Thus, from (8) and (12) the Gaussian curvature and the mean curvature are, respectively,

Proposition 1. Let be a surface defined as graph of the function . Then is a minimal surface if and only if

Proposition 2. Let be a surface defined as graph of the function . Then is flat if and only if

Remark 3. Some examples are satisfying the ODE (14) studied in [7]. Also, examples in Lorentz product space can be found in [10].

4. Minimal Surfaces Defined by

Let be a surface in parametrized by

for all , where and are smooth functions. We suppose that is a minimal surface. Then, from (14) we have the following minimal surface equation:

In order to solve it, divide first by ; then we get

for all . Differentiating with respect to , we obtain

First of all, we suppose that on an open interval; that is, . In this case, from (17) we obtain

We put . Then the last equation can be written as

Its general solution is given by

From this, we thus have

where .

Now, we assume that on an open interval, and divide (19) by . It follows that

Hence we deduce the existence of a real number such that

Let us distinguish the following cases according to .

Case 1. If , then and . It follows that . If , then . In this case, from (17) we obtain ; it is a contradiction. If , then we get . In such case, (17) is polynomial equation on and . From the coefficients of and the constant term we have and , which imply and . It is a contradiction.

Case 2. If , then from the first equation in (25) we have

where . Let

be any solution of (26), where is a smooth function. Then (26) can be rewritten as

We put . Then, we have

We again put . In this case the above equation becomes

and its general solution is given by

Thus, we get

After an integration, we can find

where . By combining (27) and (33), we thus have

Now, we consider the second equation in (25). Since , we yield

We put . Then, the above equation becomes

Since , without loss of generality we take or .

Subcase i. Let . We do the change

where is a nonzero smooth function. Then, (36) can be rewritten as the form

Thus, its general solution is

where . So, and from its integration we can obtain

where .

Subcase ii. Let . We put

where is a nonzero smooth function. Then, (36) becomes

and its general solution is given by

where . Thus, we have

where . The surface given by (34) and (44) is shown in Figure 1.

Figure 1: A minimal surface defined by (34) and (44).

Consequently, we have the following.

Theorem 4. Let be a surface defined as graph of the function . If is a minimal surface, then is parametrized as

where

(1) and with , or(2) and with .

5. Flat Surfaces Defined by

Let be a surface defined by (16). Assume that is a flat surface. Then, from (15) we have the following flat surface equation:

In order to solve it, differentiating with respect to , we have

Thus, there exists a nonzero real number such that

From the first equation in (48), we get

where . We put , and it follows that we yield

From this, the general solution is

where . We can assume that . From the last equation we can easily obtain (see Figure 2)

where .

Figure 2: A flat surface defined by (52) and (55).

In order to solve the second equation in (48), divide by and put . Then, we get

and its general solution is given by

where . From this, we thus obtain (see Figure 2)

where .

As a conclusion, we have the following.

Theorem 5. Let be a surface defined as graph of the function . If is a flat surface, then is parametrized as

where and with .

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This paper was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A1A2003994).

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