Some Surfaces with Zero Curvature in
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.
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 . 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 . 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.
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
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.
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 .
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.
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).
W. Thurston, Three-Dimensional Geometry and Topology, vol. 35 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, USA, 1997.View at: MathSciNet