Some Surfaces with Zero Curvature in H 2 × R

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 eightmaximal simple connected homogenous Riemannian 3-dimensional geometries [2]. The Riemannian product space H ×R 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 H × R 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 z = f(x, y) in the product space H × R. In Sections 4 and 5 we classify minimal and flat surfaces defined as f(x, y) = u(x) + V(y), where u(x) and V(y) are smooth functions.


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 H 2 × R 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 H 2 × R is growing very rapidly, and the interest is mainly focused on minimal and constant mean curvature surfaces [3][4][5][6][7][8][9].
The purpose of this paper is to study surfaces defined as graph of the function  = (, ) in the product space H 2 × R. In Sections 4 and 5 we classify minimal and flat surfaces defined as (, ) = () + V(), where () and V() are smooth functions.

Preliminaries
Let H 2 = {(, ) ∈ R 2 |  > 0} be the upper half plane model of the hyperbolic plane endowed with the metric, of constant Gaussian curvature −1, given by The hyperbolic space H 2 , with the group structure derived by the composition of proper affine maps, is a Lie group and the metric  H is left invariant.Therefore, the product space H 2 ×R is a Lie group with the left invariant product metric On the other hand, an orthonormal basis of left invariant vector fields on H 2 × R is with the only nontrivial commutator relation For any vectors  =

Graphs in H 2 × R
Let us consider a surface Σ parametrized by  (, ) = (, ,  (, )) , (, ) ∈ Ω, where Ω is a domain in H 2 and  : Ω → R is a smooth function.Then Σ is a surface defined as graph of the function  defined on Ω ⊂ H 2 .In this case, we have It follows that the coefficients of the first fundamental form of Σ are given by Also, the unit normal vector field  to Σ is given by where By a straightforward calculation, we obtain which imply that the coefficients of the second fundamental form of Σ are Thus, from ( 8) and ( 12) the Gaussian curvature  and the mean curvature  are, respectively, Proposition 1.Let Σ be a surface defined as graph of the function  : Ω ⊂ H 2 → R. Then Σ is a minimal surface if and only if Proposition 2. Let Σ be a surface defined as graph of the function  : Ω ⊂ H 2 → R. Then Σ is flat if and only if Remark 3. Some examples are satisfying the ODE (14) studied in [7].Also, examples in Lorentz product space H 2 × R 1 can be found in [10].

Minimal Surfaces Defined by 𝑓(𝑥,𝑦) = 𝑢(𝑥) + V(𝑦)
Let Σ be a surface in H 2 × R parametrized by  (, ) = (, ,  () + V ()) for all  > 0, where () and V() are smooth functions.We suppose that Σ is a minimal surface.Then, from (14) we have the following minimal surface equation: In order to solve it, divide first by 1 +  2 (V  ) 2 ̸ = 0; then we get for all ,  ∈ Ω. Differentiating with respect to , we obtain First of all, we suppose that   = 0 on an open interval; that is, () =  + , ,  ∈ R. In this case, from (17) we obtain We put V  () = ().Then the last equation can be written as Its general solution is given by From this, we thus have where  1 ∈ R. Now, we assume that   ̸ = 0 on an open interval, and divide (19) by     .It follows that Hence we deduce the existence of a real number  ∈ R such that Let us distinguish the following cases according to .

Case 2. If 𝑘 ̸
= 0, then from the first equation in (25) we have where be any solution of (26), where  is a smooth function.Then (26) can be rewritten as We put  =   .Then, we have We again put  =  2 .In this case the above equation becomes and its general solution is given by Thus, we get After an integration, we can find where  2 ∈ R. By combining ( 27) and (33), we thus have Now, we consider the second equation in (25).Since  > 0, we yield We put  = V  .Then, the above equation becomes Since  ̸ = 0, without loss of generality we take  = 1 or  = −1.
Subcase i.Let  = 1.We do the change where ℎ is a nonzero smooth function.Then, (36) can be rewritten as the form Thus, its general solution is where  1 ∈ R. So,  = (1/) + (1/(ln  +  1 )) and from its integration we can obtain where  2 ∈ R.
Subcase ii.Let  = −1.We put where ℎ is a nonzero smooth function.Then, (36) becomes and its general solution is given by ℎ () = − (ln  +  1 ) , where  1 ∈ R. Thus, we have where  2 ∈ R. The surface given by (34) and ( 44) is shown in Figure 1.Consequently, we have the following.
Theorem 4. Let Σ be a surface defined as graph of the function (, ) = () + V().If Σ is a minimal surface, then Σ is parametrized as where

Flat Surfaces Defined by 𝑓(𝑥,𝑦) = 𝑢(𝑥) + V(𝑦)
Let Σ be a surface defined by ( 16).Assume that Σ 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  1 ∈ R. We put  =   , and it follows that we yield From this, the general solution is where  2 ∈ R. We can assume that  1 = 0. From the last equation we can easily obtain (see Figure 2) where  3 ∈ R.
In order to solve the second equation in (48), divide by  2 and put  = V  .Then, we get and its general solution is given by where  1 ∈ R. From this, we thus obtain (see Figure 2) where  2 ∈ R.
As a conclusion, we have the following.