SURFACES AS GRAPHS OF FINITE TYPE IN H × R

In this paper, we prove that ∆X = 2H where ∆ is the Laplacian operator, r = (x, y, z) the position vector field and H is the mean curvature vector field of a surface S in H2×R and we study surfaces as graphs in H2 × R which has finite type immersion.

The Riemannian manifold (M, g) is called homogeneous if for any x, y ∈ M there exists an isometry φ : M → M such that y = φ(x). The two and three-dimensional homogeneous geometries are discussed in detail in [6] .
A Euclidean submanifold is said to be of finite Chen-type if its coordinate functions are a finite sum of eigenfunctions of its Laplacian [3]. B. Y. Chen posed the problem of classifying the finite type surfaces in the 3-dimensional Euclidean space E 3 . Further, the notion of finite type can be extended to any smooth function on a submanifold of a Euclidean space or a pseudo-Euclidean space.
Let S be a 2-dimensional surface of the Euclidean 3-space E 3 . If we denote by r, H and ∆ the position vector field, the mean curvature vector field and the Laplace operator of S respectively, then it is well-known that [3] (1.1) ∆r = −2H.
A well-known result due to Takahashi states that minimal surfaces and spheres are the only surfaces in E 3 satisfying the condition ∆r = λr for a real constant λ. From (1.1), we know that minimal surfaces and spheres also verify the condition Equation (1.1) shows that S is a minimal surface of E 3 if and only if its coordinate functions are harmonic.
In 2012, M. Bekkar and B. Senoussi [1] studied the translation surfaces in the 3-dimensional Euclidean and Lorentz-Minkowski space under the condition where ∆ III denotes the Laplacian of the surface with respect to the third fundamental form III.
A surface S in the Euclidean 3-space E 3 is called minimal when locally each point on the surface has a neighborhood which is the surface of least area with respect to its boundary [5]. In 1775, J. B. Meusnier showed that the condition of minimality of a surface in E 3 is equivalent with the vanishing of its mean curvature function, H = 0.
Let z = f (x, y) define a graph S in the Euclidean 3-space E 3 . If S is minimal, the function f satisfies which was obtained by J. L. Lagrange in 1760.
In 1835, H. F. Scherk studied translation surfaces in E 3 and proved that, besides the planes, the only minimal translation surfaces are given by where λ is a non-zero constant. In 1991, F. Dillen, L. Verstraelen and G. Zafindratafa. [4] generalized this result to higher-dimensional Euclidean space.
In 2015, D. W. Yoon [8] studied translation surfaces in the product space H 2 × R and classified translation surfaces with zero Gaussian curvature in H 2 × R.
In 2019, B. Senoussi, M. Bekkar [7] studied translation surfaces of finite type in H 3 and Sol 3 and the authors gived some theorems.
where γ 1 and γ 2 are any generating curves in R 3 . Since the multiplication * is not commutative.
In this work we study the surfaces as graphs of functions ϕ = f (s, t)) in H 2 × R satisfy the condition

Preliminaries
Let H 2 be represented by the upper half-plane model {(x, y) ∈ R | y > 0} equipped with the metric The space H 2 , with the group structure derived by the composition of proper affine maps, is a Lie group and the metric g H is left invariant.
Therefore, the product space H 2 × R is a Lie group with the left invariant product metric we can define the multiplication law on H 2 × R as follows (x, y, z) * (x,ȳ,z) = (yx + x, yȳ, z +z).
The left identity is (0, 1, 0) and the inverse of ( where is the orthonormal coframe associated with the orthonormal frame The corresponding Lie brackets are The Levi-Civita connection ∇ of H 2 × R is given by Let S be an immersed surface in H 2 ×R given as the graph of the function z = f (x, y). Hence, the position vector is described by r(x, y) = (x, y, f (x, y)) and the tangent vectors r x = ∂r ∂x and r y = ∂r ∂y in terms of the orthonormal frame (e 1 , e 2 , e 3 ) are described by The immersion (S, r) is said to be of finite Chen-type k if the position vector X admits the following spectral decomposition For the matrix G = (g ij ) consisting of the components of the induced metric on S, we denote by G −1 = (g ij ) (resp. D = det(g ij )) the inverse matrix (resp. the determinant) of the matrix (g ij ). The Laplacian ∆ on S is, in turn, given by y)) is a function of class C 2 then we set ∆r = (∆r 1 , ∆r 2 , ∆r 3 ).
3. Surfaces as graphs of finite type in H 2 × R Let S be a graph of a smooth function We consider the following parametrization of S r(x, y) = (x, y, f (x, y)), (x, y) ∈ Ω.
Theorem 3.1. A Beltrami formula in H 2 × R is given by the following: where ∆ is the Laplacian of the surface and H is the mean curvature vector field of S.
Proof. A basis of the tangent space T p S associated to this parametrization is given by The coefficients of the first fundamental form of S are given by The unit normal vector field N on S is given by To compute the second fundamental form of S, we have to calculate the following r xx = ∇ rx r x = 1 y 2 e 2 + f xx e 3 , r xy = ∇ rx r y = ∇ ry r x = − 1 y 2 e 1 + f xy e 3 , (3.1) r yy = ∇ ry r y = − 1 y 2 e 2 + f yy e 3 .
So, the coefficients of the second fundamental form of S are given by Thus, the mean curvature H of S is given by By (2.3), the Laplacian operator ∆ of r can be expressed as By a straightforward computation, the Laplacian operator ∆ of r with the help of (3.1) and (3.2) turns out to be thus we get where H is the mean curvature vector field of S.
S is a minimal surfaces in H 2 × R if and only if its coordinate functions are harmonic .

Surfaces as graphs in
Let S be an immersed surface in H 2 × R given as the graph of function z = f (x, y). Hence, the vector position is described by r(x, y) = (x, y, f (x, y)).
We have where r x = ∂r ∂x , r y = ∂r ∂x , and f x = ∂f ∂x , f y = ∂f ∂y .
From an earlier results the mean curvature H of S and the unit normal vector field N on S are given by where W = 1 If the vector position on the tangent space T p S is described by r = (x, y, f (x, y)) The equation (1.3) by means of (3.3), (4.1) and (4.2) gives rise to the following system of ordinary differential equations = λ 3 y 2 f. (4.5) ordinary differential equations.
Next we study it according to the constants λ 1 , λ 2 and λ 3 . Case 1. Let λ 3 = 0. In this case the system (4.3), (4.4) and (4.5) is reduced equivalently to The equation (4.8) implies that the mean curvature H is identically zero. Thus, the surface S is minimal; and we get also λ 1 = λ 2 = 0.
A differentiation with respect to y gives λ 3 y = 0, this implies that λ 3 = 0 and from (4.8) we get the mean curvature H is identically zero. From (4.6) and (4.7) we obtain λ 1 = λ 2 = 0, which leads to a contradiction.
Therefore, we have the following theorem, Theorem 4.1. Let S be a surface as graph of function parametrized by r(x, y) = (x, y, f (x, y)) in H 2 × R Then, S satisfies the equation ∆r i = λ i r i , λ i ∈ R if and only if S is minimal surfaces or parametrized as S : r(x, y) = x, y, ± λ 2 λ 3 ln 1 y 2 + c ; λ 2 = 0, λ 3 = 0, c ∈ R.

Conflicts of Interest:
The author(s) declare that there are no conflicts of interest regarding the publication of this paper.