Classification of space-like translation surfaces in the 3-dimensional Lorentz Heisenberg group H 3

Purpose – In the Lorentz Heisenberg space H3 endowed with flat metric g3, a translation surface is parametrized by r ( x , y ) 5 γ 1 ( x )* γ 2 ( y ), where γ 1 and γ 2 are two planar curves lying in planes, which are not orthogonal. In this article, we classify translation surfaces in H3, which satisfy some algebraic equations in terms of the coordinate functions and the Laplacian operator with respect to the first fundamental form of the surface. Design/methodology/approach – In this paper, we classify some type of space-like translation surfaces of H3 endowed with flat metric g3 under the condition Δ r i 5 λ i r i . We will develop the system which describes surfacesoftypefiniteinH3.Forsolvethesystemthusobtained,wewillusethecalculationvariational.Finally,wewilltrytogiveperformancesgeometricsurfacesthatmeettheconditionimposed. Findings – Classification of six types of translation surfaces of finite type in the three-dimensional Lorentz Heisenberg group H3. Originality/value – The subject of this paper lies at the border of geometry differential and spectral analysis on manifolds. Historically, the first research on the study of sub-finite type varieties began around the 1970 by B.Y.Chen. The idea was to find a better estimate of the mean total curvature of a compact subvariety of a Euclidean space. In fact, the notion of finite type subvariety is a natural extension of the notion of a minimal subvariety or surface, a notion directly linked to the calculation of variations. The goal of this work is the classificationofsurfacesinH3,inotherwordsthesurfaceswhichsatisfythecondition/Delta(ri) 5 /Lambda(ri), such that the Laplacian is associated with the first,


Introduction
A Euclidean submanifold is said to be of finite Chen type if its coordinate functions are finite sum of eigenfunctions of its Laplacian [1]. Chen posed the problem of classifying the finite type surfaces in the three-dimensional Euclidean space E 3 . Further, the notion of finite type can be extended to any function on a submanifold of a Euclidean space or a pseudo-Euclidean space.
A well known result, due to Takahshi [2], states that the minimal surfaces and the spheres are the only surfaces in E 3 satisfying the condition.
Δr ¼ λr; λ ∈ R In [3] Ferrandez, Garay and Lucas proved that the surfaces of E 3 satisfying where H denotes the mean curvature vector field, are either minimal, or an open piece of a sphere or of a right circular cylinder.
In [4], F. Dillen, J. Pas and L. Verstraelen proved that the only surfaces in E 3 satisfying Δr ¼ Ar þ B; A ∈ M 3 ðRÞ; B ∈ M 3;1 ðRÞ are the minimal surfaces, the spheres and the circular cylinders.
In [5], Baba-Hamed, Bekkar and Zoubir studied coordinate finite type translation surface in a three-dimensional Minkowski Space.
In [6], Yoon classified coordinate finite type translation surfaces in a three-dimensional Galilean space.
In [7], Bekkar and Senoussi studied the translation surfaces in the three-dimensional Euclidean and Lorentz-Minkowski spaces under the condition In [8][9][10] Bukcu, Karacan and Yoon classified translation surfaces of type 1 and type 2 that satisfy the condition Δ I ; II ; III x i ¼ λ i x i in the three-dimensional simply isotropic space.
Recently, in [11] Cakmak, Karakan, Kiziltug and Yoon studied the translation surfaces in three-dimensional Galilean space under the condition.
In [12,13] Rahmani and Rahmani has showed that modulo an automorphism of the Lie algebra, the three-dimensional Lorentz Heisenberg group H 3 has the following classes of leftinvariant Lorentz metrics: They proved that the metrics g 1 , g 2 , g 3 are non-isometrics and that g 3 is flat. Let r: M 2 → H 3 be an isometric immersion of surface in H 3 . Then r is said to be semi-Riemannian surface in H 3 if the induced metric on M 2 is non-degenerate. In that case this metric is Riemannian or Lorentzian and the surface is said to be space-like or time-like, respectively.
A space-like surface is said to be maximal if H 5 0. A time-like surface is said to be extremal surface (or minimal surface) if H 5 0.
In this paper, we classify space-like translation surfaces of H 3 endowed with flat metric g 3 under the condition.
Indeed this study is closely linked to that of result, due to Dillen et al. [14] where they proved that a ruled surface M 2 in three-dimensional Minkowski space R 3 1 is of finite type if and only if M 2 is minimal, or M 2 is a part of a circular cylinder, or M 2 is a part of a hyperbolic cylinder, or M 2 is an isoparametric surface with null rules.

The Lorentz Heisenberg group H 3
In this paragraph we shortly recall that the Heisenberg group H 3 is a Lie group which is diffeomorphic to R 3 [15] is defined as ðx; y; zÞ * ð x; y; zÞ ¼ ðx þ x; y þ y; z þ z À x yÞ The identity of the group is 0; 0; 0 ð Þand the inverse of x; y; z ð Þis given by −x; − y; − xy − z ð Þ . The left invariant Lorentz metric on H 3 is The following set of left-invariant vector fields forms an pseudo-orthonormal basis for corresponding Lie-algebra The characterizing properties of this algebra are the following commutation relations: ¼0; e 3 ; e 1 ½ ¼e 2 À e 3 ; e 2 ; e 1 ½ ¼e 2 À e 3 : with g 3 e 1 ; e 1 ð Þ¼1; g 3 e 2 ; e 2 ð Þ¼1; g 3 e 3 ; e 3 ð Þ¼−1: If ∇ is the Levi-Civita connection and R is the curvature tensor of ∇, we have

The Beltrami formula
We recall that a translation surfaces M 2 γ 1 ; γ 2 ð Þin the three-dimensional Lorentz Heisenberg group H 3 is a surface parametrized by where γ 1 and γ 2 are any generating curves in R 3 . M 2 ; r À Á is said to be of finite Chen-type k if the position vector r admits the following spectral decomposition where r i are H 3 -valued eigenfunctions of the Laplacian of M 2 ; r ð Þand Δr i 5 λ i r i , λ i ∈ R, i 5 1, 2, . . ., k [1]. If λ i are different, then M 2 is said to be of k-type.
The Laplacian Δ on M 2 is given by is the matrix consisting of components of the induced metric on M 2 , and is the inverse matrix of G and D 5 det G. We set Þ is a function of class C 2 . We recall that the Heisenberg space endowed with a flat Lorentzian metric has the same properties as the Minkowski space, however it is well known that the position vector of a surface in a pseudo-Euclidean space is twice the mean curvature vector [1]. So we have the following result: Theorem 1. A Beltrami formula in ðH 3 ; g 3 Þ is given by the following: where Δ is the Laplacian of the surface and H is the mean curvature vector field of M 2 .
3. Translation surfaces in ðH 3 ; g 3 Þ In the Lorentz Heisenberg space H 3 , a translation surface is parametrized by r x; y ð Þ ¼ γ 1 x ð Þ * γ 2 y ð Þ, where γ 1 and γ 2 are two planar curves lying in planes, which are non-orthogonal and * denotes the group operation of H 3 .
We distinguish six types of translation surfaces in H 3 .

Surfaces of type 1
Let us first consider a translation surface M 2 parametrized by rðx; yÞ ¼ x; 0; uðxÞ ð Þ * ð0; y; vðyÞÞ ¼ ðx; y; uðxÞ þ vðyÞ À xyÞ; where u and v are two smooth functions. Thus, the basis of the tangent space T p M 2 is Therefore the coefficients of the first and second fundamental form are where N is a unit vector field normal on M 2 , which satisfies the following system The normal unit vector field N to M 2 is given by: and the mean curvature vector field H is given by the formula: where H is, the mean curvature of the surface M 2 . So the mean curvature of the surface M 2 parametrized by (x, y, u(x) þ v(y) À xy) is given by

Surfaces of type 2
Now the translation surface M 2 is parametrized by rðx; yÞ ¼ 0; y; vðyÞ ð Þ * ðx; 0; uðxÞÞ ¼ ðx; y; uðxÞ þ vðyÞÞ; (3.6) where u and v are two smooth functions. Therefore Therefore the coefficients of the first fundamental form are (3.8) The normal unit vector field N to M 2 is given by: Then the coefficients of the second fundamental form of M 2 are We follow the same steps as the previous type to calculate the mean curvature of the translation surface M 2 . We obtain

Surfaces of type 3
The translation surface M 2 is parametrized by rðx; yÞ ¼ x; 0; uðxÞ ð Þ * ðvðyÞ; y; 0Þ ¼ ðx þ vðyÞ; y; uðxÞ À xyÞ; (3.11) where u and v are two smooth functions. Therefore whereby the coefficients of the first fundamental form are We have ∇ rx r x ¼ u xx e 2 À u xx e 3 ; ∇ rx r y ¼ 0; ∇ ry r y ¼ v yy À 1 ð Þe 1 þ v y e 2 À v y e 3 : (3.13) The normal unit vector field N to M 2 is given by: Then the coefficients of the second fundamental form of M 2 are We follow the same steps as the previous types to calculate the main curvature of the translation surface M 2 , we obtain

Surfaces of type 4
The translation surface M 2 is parametrized by rðx; yÞ ¼ vðyÞ; y; 0 ð Þ * ðx; 0; uðxÞÞ ¼ ðvðyÞ þ x; y; uðxÞÞ; (3.16) where u and v are two smooth functions. Therefore Thus, the coefficients of the first and second fundamental form are We have ∇ rx r x ¼ u xx e 2 À u xx e 3 ; ∇ rx r y ¼ e 2 À e 3 ; ∇ ry r y ¼ v yy À 1 ð Þe 1 þ 2v y e 2 À 2v y e 3 : (3.18) The normal unit vector field N to M 2 is given by: Then the coefficients of the second fundamental form of M 2 are We follow the same steps as the previous types to calculate the main curvature of the translation surface M 2 , we obtain

Surfaces of type 5
In this case, the translation surface M 2 is parametrized by rðx; yÞ ¼ vðyÞ; y; 0 ð Þ * ðx; 0; uðxÞÞ ¼ ðy; vðyÞ þ x; uðxÞÞ; (3.21) where u and v are two smooth functions. Therefore Hence the coefficients of the first and second fundamental form are We have ∇ rx r x ¼ −e 1 þ u xx e 2 À u xx e 3 ; ∇ rx r y ¼ −v y e 1 þ e 2 À e 3 ; ∇ ry r y ¼ − v y ð Þ 2 e 1 þ 2v y þ yv yy ð Þ e 2 À 2v y þ y À 1 ð Þv yy ð Þ e 3 : (3.23) The normal unit vector field N to M 2 is given by: The coefficients of the second fundamental form of M 2 then are As in the previous types, we obtain

Classification of translation surfaces in H 3
We have ∇ rx r x ¼ −e 1 þ u xx e 2 À u xx e 3 ; ∇ rx r y ¼ −v y e 1 ; ∇ ry r y ¼ − v y ð Þ 2 e 1 þ 2v y þ yv yy ð Þ e 2 À 2v y þ yv yy À v yy ð Þ e 3 : (3.28) The normal unit vector field N to M 2 is given by: Then the coefficients of the second fundamental form of M 2 are As above, we obtain 4. Translation surfaces in Lorentz Heisenberg space ðH 3 ; g 3 Þ satisfying Δr i = λ i r i This section is devoted to classify the translation surfaces of type 1, type 2, type 3, type 4, type 5 and type 6 that satisfy the condition Δr i 5 λ i r i , λ i ∈ R. The result is: Let M 2 be a translation surface of type 1, type 2, type 3, type 4, type 5 or type 6 in the three-dimensional Lorentz Heisenberg space ðH 3 ; g 3 Þ. Then M 2 satisfies the condition Δr i 5 λ i r i , i 5 1, 2, 3, λ i ∈ R, if and only if M 2 has zero mean curvature.
Proof of Theorem 2.
In the following we will prove Theorem 2.

Surfaces of type 1
By (2.2) and (3.4), the Laplacian operator Δr of r can be expressed as Next, suppose M 2 satisfies condition (1.1). Then, from (3.1) and (4.1), we obtain the following system of ordinary differential equations.
Therefore, the problem of classifying the translation surfaces M 2 satisfying (1.1) is reduced to the integration of the system (4.2), (4.3) and (4.4) of ordinary differential equations. Next we study it according to the constants λ 1 , λ 2 and λ 3 . Combining Equations (4.3) and (4.4) yields Case 1: λ 1 λ 2 5 0. Then H 5 0 and λ 3 5 0. Thus, the surface M 2 is minimal. Case 2: λ 1 λ 2 ≠ 0. Substituting (4.5) into (4.6), we get Taking the partial derivative of (4.7) with respect to x gives The left-hand side in (4.8) is either a constant or a function of x, while other side is a function of y. That is not possible.

Surfaces of type 2
We follow the same steps as the previous type. By (2.2) and (3.9), the Laplacian Δr of r can be expressed as Suppose that M 2 satisfies the condition (1.1). Then, from (3.6) and (4.9), we obtain the following system of ordinary differential equations.
Therefore, the problem of classifying the translation surfaces M 2 satisfying (1.1) is reduced to the integration of the system (4.10), (4.11) and (4.12) of ordinary differential equations.
Applying similar algebraic methods, used in the case of surface of type 1, we will study this system according to the values of the constants λ 1 , λ 2 , λ 3 . By combining Equations (4.11) and (4.12) we obtain Case 1: λ 1 λ 2 5 0.
(2) If λ 2 ≠ 0, from (4.13), we obtain u x 5 0. Substituting (3.10) into (4.14), with Taking the partial derivative of (4.15) with respect to x gives The left-hand side in (4.16) is either a constant or a function of y, whereas other side is a function of x. That is not possible. Case 2: λ 1 λ 2 ≠ 0. Substituting (4.13) into (4.14), we get Taking the partial derivative of (4.17) with respect to x gives The left-hand side in (4.18) is a constant or a function of x, whilst other side is a function of y.
That is not possible.

Surfaces of type 3
By following the same steps as the previous types, by (2.2) and (3.14), we obtain Suppose that M 2 satisfies condition (1.1). Then, from (3.11) and (4.19), we obtain the following system of ordinary differential equations.
Combining Equations (4.21) and (4.22) we have Case 1: λ 1 λ 2 5 0. Then H 5 0 and λ 3 5 0. Thus, the surface M 2 is minimal. Case 2: λ 1 λ 2 ≠ 0. Substituting (4.23) into (4.24), we get Taking the partial derivative of (4.25) with respect to x gives We have an identity of two functions, one depending only on x and the other one depending only on y. That is not possible.

Surfaces of type 4
By (2.2) and (3.19), we obtain Suppose that M 2 satisfies the condition (1.1). Then, from (3.16) and (4.27), we obtain the following system of ordinary differential equations.
(1) If λ 1 λ 2 5 0. from (4.52), we have u x v y À yv y þ x ¼ 0 (4.57) Taking the partial derivative of (4.57) with respect to x, we obtain Then both sides have to equal a nonzero constant, namely Which implies that v y ¼ − 1 a . Considering it into (4.57) gives − u x a þ x ¼ − y a (4.59) The left-hand side in (4.59) is either constant or a function of x while other side is a function of y. That is impossible.

AJMS
Taking partial derivative of (4.61) with respect to x and y leads to u xxx v yy ¼ 0 We have two situations: (1) u xxx 5 0 i.e.
(2) v yy 5 0, i.e. v ¼ c 1 y þ c 2 : Then (4.61) reduces to 2x À u xx c 1 À λ 3 The left-hand side in (4.63) is either constant or a function of x while other side is a function of y. That is impossible.

Conclusion
In this work, we give another characterization of translation surfaces in the Lorentz Heisenberg space H 3 equipped with a flat metric. This is done by showing that these translation surfaces satisfy Condition Δr i 5 λ i r i , i 5 1, 2, 3, λ i ∈ R, if and only if they have zero mean curvature.