Classification of space-like translation surfaces in the 3-dimensional Lorentz Heisenberg group H3

Medjati, Rafik; Zoubir, Hanifi; Medjahdi, Brahim

doi:10.1108/AJMS-03-2021-0071

Purpose

In the Lorentz Heisenberg space H3 endowed with flat metric g3, a translation surface is parametrized by r(x, y) = γ₁(x)*γ₂(y), where γ₁ and γ₂ 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 conditionΔr_i = λ_ir_i. We will develop the system which describes surfaces of type finite in H3. For solve the system thus obtained, we will use the calculation variational. Finally, we will try to give performances geometric surfaces that meet the condition imposed.

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 classification of surfaces in H3, in other words the surfaces which satisfy the condition/Delta (ri) = /Lambda (ri), such that the Laplacian is associated with the first, fundamental form.

1. 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.

Δ r = λ r, λ \in R

In [3] Ferrandez, Garay and Lucas proved that the surfaces of $E^{3}$ satisfying

Δ H = A H, A \in M_{3} (R)

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

Δ r = A r + B, A \in M_{3} (R), B \in M_{3,1} (R)

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

Δ^{I I I} r_{i} = μ_{i} r_{i}, μ_{i} \in R

In [8–10] Bukcu, Karacan and Yoon classified translation surfaces of type 1 and type 2 that satisfy the condition

Δ^{I, I I, I I I} x_{i} = 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.

Δ^{I I} x_{i} = λ_{i} x_{i}

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 left-invariant Lorentz metrics:

\begin{matrix} g_{1} & = & - d x^{2} + d y^{2} + {(x d y + d z)}^{2} \\ g_{2} & = & d x^{2} + d y^{2} - {(x d y + d z)}^{2} \\ g_{3} & = & d x^{2} + {(x d y + d z)}^{2} - {[(1 - x) d y - d z]}^{2} . \end{matrix}

They proved that the metrics g₁, g₂, g₃ are non-isometrics and that g₃ is flat.

Let r: $M^{2} \to H_{3}$ 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 M² 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 = 0.

A time-like surface is said to be extremal surface (or minimal surface) if H = 0.

In this paper, we classify space-like translation surfaces of $H_{3}$ endowed with flat metric g₃ under the condition.

Δ r_{i} = λ_{i} r_{i}, λ_{i} \in R

(1.1)

Indeed this study is closely linked to that of result, due to Dillen et al. [14] where they proved that a ruled surface M² in three-dimensional Minkowski space $R_{1}^{3}$ is of finite type if and only if M² is minimal, or M² is a part of a circular cylinder, or M² is a part of a hyperbolic cylinder, or M² is an isoparametric surface with null rules.

2. Preliminaries

2.1 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

(x, y, z) * (\bar{x}, \bar{y}, \bar{z}) = (x + \bar{x}, y + \bar{y}, z + \bar{z} - x \bar{y})

The identity of the group is $(0,0,0)$ and the inverse of $(x, y, z)$ is given by $(- x, - y, - x y - z)$ ⁠. The left invariant Lorentz metric on $H_{3}$ is

g_{3} = d x^{2} + {(x d y + d z)}^{2} - {[(1 - x) d y - d z]}^{2}

The following set of left-invariant vector fields forms an pseudo-orthonormal basis for corresponding Lie-algebra

B = \{e_{1} = \frac{\partial}{\partial x}, e_{2} = \frac{\partial}{\partial y} + (1 - x) \frac{\partial}{\partial z}, e_{3} = \frac{\partial}{\partial y} - x \frac{\partial}{\partial z}\}

The characterizing properties of this algebra are the following commutation relations:

[e_{2}, e_{3}] = 0, [e_{3}, e_{1}] = e_{2} - e_{3}, [e_{2}, e_{1}] = e_{2} - e_{3} .

with

g_{3} (e_{1}, e_{1}) = 1, g_{3} (e_{2}, e_{2}) = 1, g_{3} (e_{3}, e_{3}) = - 1 .

If ∇ is the Levi–Civita connection and R is the curvature tensor of ∇, we have

\begin{matrix} \nabla_{e_{1}} e_{1} & = & \nabla_{e_{1}} e_{2} = \nabla_{e_{1}} e_{3} = 0 \\ \nabla_{e_{2}} e_{1} & = & \nabla_{e_{3}} e_{1} = e_{2} - e_{3} \\ \nabla_{e_{2}} e_{2} & = & \nabla_{e_{2}} e_{3} = \nabla_{e_{3}} e_{2} = \nabla_{e_{3}} e_{3} = - e_{1} . \end{matrix}

2.2 The Beltrami formula

We recall that a translation surfaces $M^{2} (γ_{1}, γ_{2})$ in the three-dimensional Lorentz Heisenberg group $H_{3}$ is a surface parametrized by

r : M^{2} \to H_{3}, r (x, y) = γ_{1} (x) * γ_{2} (y),

where γ₁ and γ₂ are any generating curves in $R^{3}$ ⁠.

$(M^{2}, r)$ is said to be of finite Chen-type k if the position vector r admits the following spectral decomposition

r = r_{0} + \sum_{i = 1}^{k} r_{i}

where r_i are $H_{3}$ -valued eigenfunctions of the Laplacian of $(M^{2}, r)$ and Δr_i = λ_ir_i, $λ_{i} \in R$ ⁠, i = 1, 2, …, k [1]. If λ_i are different, then M² is said to be of k-type.

The Laplacian Δ on M² is given by

Δ = \frac{- 1}{\sqrt{|D|}} \sum_{i j} \frac{\partial}{\partial x^{i}} (\sqrt{|D|} g^{i j} \frac{\partial}{\partial x^{j}}) .

(2.1)

where $G = (g_{i j})$ is the matrix consisting of components of the induced metric on M², and $G^{- 1} = (g^{i j})$ is the inverse matrix of G and D = det G.

We set

Δ r = (Δ r_{1}, Δ r_{2}, Δ r_{3})

such that $r = r (u, v) = (r_{1} (u, v), r_{2} (u, v), r_{3} (u, v))$ is a function of class C².

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 $(H_{3}, g_{3})$ is given by the following:

Δ r = 2 H

(2.2)

where Δ is the Laplacian of the surface and H is the mean curvature vector field of M².

3. Translation surfaces in $(H_{3}, g_{3})$

In the Lorentz Heisenberg space $H_{3}$ ⁠, a translation surface is parametrized by $r (x, y) = γ_{1} (x) * γ_{2} (y)$ ⁠, where γ₁ and γ₂ are two planar curves lying in planes, which are non-orthogonal and * denotes the group operation of $H_{3}$ ⁠.

Definition 1.

A translation surface M² in $H_{3}$ is surface parametrized by $γ_{1} (x) * γ_{2} (y)$ ⁠, where $γ_{1} : I \subset R \to H_{3}$ , $γ_{2} : J \subset R \to H_{3}$ are curves in two coordinate planes of $R^{3}$ ⁠.

We distinguish six types of translation surfaces in $H_{3}$ ⁠.

3.1 Surfaces of type 1

Let us first consider a translation surface M² parametrized by

r (x, y) = (x, 0, u (x)) * (0, y, v (y)) = (x, y, u (x) + v (y) - x y),

(3.1)

where u and v are two smooth functions.

Thus, the basis of the tangent space T_pM² is

\begin{matrix} r_{x} & = & (1,0, u_{x} - y) = \partial_{x} + (u_{x} - y) \partial_{z} \\ r_{y} & = & (0,1, v_{y} - x) = \partial_{y} + (v_{y} - x) \partial_{z} \end{matrix}

That is

r_{x} = e_{1} + (u_{x} - y) e_{2} - (u_{x} - y) e_{3}, r_{y} = v_{y} e_{2} + (1 - v_{y}) e_{3}

(3.2)

Therefore the coefficients of the first and second fundamental form are

\begin{matrix} E = g_{3} (r_{x}, r_{x}) = 1, F = g_{3} (r_{x}, r_{y}) = u_{x} - y, G = g_{3} (r_{y}, r_{y}) = 2 v_{y} - 1 \end{matrix}

and

L = g_{3} (\nabla_{r_{x}} r_{x}, N), M = g_{3} (\nabla_{r_{x}} r_{y}, N), N = g_{3} (\nabla_{r_{y}} r_{y}, N),

where $N$ is a unit vector field normal on M², which satisfies the following system

\{\begin{cases} g_{3} (r_{x}, N) = 0, \\ g_{3} (r_{y}, N) = 0, \\ g_{3} (N, N) = - 1 . \end{cases}

and so,

\begin{matrix} \begin{matrix} \nabla_{r_{x}} r_{x} = u_{x x} e_{2} - u_{x x} e_{3}, \end{matrix} \\ \begin{matrix} \nabla_{r_{x}} r_{y} = 0, \end{matrix} \\ \begin{matrix} \nabla_{r_{y}} r_{y} = - e_{1} + v_{y y} e_{2} - v_{y y} e_{3} . \end{matrix} \end{matrix}

(3.3)

The normal unit vector field $N$ to M² is given by:

N = - \frac{(u_{x} - y)}{W} e_{1} + \frac{(1 - v_{y})}{W} e_{2} + \frac{v_{y}}{W} e_{3}

(3.4)

with

W = \sqrt{E G - F^{2}} = \sqrt{2 v_{y} - {(u_{x} - y)}^{2} - 1} .

Therefore

\begin{matrix} L = \frac{u_{x x}}{W}, & M = 0, & N = \frac{v_{y y} + u_{x} - y}{W} . \end{matrix}

and the mean curvature vector field H is given by the formula:

H = H N,

with

H = \frac{E N + G L - 2 F M}{2 (E G - F^{2})}

where H is, the mean curvature of the surface M².

So the mean curvature of the surface M² parametrized by (x, y, u(x) + v(y) − xy) is given by

H = \frac{1}{2 W^{3}} [v_{y y} + (2 v_{y} - 1) u_{x x} + (u_{x} - y)]

(3.5)

3.2 Surfaces of type 2

Now the translation surface M² is parametrized by

r (x, y) = (0, y, v (y)) * (x, 0, u (x)) = (x, y, u (x) + v (y)),

(3.6)

where u and v are two smooth functions. Therefore

\begin{matrix} r_{x} & = & (1,0, u_{x}) = \partial_{x} + u_{x} \partial_{z} \\ r_{y} & = & (0,1, v_{y}) = \partial_{y} + v_{y} \partial_{z} \end{matrix}

and so,

r_{x} = e_{1} + u_{x} e_{2} - u_{x} e_{3}, r_{y} = (x + v_{y}) e_{2} + (1 - x - v_{y}) e_{3}

(3.7)

Therefore the coefficients of the first fundamental form are

\begin{matrix} E = 1 & F = u_{x} & G = (2 v_{y} + 2 x - 1) \end{matrix}

We have

\begin{matrix} \begin{matrix} \nabla_{r_{x}} r_{x} = u_{x x} e_{2} - u_{x x} e_{3}, \end{matrix} \\ \begin{matrix} \nabla_{r_{x}} r_{y} = e_{2} - e_{3}, \end{matrix} \\ \begin{matrix} \nabla_{r_{y}} r_{y} = - e_{1} + v_{y y} e_{2} - v_{y y} e_{3} . \end{matrix} \end{matrix}

(3.8)

The normal unit vector field $N$ to M² is given by:

N = - \frac{u_{x}}{W} e_{1} + \frac{(1 - x - v_{y})}{W} e_{2} + \frac{(x + v_{y})}{W} e_{3}

(3.9)

with

W = \sqrt{E G - F^{2}} = \sqrt{2 (v_{y} + x) - 1 - {(u_{x})}^{2}} .

Then the coefficients of the second fundamental form of M² are

\begin{matrix} L = \frac{u_{x x}}{W}, & M = \frac{1}{W}, & N = \frac{u_{x} + v_{y y}}{W} . \end{matrix}

We follow the same steps as the previous type to calculate the mean curvature of the translation surface M². We obtain

H = \frac{1}{2 W^{3}} [v_{y y} + (2 v_{y} + 2 x - 1) u_{x x} - u_{x}]

(3.10)

3.3 Surfaces of type 3

The translation surface M² is parametrized by

r (x, y) = (x, 0, u (x)) * (v (y), y, 0) = (x + v (y), y, u (x) - x y),

(3.11)

where u and v are two smooth functions. Therefore

\begin{matrix} r_{x} & = & (1,0, u_{x} - y) = \partial_{x} + (u_{x} - y) \partial_{z} \\ r_{y} & = & (v_{y}, 1, - x) = v_{y} \partial_{x} + \partial_{y} - x \partial_{z} \end{matrix}

and so,

r_{x} = e_{1} + (u_{x} - y) e_{2} - (u_{x} - y) e_{3}, r_{y} = v_{y} e_{1} + e_{3},

(3.12)

whereby the coefficients of the first fundamental form are

\begin{matrix} E = g_{3} (r_{x}, r_{x}) = 1, F = g_{3} (r_{x}, r_{y}) = u_{x} + v_{y} - y, \\ G = g_{3} (r_{y}, r_{y}) = {(v_{y})}^{2} - 1 . \end{matrix}

We have

\begin{matrix} \begin{matrix} \nabla_{r_{x}} r_{x} = u_{x x} e_{2} - u_{x x} e_{3}, \end{matrix} \\ \begin{matrix} \nabla_{r_{x}} r_{y} = 0, \end{matrix} \\ \begin{matrix} \nabla_{r_{y}} r_{y} = (v_{y y} - 1) e_{1} + v_{y} e_{2} - v_{y} e_{3} . \end{matrix} \end{matrix}

(3.13)

The normal unit vector field $N$ to M² is given by:

N = \frac{1}{W} [(u_{x} - y) e_{1} - (v_{y} (u_{x} - y) + 1) e_{2} + v_{y} (u_{x} - y) e_{3}]

(3.14)

with the space-like case $W = \sqrt{- {(u_{x} - y)}^{2} - 2 v_{y} (u_{x} - y) - 1}$ ⁠.

Then the coefficients of the second fundamental form of M² are

\begin{matrix} L = - \frac{u_{x x}}{W}, & M = 0, & N = \frac{[(v_{y y} - 1) (u_{x} - y) - v_{y}]}{W} . \end{matrix}

We follow the same steps as the previous types to calculate the main curvature of the translation surface M², we obtain

H = \frac{1}{2 W^{3}} [(u_{x} - y) (v_{y y} - 1) - v_{y} - u_{x x} ({(v_{y})}^{2} - 1)]

(3.15)

3.4 Surfaces of type 4

The translation surface M² is parametrized by

r (x, y) = (v (y), y, 0) * (x, 0, u (x)) = (v (y) + x, y, u (x)),

(3.16)

where u and v are two smooth functions. Therefore

\begin{matrix} r_{x} & = & (1,0, u_{x}) = \partial_{x} + u_{x} \partial_{z} \\ r_{y} & = & (v_{y}, 1,0) = v_{y} \partial_{x} + \partial_{y} \end{matrix}

and so,

r_{x} = e_{1} + u_{x} e_{2} - u_{x} e_{3}, r_{y} = v_{y} e_{1} + (v + x) e_{2} + (1 - v - x) e_{3}

(3.17)

Thus, the coefficients of the first and second fundamental form are

\begin{matrix} E = g_{3} (r_{x}, r_{x}) = 1, F = g_{3} (r_{x}, r_{y}) = v_{y} + u_{x}, \\ G = g_{3} (r_{y}, r_{y}) = {(v_{y})}^{2} + 2 v + 2 x - 1 . \end{matrix}

We have

\begin{matrix} \begin{matrix} \nabla_{r_{x}} r_{x} = u_{x x} e_{2} - u_{x x} e_{3}, \end{matrix} \\ \begin{matrix} \nabla_{r_{x}} r_{y} = e_{2} - e_{3}, \end{matrix} \\ \begin{matrix} \nabla_{r_{y}} r_{y} = (v_{y y} - 1) e_{1} + 2 v_{y} e_{2} - 2 v_{y} e_{3} . \end{matrix} \end{matrix}

(3.18)

The normal unit vector field $N$ to M² is given by:

N = \frac{1}{W} [- u_{x} e_{1} + (u_{x} v_{y} + 1 - v - x) e_{2} + (v + x - u_{x} v_{y}) e_{3}]

(3.19)

with the space-like case $W = \sqrt{2 (v + x - u_{x} v_{y}) - {(u_{x})}^{2} - 1}$ ⁠.

Then the coefficients of the second fundamental form of M² are

\begin{matrix} L = \frac{u_{x x}}{W}, & M = \frac{1}{W}, & N = \frac{2 v_{y} - (v_{y y} - 1) u_{x}}{W} . \end{matrix}

We follow the same steps as the previous types to calculate the main curvature of the translation surface M², we obtain

H = \frac{1}{2 W^{3}} [(v_{y y} + 1) u_{x} + [{(v_{y})}^{2} + 2 v + 2 x - 1] u_{x x}]

(3.20)

3.5 Surfaces of type 5

In this case, the translation surface M² is parametrized by

r (x, y) = (v (y), y, 0) * (x, 0, u (x)) = (y, v (y) + x, u (x)),

(3.21)

where u and v are two smooth functions. Therefore

\begin{matrix} r_{x} & = & (0,1, u_{x}) = \partial_{y} + u_{x} \partial_{z} \\ r_{y} & = & (1, v_{y}, 0) = \partial_{x} + v_{y} \partial_{y} \end{matrix}

and so,

r_{x} = (y + u_{x}) e_{2} - (u_{x} + y - 1) e_{3}, r_{y} = e_{1} + y v_{y} e_{2} + (1 - y) v_{y} e_{3}

(3.22)

Hence the coefficients of the first and second fundamental form are

\begin{matrix} E = g_{3} (r_{x}, r_{x}) = 2 (u_{x} + y) - 1, F = g_{3} (r_{x}, r_{y}) = (u_{x} + 2 y - 1) v_{y}, \\ G = g_{3} (r_{y}, r_{y}) = (2 y - 1) {(v_{y})}^{2} + 1 . \end{matrix}

We have

\begin{matrix} \begin{matrix} \nabla_{r_{x}} r_{x} = - e_{1} + u_{x x} e_{2} - u_{x x} e_{3}, \end{matrix} \\ \begin{matrix} \nabla_{r_{x}} r_{y} = - v_{y} e_{1} + e_{2} - e_{3}, \end{matrix} \\ \begin{matrix} \nabla_{r_{y}} r_{y} = - {(v_{y})}^{2} e_{1} + (2 v_{y} + y v_{y y}) e_{2} - (2 v_{y} + (y - 1) v_{y y}) e_{3} . \end{matrix} \end{matrix}

(3.23)

The normal unit vector field $N$ to M² is given by:

N = \frac{1}{W} [u_{x} v_{y} e_{1} - (u_{x} + y - 1) e_{2} + (u_{x} + y) e_{3}]

(3.24)

with $W = \sqrt{2 (y + u_{x}) - {(u_{x})}^{2} {(v_{y})}^{2} - 1}$ ⁠.

The coefficients of the second fundamental form of M² then are

\begin{matrix} L = \frac{u_{x x} - u_{x} v_{y}}{W}, & M = \frac{1 - u_{x} {(v_{y})}^{2}}{W}, & N = \frac{2 v_{y} - v_{y y} u_{x} - u_{x} {(v_{y})}^{3}}{W} . \end{matrix}

As in the previous types, we obtain

H = \frac{1}{2 W^{3}} [\begin{matrix} [(2 y - 1) {(v_{y})}^{2} + 1] u_{x x} - [2 (y + u_{x}) - 1] v_{y y} u_{x} \\ + [1 - (2 y - 1 + u_{x}) {(v_{y})}^{2}] u_{x} v_{y} \end{matrix}]

(3.25)

3.6 Surfaces of type 6

The translation surface M² is parametrized by

r (x, y) = (y, v (y), 0) * (0, x, u (x)) = (y, v (y) + x, u (x) - x y),

(3.26)

where u and v are two smooth functions. Therefore

r_{x} = (0,1, u_{x}) = \partial_{y} + u_{x} \partial_{z}, r_{y} = (1, v_{y}, 0) = \partial_{x} + v_{y} \partial_{y}

and so,

\begin{align} r_{x} = u_{x} e_{2} + (1 - u_{x}) e_{3}, \\ r_{y} = e_{1} + (y v_{y} - x) e_{2} + (x - y v_{y} + v_{y}) e_{3} \end{align}

(3.27)

Therefore the coefficients of the first and second fundamental form are

\begin{matrix} E = g_{3} (r_{x}, r_{x}) = 2 u_{x} - 1, F = g_{3} (r_{x}, r_{y}) = v_{y} u_{x} + y v_{y} - v_{y} - x, \\ G = g_{3} (r_{y}, r_{y}) = 1 - {(v_{y})}^{2} + 2 v_{y} (y v_{y} - x) . \end{matrix}

We have

\begin{matrix} \begin{matrix} \nabla_{r_{x}} r_{x} = - e_{1} + u_{x x} e_{2} - u_{x x} e_{3}, \end{matrix} \\ \begin{matrix} \nabla_{r_{x}} r_{y} = - v_{y} e_{1}, \end{matrix} \\ \begin{matrix} \nabla_{r_{y}} r_{y} = - {(v_{y})}^{2} e_{1} + (2 v_{y} + y v_{y y}) e_{2} - (2 v_{y} + y v_{y y} - v_{y y}) e_{3} . \end{matrix} \end{matrix}

(3.28)

The normal unit vector field $N$ to M² is given by:

N = \frac{1}{W} [(u_{x} v_{y} - y v_{y} + x) e_{1} + (1 - u_{x}) e_{2} + u_{x} e_{3}]

(3.29)

With $W = \sqrt{2 u_{x} - {(u_{x} v_{y} - y v_{y} + x)}^{2} - 1}$ ⁠.

Then the coefficients of the second fundamental form of M² are

\begin{matrix} L = \frac{u_{x x} - u_{x} v_{y} + y v_{y} - x}{W}, M = - \frac{(u_{x} v_{y} - y v_{y} + x) v_{y}}{W}, \\ N = \frac{- (u_{x} v_{y} - y v_{y} + x) {(v_{y})}^{2} + 2 v_{y} + y v_{y y} - v_{y y} u_{x}}{W} \end{matrix}

As above, we obtain

H = \frac{1}{2 W^{3}} [\begin{matrix} (2 u_{x} - 1) (y - u_{x}) v_{y y} + [1 - {(v_{y})}^{2} + 2 v_{y} (y v_{y} - x)] u_{x x} \\ + (3 u_{x} + y - 2) v_{y} - x \end{matrix}]

(3.30)

4. Translation surfaces in Lorentz Heisenberg space $(H_{3}, g_{3})$ satisfying Δr_i = λ_ir_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 = λ_ir_i, $λ_{i} \in R$ ⁠.

The result is:

Theorem 2.

Let M² 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 $(H_{3}, g_{3})$ ⁠. Then M² satisfies the condition Δr_i = λ_ir_i, i = 1, 2, 3, $λ_{i} \in R$ ⁠, if and only if M² has zero mean curvature.

Proof of Theorem 2.

In the following we will prove Theorem 2.

4.1 Surfaces of type 1

By (2.2) and (3.4), the Laplacian operator Δr of r can be expressed as

Δ r = - \frac{2 (u_{x} - y)}{W} H e_{1} + \frac{2 (1 - v_{y})}{W} H e_{2} + \frac{2 v_{y}}{W} H e_{3}

(4.1)

Next, suppose M² satisfies condition (1.1). Then, from (3.1) and (4.1), we obtain the following system of ordinary differential equations.

- \frac{2 (u_{x} - y)}{W} H = λ_{1} x

(4.2)

\frac{2 (1 - v_{y})}{W} H = λ_{2} x y + λ_{3} (u + v - x y)

(4.3)

\frac{2 v_{y}}{W} H = λ_{2} (1 - x) y - λ_{3} (u + v - x y)

(4.4)

Therefore, the problem of classifying the translation surfaces M² 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 λ₁, λ₂ and λ₃.

Combining Equations (4.3) and (4.4) yields

- \frac{2 (u_{x} - y)}{W} H = λ_{1} x

(4.5)

\frac{2}{W} H = λ_{2} x y

(4.6)

Case 1: λ₁λ₂ = 0. Then H = 0 and λ₃ = 0. Thus, the surface M² is minimal.

Case 2: λ₁λ₂ ≠ 0. Substituting (4.5) into (4.6), we get

- (u_{x} - y) = \frac{λ_{1}}{λ_{2}} \frac{x}{y}

(4.7)

Taking the partial derivative of (4.7) with respect to x gives

- u_{x x} = \frac{λ_{1}}{λ_{2}} \frac{1}{y}

(4.8)

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.

4.2 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

Δ r = - \frac{2 u_{x}}{W} H e_{1} + \frac{2 (1 - v_{y} - x)}{W} H e_{2} + \frac{2 (v_{y} + x)}{W} H e_{3}

(4.9)

Suppose that M² satisfies the condition (1.1). Then, from (3.6) and (4.9), we obtain the following system of ordinary differential equations.

- \frac{2 u_{x}}{W} H = λ_{1} x

(4.10)

\frac{2 (1 - v_{y} - x)}{W} H = λ_{2} x y + λ_{3} (u + v)

(4.11)

\frac{2 (v_{y} + x)}{W} H = λ_{2} (1 - x) y - λ_{3} (u + v)

(4.12)

Therefore, the problem of classifying the translation surfaces M² 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 λ₁, λ₂, λ₃.

By combining Equations (4.11) and (4.12) we obtain

- \frac{2 u_{x}}{W} H = λ_{1} x

(4.13)

\frac{2}{W} H = λ_{2} y

(4.14)

Case 1: λ₁λ₂ = 0.

If λ₂ = 0, then we have H = 0. Thus, the surface M² is minimal.
If λ₂ ≠ 0, from (4.13), we obtain u_x = 0. Substituting (3.10) into (4.14), with $W = \sqrt{2 (v_{y} + x) - 1 - {(u_{x})}^{2}}$ ⁠, we get

v_{y y} = λ_{2} y {[2 (v_{y} + x) - 1]}^{2}

(4.15)

Taking the partial derivative of (4.15) with respect to x gives

v_{y} = \frac{1 - 2 x}{2}

(4.16)

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: λ₁λ₂ ≠ 0. Substituting (4.13) into (4.14), we get

- u_{x} = \frac{λ_{1}}{λ_{2}} \frac{x}{y}

(4.17)

Taking the partial derivative of (4.17) with respect to x gives

- u_{x x} = \frac{λ_{1}}{λ_{2}} \frac{1}{y}

(4.18)

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.

4.3 Surfaces of type 3

By following the same steps as the previous types, by (2.2) and (3.14), we obtain

Δ r = \frac{2 (u_{x} - y)}{W} H e_{1} - \frac{2 [v_{y} (u_{x} - y) + 1]}{W} H e_{2} + \frac{2 v_{y} (u_{x} - y)}{W} H e_{3}

(4.19)

Suppose that M² satisfies condition (1.1). Then, from (3.11) and (4.19), we obtain the following system of ordinary differential equations.

\frac{2 (u_{x} - y)}{W} H = λ_{1} (x + v)

(4.20)

- \frac{2 [v_{y} (u_{x} - y) + 1]}{W} H = λ_{2} (x + v) y + λ_{3} (u - x y)

(4.21)

\frac{2 v_{y} (u_{x} - y)}{W} H = (1 - (x + v)) λ_{2} y - λ_{3} (u - x y)

(4.22)

Combining Equations (4.21) and (4.22) we have

\frac{2 (u_{x} - y)}{W} H = λ_{1} (x + v)

(4.23)

- \frac{2}{W} H = λ_{2} y

(4.24)

Case 1: λ₁λ₂ = 0. Then H = 0 and λ₃ = 0. Thus, the surface M² is minimal.

Case 2: λ₁λ₂ ≠ 0. Substituting (4.23) into (4.24), we get

- (u_{x} - y) = \frac{λ_{1}}{λ_{2}} \frac{(x + v)}{y}

(4.25)

Taking the partial derivative of (4.25) with respect to x gives

- u_{x x} = \frac{λ_{1}}{λ_{2}} \frac{1}{y}

(4.26)

We have an identity of two functions, one depending only on x and the other one depending only on y. That is not possible.

4.4 Surfaces of type 4

By (2.2) and (3.19), we obtain

Δ r = - \frac{2 u_{x}}{W} {H e}_{1} + \frac{2 (u_{x} v_{y} + 1 - v - x)}{W} {H e}_{2} + \frac{2 (v + x - u_{x} v_{y})}{W} {H e}_{3}

(4.27)

Suppose that M² satisfies the condition (1.1). Then, from (3.16) and (4.27), we obtain the following system of ordinary differential equations.

- \frac{2 u_{x}}{W} H = λ_{1} (v + x)

(4.28)

\frac{2 (u_{x} v_{y} + 1 - v - x)}{W} H = λ_{2} (v + x) y + λ_{3} u

(4.29)

\frac{2 (v + x - u_{x} v_{y})}{W} H = λ_{2} ((1 - (v + x)) y - λ_{3} u)

(4.30)

Combining Equations (4.29) and (4.30) we have

- \frac{2 u_{x}}{W} H = λ_{1} (v + x)

(4.31)

\frac{2}{W} H = λ_{2} y

(4.32)

Case 1: λ₁λ₂ = 0.

If λ₂ = 0, then we have H = 0. Thus, the surface M² is minimal.
If λ₂ ≠ 0, from (4.31), we obtain u_x = 0. Then from (3.20) we have H = 0 and λ₂ = 0, which is contradiction.

Case 2: λ₁λ₂ ≠ 0. Substituting (4.31) into (4.32), we get

- u_{x} = \frac{λ_{1}}{λ_{2}} \frac{(v + x)}{y}

(4.33)

Taking the partial derivative of (4.33) with respect to x gives

- u_{x x} = \frac{λ_{1}}{λ_{2}} \frac{1}{y}

(4.34)

We have an identity of two functions, one depending only on x and the other one depending only on y. That is impossible.

4.5 Surfaces of type 5

By (2.2) and (3.24), the Laplacian Δ of r can be expressed as

Δ r = \frac{2 u_{x} v_{y}}{W} H e_{1} - \frac{2 (u_{x} + y - 1)}{W} H e_{2} + \frac{2 (u_{x} + y)}{W} H e_{3}

(4.35)

Suppose that M² satisfies condition (1.1). Then, from (3.21) and (4.35), we obtain the following system of ordinary differential equations.

\frac{2 u_{x} v_{y}}{W} H = λ_{1} y

(4.36)

- \frac{2 (u_{x} + y - 1)}{W} H = λ_{2} y (v + x) + λ_{3} u

(4.37)

\frac{2 (u_{x} + y)}{W} H = λ_{2} (1 - y) (v + x) - λ_{3} u

(4.38)

Combining the Equations (4.37) and (4.38) we have

\frac{2 u_{x} v_{y}}{W} H = λ_{1} y

(4.39)

\frac{2}{W} H = λ_{2} (v + x)

(4.40)

\frac{2 (u_{x} + 2 y - 1)}{W} H = - λ_{3} u

(4.41)

Case 1: λ₃ = 0. Since u depends only on x, then from (4.41) H = 0 and λ₁ = λ₂ = λ₃ = 0. Thus, the surface M² is minimal.

Case 2: λ₃ ≠ 0.

If λ₂ = 0, from (4.40) H = 0, and λ₃ = 0, which is contradiction.
If λ₁ = 0 and λ₂ ≠ 0 from (4.39) we obtain u_xv_y = 0. Then
- If u_x = 0. From (3.25) we obtain H = 0 and λ₃ = 0, which is a contradiction.
- If v_y = 0, then v is constant $(v = a)$ ⁠. By combining Equations (3.25) and (4.40) and the fact that $W = \sqrt{2 (y + u_{x}) - {(u_{x})}^{2} {(v_{y})}^{2} - 1}$ ⁠, we conclude

u_{x x} = λ_{2} (a + x) {[2 y + 2 u_{x} - 1]}^{2}

(4.42)

Taking the partial derivative of (4.42) with respect to y gives

u_{x} = \frac{1 - 2 y}{2}

(4.43)

The right-hand side in (4.43) is either a function of y while other side is a constant or a function of x. That is impossible.

If λ₁λ₂ ≠ 0. Substituting (4.39) into (4.40) and (4.41) into (4.40) we get

u_{x} v_{y} = \frac{λ_{1}}{λ_{2}} \frac{y}{(v + x)}

(4.44)

(u_{x} + 2 y - 1) = - \frac{λ_{3}}{λ_{2}} \frac{u}{(v + x)}

(4.45)

Combining Equations (4.44) and (4.45) we have

λ_{2} (2 y - 1) (v + x) v_{y} = λ_{3} u v_{y} + λ_{1} y

(4.46)

Taking the partial derivative of (4.46) with respect to x and the fact that v_y ≠ 0, we obtain

u_{x} = \frac{λ_{2}}{λ_{3}} (2 y - 1)

(4.47)

That is impossible because (4.47) is an identity of two functions, one depending only on x and the other one depending only on y.

4.6 Surfaces of type 6

By (2.2) and (3.29), we have

Δ r = \frac{2 (u_{x} v_{y} - y v_{y} + x)}{W} H e_{1} + \frac{2 (1 - u_{x})}{W} H e_{2} + \frac{2 u_{x}}{W} H e_{3}

(4.48)

Suppose that M² satisfies condition (1.1). Then, from (3.26) and (4.48), we obtain the following system of ordinary differential equations.

\frac{2 (u_{x} v_{y} - y v_{y} + x)}{W} H = λ_{1} y

(4.49)

\frac{2 (1 - u_{x})}{W} H = λ_{2} y (v + x) + λ_{3} (u - x y)

(4.50)

\frac{2 (x + u_{x} - y)}{W} H = λ_{2} (1 - y) (v + x) - λ_{3} (u - x y)

(4.51)

Combining the Equations (4.50) and (4.51) we have

\frac{2 (u_{x} v_{y} - y v_{y} + x)}{W} H = λ_{1} y

(4.52)

\frac{2 (x - y + 1)}{W} H = λ_{2} (v + x)

(4.53)

\frac{2 [1 - u_{x} - (x - y + 1) y]}{W} H = λ_{3} (u - x y)

(4.54)

Case 1: λ₃ = 0 and H ≠ 0. Then from (4.54) we obtain

1 - u_{x} - (x - y + 1) y = 0

(4.55)

Taking the partial derivative of (4.55) with respect to x, we obtain

u_{x x} = y

(4.56)

That is impossible because u_xx is function of x. Then H = 0 and λ₁ = λ₂ = λ₃ = 0. Thus, the surface M² is minimal.

Case 2: λ₃ ≠ 0.

If λ₁λ₂ = 0. from (4.52), we have

u_{x} v_{y} - y v_{y} + x = 0

(4.57)

Taking the partial derivative of (4.57) with respect to x, we obtain

u_{x x} = - \frac{1}{v_{y}}

(4.58)

Then both sides have to equal a nonzero constant, namely

u_{x x} = a = - \frac{1}{v_{y}}

Which implies that $v_{y} = - \frac{1}{a}$ ⁠. Considering it into (4.57) gives

- \frac{u_{x}}{a} + x = - \frac{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.

If λ₁λ₂ ≠ 0. Substituting (4.53) into (4.54), we get

[1 - u_{x} - (x - y + 1) y] (v + x) = \frac{λ_{3}}{λ_{2}} (u - x y) (x - y + 1)

(4.60)

The partial derivative of (4.60) with respect to x and y yields

2 x - 2 y + 1 + v - (u_{x x} - y) v_{y} = \frac{λ_{3}}{λ_{2}} (2 x - 2 y + 1 + u_{x})

(4.61)

Taking partial derivative of (4.61) with respect to x and y leads to

u_{x x x} v_{y y} = 0

We have two situations:

u_xxx = 0 i.e.

u_{x} = b_{1} x + b_{2} .

Then (4.61) reduces to

2 x + 1 - \frac{λ_{3}}{λ_{2}} (2 x + 1 + b_{1} x + b_{2}) = (b_{1} - y) v_{y} - v + 2 y (1 - \frac{λ_{3}}{λ_{2}})

(4.62)

We have an identity of two functions, one depending only on x and the other one depending only on y. That is impossible.

v_yy = 0, i.e.

v = c_{1} y + c_{2} .

Then (4.61) reduces to

2 x - u_{x x} c_{1} - \frac{λ_{3}}{λ_{2}} (2 x + 1 + u_{x}) + c_{2} + 1 = 2 (1 - c_{1} - \frac{λ_{3}}{λ_{2}}) y

(4.63)

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.

5. 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 = λ_ir_i, i = 1, 2, 3, $λ_{i} \in R$ ⁠, if and only if they have zero mean curvature.

The authors would like to thank anonymous referees for their valuable comments and careful corrections to the original version of this paper.

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Rafik Medjati, Hanifi Zoubir and Brahim Medjahdi

Published in Arab Journal of Mathematical Sciences. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence may be seen at http://creativecommons.org/licences/by/4.0/legalcode

Classification of space-like translation surfaces in the 3-dimensional Lorentz Heisenberg group $H_{3}$

1. Introduction

2. Preliminaries

2.1 The Lorentz Heisenberg group $H_{3}$

2.2 The Beltrami formula

3. Translation surfaces in $(H_{3}, g_{3})$

3.1 Surfaces of type 1

3.2 Surfaces of type 2

3.3 Surfaces of type 3

3.4 Surfaces of type 4

3.5 Surfaces of type 5

3.6 Surfaces of type 6

4. Translation surfaces in Lorentz Heisenberg space $(H_{3}, g_{3})$ satisfying Δr_i = λ_ir_i

4.1 Surfaces of type 1

4.2 Surfaces of type 2

4.3 Surfaces of type 3

4.4 Surfaces of type 4

4.5 Surfaces of type 5

4.6 Surfaces of type 6

5. Conclusion

References

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Classification of space-like translation surfaces in the 3-dimensional Lorentz Heisenberg group H3

1. Introduction

2. Preliminaries

2.1 The Lorentz Heisenberg group H3

2.2 The Beltrami formula

3. Translation surfaces in (H3,g3)

3.1 Surfaces of type 1

3.2 Surfaces of type 2

3.3 Surfaces of type 3

3.4 Surfaces of type 4

3.5 Surfaces of type 5

3.6 Surfaces of type 6

4. Translation surfaces in Lorentz Heisenberg space (H3,g3) satisfying Δri = λiri

4.1 Surfaces of type 1

4.2 Surfaces of type 2

4.3 Surfaces of type 3

4.4 Surfaces of type 4

4.5 Surfaces of type 5

4.6 Surfaces of type 6

5. Conclusion

References

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Classification of space-like translation surfaces in the 3-dimensional Lorentz Heisenberg group $H_{3}$

2.1 The Lorentz Heisenberg group $H_{3}$

3. Translation surfaces in $(H_{3}, g_{3})$

4. Translation surfaces in Lorentz Heisenberg space $(H_{3}, g_{3})$ satisfying Δr_i = λ_ir_i