Surfaces of Constant Curvature in the Pseudo-Galilean Space

Abstract

We develop the local theory of surfaces immersed in the pseudo-Galilean space, a special type of Cayley-Klein spaces. We define principal, Gaussian, and mean curvatures. By this, the general setting for study of surfaces of constant curvature in the pseudo-Galilean space is provided. We describe surfaces of revolution of constant curvature. We introduce special local coordinates for surfaces of constant curvature, so-called the Tchebyshev coordinates, and show that the angle between parametric curves satisfies the Klein-Gordon partial differential equation. We determine the Tchebyshev coordinates for surfaces of revolution and construct a surface with constant curvature from a particular solution of the Klein-Gordon equation.

1. Introduction

Study of differential geometry of curves and surfaces in Euclidean, as well as in other non-Euclidean ambient spaces, has a long history. Classical context of the Euclidean space is a source of results which could be transferred to some other geometries. One way of defining new geometries is through Cayley-Klein spaces. They are defined as projective spaces 𝒫_𝑛(𝐑) with an absolute figure, a subset of 𝒫_𝑛(𝐑) consisting of a sequence of quadrics and planes [1]. Projectivities of the projective space 𝒫_𝑛(𝐑) which leave invariant the absolute figure define the subgroup of projectivities called the group of motions of a Cayley-Klein space. By means of the absolute figure, metric relations are also defined and they are invariant under the group of motions.

In three-dimensional projective space 𝒫₃(𝐑) various types of Cayley-Klein spaces can be defined, such as elliptic and hyperbolic space, Euclidean and pseudo-Euclidean (Minkowski) space, simple and double isotropic space, Galilean and pseudo-Galilean space, and quasielliptic and quasihyperbolic space. General theory of differential geometry of curves and surfaces in Cayley-Klein spaces can be found in [1]. Foundations of these areas in the pseudo-Galilean space were established in [2], as well as in the papers [3–7]. Geometry of the Galilean space 𝐺₃ was studied in [8–11]. The four-dimensional Galilean space appears in connection with classical Newtonian mechanics, where first coordinate describes time and other three coordinates are space coordinates.

The main interest of this paper is to develop the local theory of surfaces in the pseudo-Galilean space and to study surfaces of constant curvatures. As in the Minkowski space [12], two classes of surfaces are introduced, spacelike and timelike surfaces, and for them the Gaussian curvature is defined. The obtained results are compared to the well-known results in the Euclidean and Minkowski geometry. The results can easily be transferred to the Galilean space. Furthermore, we define the Tchebyshev coordinates on a surface and show that the asymptotic lines form the Tchebyshev net if and only if this surface is a spacelike surface of constant negative curvature or timelike surface of constant positive curvature. We study the angle between the Tchebyshev curves on a surface of constant curvature and show that the angle satisfies the Klein-Gordon partial differential equation. In this respect, the Klein-Gordon equation plays the analogous role in the pseudo-Galilean space as the sine-Gordon equation in Euclidean space. We construct a surface with constant Gaussian curvature from a known solution of a Klein-Gordon equation. Similar problem is treated in [13] for the Galilean space and in wider context in [14] by means of Cartan frames.

2. Preliminaries

The absolute figure of the pseudo-Galilean space is the ordered triple {𝜔,𝑓,𝐼}, where 𝜔 is the ideal (absolute) plane in the real three-dimensional projective space 𝒫₃(𝐑), 𝑓 the line (absolute line) in 𝜔, and 𝐼 the fixed hyperbolic involution of points of 𝑓. Homogeneous coordinates in 𝐺¹₃ are introduced in such a way that the absolute plane 𝜔 is given by 𝑥₀=0, the absolute line 𝑓 by 𝑥₀=𝑥₁=0, and the hyperbolic involution by (0∶0∶𝑥₂∶𝑥₃)↦(0∶0∶𝑥₃∶𝑥₂). The last condition is equivalent to the requirement that the conic 𝑥²₂−𝑥²₃=0 is the absolute conic. Metric relations are introduced with respect to the absolute figure. In affine coordinates defined by (𝑥₀∶𝑥₁∶𝑥₂∶𝑥₃)=(1∶𝑥∶𝑦∶𝑧), distance between points 𝑃_𝑖=(𝑥_𝑖,𝑦_𝑖,𝑧_𝑖),  𝑖=1,2, is defined by 𝑑𝑃₁,𝑃₂=𝑥₂−𝑥₁,if𝑥₁≠𝑥₂,𝑦₂−𝑦₁²−𝑧₂−𝑧₁²,if𝑥₁=𝑥₂. (2.1)

The group of motions of 𝐺¹₃ is a six-parameter group given (in affine coordinates) by 𝑥=𝑎+𝑥,𝑦=𝑏+𝑐𝑥+𝑦cosh𝜑+𝑧sinh𝜑,𝑧=𝑑+𝑒𝑥+𝑦sinh𝜑+𝑧cosh𝜑. (2.2) It leaves invariant the absolute figure as well as the pseudo-Galilean distance (2.1) of points.

In the pseudo-Galilean space, a vector is called isotropic if it is of the form (0,𝑦,𝑧). Among these vectors, there are also vectors with supplementary norm |𝑦²−𝑧²| equal to zero; they are called lightlike vectors. Isotropic vectors satisfying 𝑦²−𝑧²>0 are said to be spacelike vectors and vectors satisfying 𝑦²−𝑧²<0 timelike vectors. A plane of the form 𝑥=const. is called a pseudo-Euclidean plane (since its induced geometry is pseudo-Euclidean, i.e., Minkowski plane geometry), otherwise it is called isotropic (since its induced geometry is isotropic, i.e., Galilean plane geometry). In the pseudo-Euclidean plane distance between points 𝑃_𝑖,  𝑖=1,2, given by their affine coordinates 𝑃_𝑖=(𝑥_𝑖,𝑦_𝑖) is defined by 𝑑𝑃₁,𝑃₂=𝑥₂−𝑥₁²−𝑦₂−𝑦₁², (2.3) while in the isotropic plane 𝑑𝑃₁,𝑃₂=𝑥₂−𝑥₁,if𝑥₁≠𝑥₂,𝑦₂−𝑦₁,if𝑥₁=𝑥₂. (2.4)

The pseudo-Galilean space 𝐺¹₃ can be also regarded as a Cayley-Klein space equipped with the projective metric of signature (0,0,+,−), as explained in [15]. According to the description of the Cayley-Klein spaces in [1], it is denoted by 𝑃³_11|001 and also called the Galilean space of index 1.

3. The Gaussian Curvature of Surfaces in 𝐺¹₃

We will treat a 𝐶^𝑟-surface, 𝑟≥2, as a subset Φ⊂𝐺¹₃ for which there exists an open subset 𝐷 of 𝐑² and 𝐶^𝑟-mapping 𝐱∶𝐷→𝐺¹₃ satisfying Φ=𝐱(𝐷). A 𝐶^𝑟-surface Φ⊂𝐺¹₃ is called regular if 𝐱 is an immersion and simple if 𝐱 is an embedding. It is admissible if it does not have pseudo-Euclidean tangent planes. Let us denote 𝐱=𝐱(𝑥(𝑢₁,𝑢₂),𝑦(𝑢₁,𝑢₂),𝑧(𝑢₁,𝑢₂)), 𝑥_,𝑖=𝜕𝑥/𝜕𝑢_𝑖, 𝑦_,𝑖=𝜕𝑦/𝜕𝑢_𝑖, 𝑧_,𝑖=𝜕𝑧/𝜕𝑢_𝑖, 𝑖=1,2. Then a surface is admissible if and only if 𝑥_,𝑖≠0, for some 𝑖=1,2. If we assume 𝑥_,1𝑦_,1𝑥_,2𝑦_,2≠0, (3.1) then such a surface is admissible and can be locally expressed in the form 𝑧=𝑧𝑥,𝑦. (3.2)

Let Φ⊂𝐺¹₃ be a regular admissible surface. We define a side tangential vector by 1𝜎=𝑊𝑥_,1𝐱_,2−𝑥_,2𝐱_,1. (3.3) The vector 𝜎 is a vector in a tangent plane and we assume that it is not an isotropic lightlike vector, but a unit isotropic spacelike or timelike vector. The function 𝑊, 𝑊>0, defined by 𝑊=𝑥_,1𝑦_,2−𝑥_,2𝑦_,1²−𝑥_,1𝑧_,2−𝑥_,2𝑧_,1², (3.4) is equal to the pseudo-Galilean norm of the isotropic vector 𝑥_,1𝐱_,2−𝑥_,2𝐱_,1. In particular, in the parametrization (3.2) we have 1𝜎=𝑊0,1,𝑧_,𝑦,𝑊=1−𝑧²_,𝑦. (3.5) In the following, we will not consider surfaces with 𝑊=0, that is, surfaces having lightlike side tangential vectors (lightlike surfaces).

In a tangent plane in a point 𝑃₀ of a regular admissible surface, there is a unique isotropic direction defined by the condition 𝑥_,1d𝑢₁+𝑥_,2d𝑢₂=0. This isotropic line in a tangent plane in a point 𝑃₀ of the surface meets the absolute line 𝑓 in a point 𝑆. If we denote by 𝑆^⟂ a point on 𝑓 obtained from 𝑆 by the hyperbolic involution 𝐼, then a line connecting 𝑃₀ and 𝑆^⟂ is perpendicular to the tangent plane. Therefore a unit surface normal field is defined by 1𝐍=𝑊0,𝑥_,1𝑧_,2−𝑥_,2𝑧_,1,𝑥_,1𝑦_,2−𝑥_,2𝑦_,1. (3.6) We introduce a pseudo-Galilean cross-product in the following way: 𝐚×_𝐺𝐛=𝟎−𝐞₂𝐞₃𝑎₁𝑎₂𝑎₃𝑏₁𝑏₂𝑏₃, (3.7) where 𝐞₂=(0,1,0), 𝐞₃=(0,0,1), 𝐞₂ is a unit spacelike, and 𝐞₃ is a unit timelike vector, 𝐚=(𝑎₁,𝑎₂,𝑎₃), 𝐛=(𝑏₁,𝑏₂,𝑏₃). Now we can write 1𝐍=𝑊𝐱₁×_𝐺𝐱₂ (3.8) which for (3.2) turns to 𝐍=(1/𝑊)(0,𝑧_𝑦,1). Furthermore, we can notice that the pseudo-Galilean cross product can be defined by means of the pseudo-Galilean scalar product so that 𝐚×_𝐺𝐛 is the isotropic vector defined by the relation 𝐚×_𝐺𝐛⋅𝐜=−det(𝐚,𝐛,𝐜̃), (3.9) for any vector 𝐜. By∼above a vector, the projection of a vector on the pseudo-Euclidean 𝑦𝑧-plane is denoted and by ⋅ the pseudo-Euclidean scalar product in the same plane, (𝑦₁,𝑦₂)⋅(𝑧₁,𝑧₂)=𝑦₁𝑧₁−𝑦₂𝑧₂.

Obviously the following proposition holds.

Proposition 3.1. The side tangential vector 𝜎 and the normal vector 𝐍 are unit isotropic vectors that satisfy 𝜎⋅𝐍=0,𝐍²∶=𝜖=±1,𝜎²=−𝜖. (3.10)

Since the normal vector field satisfies 𝐍⋅𝐍=𝜖=±1, we distinguish two basic types of admissible surfaces: spacelike surfaces having timelike surface normals (𝜖=−1) and timelike surfaces having spacelike normals (𝜖=1). A surface is spacelike if (𝑥_,1𝐱_,2−𝑥_,2𝐱_,1)²=(𝑥_,1𝑦_,2−𝑥_,2𝑦_,1)²−(𝑥_,1𝑧_,2−𝑥_,2𝑧_,1)²>0 in all of its points, timelike otherwise. In the parametrization (3.2) a surface is spacelike if 1−𝑧²_,𝑦>0.

The first fundamental form of a surface is induced from the metric of the ambient space 𝐺¹₃d𝑠²=𝑥_,1d𝑢₁+𝑥_,2d𝑢₂²+𝛿𝐱_,1d𝑢₁+𝐱_,2d𝑢₂², (3.11) where 𝛿=0,ifdirectiond𝑢₁∶d𝑢₂isnonisotropic,1,ifdirectiond𝑢₁∶d𝑢₂isisotropic. (3.12) We introduce the fundamental coefficients 𝑔_𝑖=𝑥_,𝑖,ℎ_𝑖𝑗=𝐱_,𝑖⋅𝐱_,𝑗,𝑖,𝑗=1,2, (3.13) by which the first fundamental form can be written as d𝑠²=𝑔₁d𝑢₁+𝑔₂d𝑢₂²+𝛿ℎ₁₁d𝑢²₁+2ℎ₁₂d𝑢₁d𝑢₂+ℎ₂₂d𝑢²₂. (3.14) Notice again that the indices 𝑖 or 𝑖𝑗 on variables denote different symbols, whereas indices,𝑖 or,𝑖,𝑗 denote the partial derivatives with respect to the 𝑖th and 𝑗th parameter. Furthermore ℎ_𝑖𝑗=ℎ_𝑗𝑖, 𝑔_𝑖,𝑗=𝑔_𝑗,𝑖, 𝑖=1,2.

Now the function 𝑊 has the form 𝑊²=−𝜖𝑔₁𝐱_,2−𝑔₂𝐱_,1²=−𝜖𝑔²₁ℎ₂₂−2𝑔₁𝑔₂ℎ₁₂+𝑔²₂ℎ₁₁>0, (3.15) and the surface is spacelike if 𝑔²₁ℎ₂₂−2𝑔₁𝑔₂ℎ₁₂+𝑔²₂ℎ₁₁>0, timelike otherwise. We can notice that if a surface is spacelike, both parts of the first fundamental form, d𝑠²₁=(𝑔₁d𝑢₁+𝑔₂d𝑢₂)² and d𝑠²₂=ℎ₁₁d𝑢²₁+2ℎ₁₂d𝑢₁d𝑢₂+ℎ₂₂d𝑢²₂𝑊=−𝜖²𝑔²₁d𝑢²₂,𝑔₁≠0, (3.16) are positive definite, while for the timelike surfaces, the form d𝑠²₁ is positive definite whereas d𝑠²₂is negative definite. We have assumed here, without loss of generality, 𝑔₁≠0. In the latter case this means that the matrix of the first fundamental form d𝑠²₁00d𝑠²₂ (3.17) is indefinite, analogously to the timelike surfaces in the Minkowski space 𝐑³₁ [12].

In particular, for the parametrization (3.2) we have d𝑠²=d𝑥²+𝛿1−𝑧²_,𝑦d𝑦², (3.18) sinced𝑥=0 when 𝛿=1.

Example 3.2. Hyperbolic cylinders 𝑧²−𝑦²=𝑅²(𝑦²−𝑧²=𝑅²) are surfaces which are everywhere spacelike (timelike). They are spheres of the space 𝐺¹₃, called hyperbolic spheres. Planes 𝑦²−𝑧²=0 are everywhere lightlike surfaces, see Figure 1.

Figure 1 

Lightlike planes, a spacelike and a timelike hyperbolic sphere.

The Gaussian curvature of a surface is defined by means of the coefficients of the second fundamental form 𝐿_𝑖𝑗, 𝑖,𝑗=1,2, which are the normal components of 𝐱_,𝑖,𝑗, 𝑖=1,2, respectively. If we put 𝐱_,𝑖,𝑗=²_𝑘=1Γ^𝑘_𝑖𝑗𝐱_,𝑘+𝐿_𝑖𝑗𝐍, (3.19) then the following proposition follows.

Proposition 3.3. One has 𝐿_𝑖𝑗1=𝜖𝑔₁𝑔₁𝐱_,𝑖,𝑗−𝑔_𝑖,𝑗𝐱_,11⋅𝐍=𝜖𝑔₂𝑔₂𝐱_,𝑖,𝑗−𝑔_𝑖,𝑗𝐱_,2⋅𝐍, (3.20)

Proof. The first coordinate of (3.19) is given by 𝑔_𝑖,𝑗=Γ¹_𝑖𝑗𝑔₁+Γ²_𝑖𝑗𝑔₂. (3.21) Under assumption 𝑔₁≠0 we have Γ¹_𝑖𝑗=𝑔_𝑖,𝑗−Γ²_𝑖𝑗𝑔₂𝑔₁, (3.22) and therefore (3.19) turns to 𝐱_,𝑖,𝑗=1𝑔₁𝑔_𝑖,𝑗𝐱_,1+𝑊Γ²_𝑖𝑗𝜎+𝐿_𝑖𝑗𝐍. (3.23) From (3.23) it follows that 𝑔₁𝐱_,𝑖,𝑗−𝑔_𝑖,𝑗𝐱_,1 is an isotropic vector equal to 𝑊Γ²_𝑖𝑗𝜎+𝐿_𝑖𝑗𝑔₁𝐍. The coefficients 𝐿_𝑖𝑗 are obtained by scalar multiplication by 𝐍.

In particular, for the parametrization (3.2) we have 𝐿₁₁1=𝜖𝑊0,0,𝑧_,𝑥,𝑥⋅0,𝑧_,𝑦1,1=−𝜖𝑊𝑧_,𝑥,𝑥,𝐿₁₂1=−𝜖𝑊𝑧_,𝑥,𝑦,𝐿₂₂1=−𝜖𝑊𝑧_,𝑦,𝑦. (3.24)

Functions Γ^𝑘_𝑖𝑗 defined by (3.19) are called the Christoffel symbols of the second kind.

Now we can prove the following proposition.

Proposition 3.4. Derivatives of the side tangential vector 𝜎 and the normal vector 𝐍 are given by 𝜎_,𝑖𝜕∶=𝜕𝑢_𝑖1𝜎=𝑊𝑔₁𝐿_𝑖2−𝑔₂𝐿_1𝑖𝐍𝐍._,𝑖=1𝑊𝑔₁𝐿_𝑖2−𝑔₂𝐿_1𝑖𝜎,𝑖=1,2. (3.25)

Proof. Vectors 𝜎_,𝑖 and 𝐍_,𝑖 are isotropic vectors and therefore can be expressed as linear combinations of 𝜎 and 𝐍. Since 𝐍²∶=𝜖=±1, 𝜎²=−𝜖 it follows 𝐍⋅𝐍_,𝑖=0, 𝜎⋅𝜎_,𝑖=0. Also 𝜎_,𝑖⋅𝐍+𝜎⋅𝐍_,𝑖=0. Having a pseudoscalar product in the isotropic plane, we conclude 𝜎_,𝑖=𝑓𝐍, 𝐍_,𝑖=𝑓𝜎, for a 𝐶^𝑟-function 𝑓, 𝑟≥1.
Now, from the definition of 𝜎 it follows that 𝜎_,𝑖=𝑔₁𝑊_,𝑖𝐱_,2−𝑔₂𝑊_,𝑖𝐱_,1+𝑔₁𝑊𝐱_,2,𝑖−𝑔₂𝑊𝐱_,1,𝑖. (3.26) By using (3.19) it is easy to show that the component by 𝐍 of 𝜎_,𝑖 is equal to (1/𝑊)(𝑔₁𝐿_𝑖2−𝑔₂𝐿_1𝑖).

We will define the Gaussian curvature 𝐾 as the product of principal curvatures, the normal curvatures in the principal direction. The principal directions are tangent directions of a curve 𝑐 on a surface along which the normal field of the surface determines developable ruled surface, that is, det(𝑐,̇𝐍,𝐍̇)=0. This property characterizes principal directions in Euclidean space [16].

Proposition 3.5. The principal directions on a regular admissible surface Φ are given by 𝑔₁d𝑢₁+𝑔₂d𝑢₂=0 (3.27) (the isotropic direction) and 𝑔₂𝐿₁₁−𝑔₁𝐿₁₂d𝑢₁+𝑔₂L₁₂−𝑔₁L₂₂d𝑢₂=0. (3.28)

Proof. Let 𝑐(𝑡)=(𝑥(𝑡),𝑦(𝑡),𝑧(𝑡)) be a curve on a surface Φ. Since 𝐍, 𝐍̇ are isotropic vectors, det(𝑐,̇𝐍,𝐍̇)=0 if and only if 𝑥̇=𝑥_,1d𝑢₁+𝑥_,2d𝑢₂=0 which gives (3.27) or det(𝐍,𝐍̇)=0 which by using Proposition 3.4 gives (3.28).

The principal curvature is given in the next proposition.

Proposition 3.6. The principal curvature 𝑘₁ for the direction (3.27) is given by 𝑘₁𝐿=−𝜖¹¹𝑔²₂−2𝐿₁₂𝑔₁𝑔₂+𝐿₂₂𝑔²₁𝑊² (3.29) and the principal curvature 𝑘₂ for the direction (3.28) by 𝑘₂=𝐿₁₁𝐿₂₂−𝐿²₁₂𝐿₁₁𝑔²₂−2𝐿₁₂𝑔₁𝑔₂+𝐿₂₂𝑔²₁. (3.30)

Proof. Principal curvatures are calculated from 𝑘=𝐼𝐼/𝐼, where 𝐼𝐼=𝐿₁₁d𝑢²₁+2𝐿₁₂d𝑢₁d𝑢₂+𝐿₂₂d𝑢²₂ is the second and 𝐼=d𝑠² the first fundamental form. Therefore, for the direction (3.27) we have 𝑘₁=𝐿₁₁d𝑢²₁+2𝐿₁₂d𝑢₁d𝑢₂+𝐿₂₂d𝑢²₂ℎ₁₁d𝑢²₁+2ℎ₁₂d𝑢₁d𝑢₂+ℎ₂₂d𝑢²₂=𝐿₁₁𝑔²₂−2𝐿₁₂𝑔₁𝑔₂+𝐿₂₂𝑔²₁ℎ₁₁𝑔²₂−2ℎ₁₂𝑔₁𝑔₂+ℎ₂₂𝑔²₁𝐿=−𝜖¹¹𝑔²₂−2𝐿₁₂𝑔₁𝑔₂+𝐿₂₂𝑔²₁𝑊² (3.31) and for the direction (3.28) 𝑘₂=𝐿₁₁d𝑢²₁+2𝐿₁₂d𝑢₁d𝑢₂+𝐿₂₂d𝑢²₂𝑔₁d𝑢₁+𝑔₂d𝑢₂²=𝐿₁₁𝐿₂₂−𝐿²₁₂𝐿₁₁𝑔²₂−2𝐿₁₂𝑔₁𝑔₂+𝐿₂₂𝑔²₁. (3.32)

The Gaussian curvature of a regular admissible surface is defined by 𝐾=𝑘₁𝑘₂𝐿=−𝜖¹¹𝐿₂₂−𝐿²₁₂𝑊² (3.33) or for (3.2) 𝑧𝐾=−𝜖^,𝑥,𝑥𝑧_,𝑦,𝑦−𝑧²_,𝑥,𝑦𝑊⁴. (3.34)

The mean curvature of a surface is defined by 𝜖𝐻=−2𝑊²𝑔²₂𝐿₁₁−2𝑔₁𝑔₂𝐿₁₂+𝑔²₁𝐿₂₂=𝑘₁2, (3.35) and for (3.2) it turns to 𝜖𝐻=2𝑊³𝑧_,𝑦,𝑦. (3.36) Such definition of 𝐻 is motivated by Proposition 3.7

The third fundamental form 𝐼𝐼𝐼 is introduced in the analogous way as in Euclidean space. Since 𝐍 is a unit isotropic field along Φ, the end points of its associated position vectors lie on a hyperbolic unit sphere. More precisely, if 𝐍 is a timelike (spacelike) field, that is, if a surface is spacelike (timelike), then the end points of associated position vectors of 𝐍 lie on a unit spacelike sphere 𝑧²−𝑦²=1 (unit timelike sphere 𝑦²−𝑧²=1), see Figure 1. The obtained mapping is called the Gauss map (the spherical map); the set of all end points of 𝐍 is called the spherical image of a surface. The third fundamental form is the first fundamental form of the spherical image. Therefore it is defined by 𝐼𝐼𝐼=d𝐍⋅d𝐍=𝐍_,1d𝑢₁+𝐍_,2d𝑢₂²=𝐍²_,1d𝑢²₁+2𝐍_,1⋅𝐍_,2d𝑢₁d𝑢₂+𝐍²_,2d𝑢²₂, (3.37) where 𝐍²_,1𝜖=−𝑊²𝑔²₂𝐿²₁₁−2𝑔₁𝑔₂𝐿₁₁𝐿₁₂+𝑔²₁𝐿²₁₂,𝐍_,1⋅𝐍_,2𝜖=−𝑊²𝑔²₂𝐿₁₁𝐿₁₂−𝑔₁𝑔₂𝐿₁₁𝐿₂₂+𝐿²₁₂+𝑔²₁𝐿₁₂𝐿₂₂,𝐍²_,2𝜖=−𝑊²𝑔²₂𝐿²₁₂−2𝑔₁𝑔₂𝐿₁₂𝐿₂₂+𝑔²₁𝐿²₂₂. (3.38) Particularly, for (3.2) we have 𝑁=(1/𝑊)(0,𝑧_,𝑦,1), and d𝑧𝐼𝐼𝐼=−𝜖^,𝑦𝑊²1−d𝑊²1=−𝜖𝑊⁴𝑧_,𝑥,𝑦d𝑥+𝑧_,𝑦,𝑦d𝑦². (3.39) By a simple computation we can notice that the following relation holds.

Proposition 3.7. One has𝐾𝐼−2𝐻𝐼𝐼+𝐼𝐼𝐼=0.

Theorem 3.8. Minimal surfaces in a pseudo-Galielan space 𝐺¹₃ are ruled conoidal surfaces, that is, they are cones with vertices on the absolute line or ruled surfaces with the absolute line as a director curve in infinity.

Proof. We define a normal section in a point of a surface as a plane curve obtained as a section of the surface by a pseudo-Euclidean plane. It can be shown that the curvature of a normal section (parametrized by the arc-length) as a curve in a pseudo-Euclidean plane is 𝜅^∗=𝑘₁. This is obtained from the fact that, for a curve 𝑐(𝑡)=𝐱(𝑢(𝑡),𝑣(𝑡)) parametrized by the arc length, tangent vector field 𝑐′(𝑡) is equal to the side tangential field 𝜎, and therefore 𝑐^′′(𝑡)=𝑘₁𝐍.
Furthermore, 𝜅^∗=0 if and only if a curve is a line in the pseudo-Euclidean plane, and therefore 𝑐(𝑡) is an isotropic line in 𝐺¹₃. Since 𝐻=𝑘₁/2, the assertion follows.

4. Surfaces of Revolution

In the pseudo-Galilean space 𝐺¹₃ there are two types of rotations: pseudo-Euclidean rotations given by the normal form 𝑥=𝑥,𝑦=𝑦cosh𝑡+𝑧sinh𝑡,𝑧=𝑦sinh𝑡+𝑧cosh𝑡 (4.1) and isotropic rotations with the normal form 𝑥(𝑡)=𝑥+𝑏𝑡,𝑦(𝑡)𝑡=𝑦+𝑥𝑡+𝑏²2,𝑧(𝑡)=𝑧, (4.2) where 𝑡∈𝐑 and 𝑏=const>0.

The trajectory of a single point under a pseudo-Euclidean rotation is a pseudo-Euclidean circle (i.e., a rectangular hyperbola) 𝑥=const.,𝑦²−𝑧²=𝑅²,𝑅∈𝐑. (4.3) The invariant 𝑅 is the radius of the circle. Pseudo-Euclidean circles intersect the absolute line 𝑓 in the fixed points of the hyperbolic involution (𝐹₁,𝐹₂). There are three kinds of pseudo-Euclidean circles: circles of real radius, of imaginary radius, and of radius zero. Circles of real radius are timelike curves (having timelike tangent vectors) and of imaginary radius spacelike curves (having spacelike tangent vectors).

The trajectory of a point under the isotropic rotation is an isotropic circle whose normal form is 𝑥𝑧=const.,𝑦=².2𝑏 (4.4) The invariant 𝑏 is the radius of the circle. The fixed line of the isotropic rotations is the absolute line 𝑓.

By rotating a nonisotropic curve 𝑢↦(𝑓(𝑢),𝑔(𝑢),0), 𝑔>0, around the 𝑥-axis by pseudo-Euclidean rotations, we obtain a timelike surface 𝐱(𝑢,𝑣)=𝑓(𝑢),𝑔(𝑢)cosh𝑣,𝑔(𝑢)sinh𝑣, (4.5) and by rotating a curve 𝑢↦(𝑓(𝑢),0,𝑔(𝑢)), 𝑔>0, around the 𝑥-axis, we obtain a spacelike surface 𝐱(𝑢,𝑣)=𝑓(𝑢),𝑔(𝑢)sinh𝑣,𝑔(𝑢)cosh𝑣. (4.6)

The Gaussian curvature of these surfaces is given by 𝐾=−𝜖(𝑔^′/𝑓^′)^′(1/𝑓′𝑔) or if we assume that the rotated curve is parametrized by the arc length 𝑢, 𝑢↦(𝑢,𝑔(𝑢),0) (𝑢↦(𝑢,0,𝑔(𝑢))) by 𝑔𝐾=−𝜖^′′𝑔. (4.7)

Therefore, surfaces with constant curvature 𝐾₀ are described by the ordinary differential equation 𝑔^′′+𝜖𝐾₀𝑔=0,𝐾₀=const. (4.8) Their implicit equation is 𝑦²−𝑧²=𝜖𝑔²(𝑥) (4.9) and the first fundamental form d𝑠²=d𝑢²−𝜖𝛿𝑔²(𝑢)d𝑣². (4.10) The following theorem holds.

Theorem 4.1. The profile curve of surfaces of revolution of constant Gaussian curvature in pseudo-Galilean space is as follows.(1)If 𝐾=−𝜖1/𝑎² (i.e., 𝐾=1/𝑎² for spacelike surfaces and 𝐾=−1/𝑎² for timelike surfaces), then the general solution of the differential equation (4.8) is 𝑔(𝑢)𝑢=𝐴sinh𝑎𝑢+𝐵cosh𝑎,𝐴,𝐵,𝑎∈𝐑. (4.11)(2)If 𝐾=0, then the general solution of the differential equation (4.8) is 𝑔(𝑢)=𝐴𝑢+𝐵,𝐴,𝐵∈𝐑. (4.12)(3)If 𝐾=𝜖1/𝑎², then the general solution of the differential equation (4.8) is 𝑔(𝑢)𝑢=𝐴sin𝑎𝑢+𝐵cos𝑎,𝐴,𝐵,𝑎∈𝐑. (4.13)

Examples of these surfaces are given in Figures 2 and 3. Notice that if 𝐾=0, then the profile curve is a line (4.12). Among these surfaces there are also hyperbolic spheres (𝐴=0, 𝑦²−𝑧²=±𝐵²), see Figures 1 and 4. Cones (𝐴≠0, 𝑦²−𝑧²=±(𝐴𝑥+𝐵)²) are also surfaces of revolution with vanishing curvature.

Figure 2 

A spacelike (timelike) surface of revolution with positive (negative) curvature.

Figure 3 

A spacelike (timelike) surface of revolution with negative (positive) curvature.

Figure 4 

A spacelike hyperbolic sphere (𝐾=0).

Next we consider the isotropic rotations. By rotating an isotropic curve 𝑢↦(0,𝑓(𝑢),𝑔(𝑢)) about the 𝑧-axis by isotropic rotation, we obtain a surface 𝐱()=(𝑢)+𝑣𝑢,𝑣𝑣,𝑓²(𝑢)2𝑏,𝑔,𝑏≠0. (4.14) Let us assume that the rotated curve is parametrized by the arc length 𝑓^′2−𝑔^′2=−𝜖, (4.15) that is, the curve is spacelike (its tangent vectors are spacelike, 𝜖=−1) or timelike (its tangent vectors are timelike, 𝜖=1). By a simple calculation it can be shown that by revolving a spacelike (timelike) curve a spacelike (timelike) surface is obtained. From (4.15) it follows 𝑔^′′=(𝑓^′/𝑔^′)𝑓^′′, and therefore, the following expression for 𝐾 is obtained regardless of the type of the surface: 𝑓𝐾=−^′′𝑏=𝐾₀=const. (4.16) Therefore, the profile curve of a surface with constant curvature is described by the ordinary differential equation 𝑓^′′=−𝑏𝐾₀. (4.17)

Theorem 4.2. The profile curve 𝑢↦(0,𝑓(𝑢),𝑔(𝑢)) of a surface with constant curvature obtained by isotropic rotations is given by 𝑓(𝑢)=−𝑏𝐾₀𝑢²2𝑔(𝑢)1+𝐴𝑢+𝐵,𝐴,𝐵∈𝐑,=−2𝑏𝐾₀𝐴−𝑏K₀𝑢𝐴−𝑏𝐾₀𝑢²−1−Arccosh𝐴−𝑏𝐾₀𝑢+𝐶, (4.18) for a spacelike surface and 𝑔(𝑢)1=−2𝑏𝐾₀𝐴−𝑏𝐾₀𝑢𝐴−𝑏𝐾₀𝑢²+1+Arcsinh𝐴−𝑏𝐾₀𝑢+𝐶 (4.19) and for a timelike surface, where 𝐶∈𝐑 is a constant.
If 𝐾₀=0, then 𝑓(𝑢)=𝐴𝑢+𝐵,𝑔(𝑢)=𝐶𝑢+𝐷,𝐴,𝐵,𝐶,𝐷∈𝐑,𝐴²−𝐶²=−𝜖 (4.20) which implies that the profile curve is a line and obtained surface a parabolic sphere 𝐱(𝑢,𝑣)=(𝑏𝑣,(𝐴𝑢+𝐵)+𝑏𝑣²/2,𝐶𝑢+𝐷) (Figure 5).

Figure 5 

A spacelike parabolic sphere (𝐾=0).

We can notice that this situation appears much more simpler than the same situation in the Euclidean space, where the expressions of the profile curves involve elliptic integrals.

Now we treat surfaces of constant mean curvature. The mean curvature 𝐻 of the surfaces (4.5), (4.6) is given by 𝜖𝐻=−2𝑊²=1.2𝑔 (4.21) Therefore the following theorem holds.

Theorem 4.3. There are no minimal surfaces of revolution (4.5), (4.6). Surfaces with constant mean curvature 𝐻≠0 are hyperbolic timelike, respectively, spacelike spheres obtained by rotating a line (𝑢,−1/2𝐻,0), resp., (𝑢,0,−(1/2𝐻)).

The mean curvature 𝐻 of a surface (4.14) is given by 𝜖𝐻=−2𝑓^′′𝑔^′𝜖=−2𝑓^′′𝑓^′2.+𝜖 (4.22)

Theorem 4.4. The profile curve of a surface of revolution of constant mean curvature obtained by isotropic rotations (Figure 6) in pseudo-Galilean space is as follows. (1)If 𝐻=0, then 𝑓(𝑢)=𝐴𝑢+𝐵, 𝑔(𝑢)=𝐴₁𝑢+𝐵₁,  𝐴,𝐵,𝐴₁,𝐵₁, 𝐴²−𝐴²₁=−𝜖, that is, the surface is generated by an isotropic rotation of an isotropic line (a parabolic sphere). (2)If 𝐻≠0, then for a spacelike surface (𝜖=−1) 𝑓(𝑢)=1ℎ₀sinhℎ₀𝑢+𝑐+𝑐₁(𝑢)=1,𝑔ℎ₀coshℎ₀𝑢+𝑐+𝑐₂ (4.23) and for a timelike surface (𝜖=1)𝑓(𝑢)=1ℎ₀coshℎ₀𝑢+𝑐+𝑐₁(𝑢)=1,𝑔ℎ₀sinhℎ₀𝑢+𝑐+−2, (4.24) where ℎ₀=±2H𝜖=const, 𝑐,𝑐₁,𝑐₂=const. A surface is obtained by an isotropic rotation of a pseudo-Euclidean circle.

Figure 6 

A surface with constant mean curvature obtained by an isotropic rotation.

5. Klein-Gordon Equation and the Tchebyshev Coordinates in 𝐺¹₃

In the context of classical surface theory in the three-dimensional Euclidean space, the sine-Gordon equation has the geometrical interpretation in terms of surfaces with negative constant Gaussian curvature. This is shown by parametrizing a surface by coordinates that satisfy 𝐸=𝐺=1,𝐹=cos𝜑, (5.1) where 𝐸,𝐹,and𝐺 are coefficients of the first fundamental form (i.e., by Tchebyshev coordinates). Then Theorema Egregium implies 𝜑_,𝑢,𝑣=−𝐾sin𝜑. (5.2)

In particular, for a surface with constant negative curvature 𝐾=−1/𝑚², the previous equation turns to the sine-Gordon equation for the angle 𝜑 between parametric curves 𝜑_,𝑢,𝑣=1𝑚²sin𝜑. (5.3)

Similar results hold in a three-dimensional pseudo-Riemannian manifold of constant curvature (e.g., Minkowski space 𝐑¹₃). The angle between curves of the Tchebyshev net on a spacelike (timelike) surface of constant negative (positive) curvature satisfies the sine-Gordon equation or its hyperbolic analogue sinh-Gordon equation [17].

Our aim is to introduce the analogue of the Tchebyshev coordinates on a surface in the pseudo-Galilean space and to establish a partial differential equation satisfied by the angle between the parametric curves. In order to be able to consider the angle between parametric curves of a surface, it is assumed that parametric curves are non-isotropic curves, that is, 𝑔₁≠0,𝑔₂≠0.

We proceed with the following definition.

Definition 5.1. Tchebyshev net on a surface is the net of parametric curves for which the first fundamental coefficients satisfy 𝑔₁=𝑔₂=1, ℎ_11,2=ℎ_22,1.

Notice that according to our notation we have ℎ_{𝑖𝑖,𝑗}=𝜕ℎ_𝑖𝑖/𝜕𝑢_𝑗, 𝑖,𝑗=1,2.

The first condition from the definition implies that the parametric curves of this net are parametrized by the arc length. The side tangential vector in these coordinates is given by 𝜎=(1/𝑊)(𝐱_,2−𝐱_,1), where 𝑊²=−𝜖(𝐱_,2−𝐱_,𝟏)². Furthermore, since ℎ_11,2=𝜕𝐱²_,1𝜕𝑣=2𝐱_,1⋅𝐱_,1,2,ℎ_22,1=𝜕𝐱²_,2𝜕𝑢=2𝐱_,2⋅𝐱_,1,2, (5.4) it follows that 𝜎⋅𝐱_,1,2=1𝑊𝐱_,2−𝐱_,1⋅𝐱_,1,2=1ℎ2𝑊^22,1−ℎ_11,2=0. (5.5) Such definition of the Tchebyshev coordinates is motivated by the following theorem whose counterpart holds in Euclidean space. For the analogous result in simply isotropic space 𝐼¹₃ see [18].

Theorem 5.2. Asymptotic curves on a 𝐶^𝑟-surface Φ in 𝐺¹₃, 𝑟≥3, form the Tchebychev net if and only if Φ is a spacelike surface with constant negative curvature or timelike surface with constant positive curvature.

Proof. First notice that on spacelike surfaces with negative curvature and timelike surface with positive curvature there are two families of real asymptotic curves, due to the fact that the equation for the asymptotic curves 𝐼𝐼=0 has two real solutions.
Proof follows from the analogues of the Gauss and Codazzi-Mainardi equations for surfaces in the pseudo-Galilean space 𝐺¹₃. They are obtained in the following way.
Let 𝐱=𝐱(𝑥(𝑢₁,𝑢₂),𝑦(𝑢₁,𝑢₂),𝑧(𝑢₁,𝑢₂)) be a parametrization of Φ, and let 𝐍 denote its unit normal field. Multiplying (3.23) by 𝜎 (and analogous expression obtained when assuming 𝑔₂≠0), it can be shown that the Christoffel symbols of the second kind defined by (3.19) are given by Γ¹_𝑖𝑗=𝜖𝑊𝑔₂𝐱_,𝑖,𝑗−𝑔_𝑖,𝑗𝐱_,2Γ⋅𝜎,_2𝑖𝑗𝜖=−𝑊𝑔₁𝐱_,𝑖,𝑗−𝑔_𝑖,𝑗𝐱_,1⋅𝜎. (5.6)
Under assumption 𝐱_{,𝑖,𝑗,𝑘}=𝐱_{,𝑖,𝑘,𝑗} the following is obtained: 𝐿_{𝑖𝑗,𝑘}−𝐿_{𝑖𝑘,𝑗}+_𝑛Γ^𝑛_𝑖𝑗𝐿_𝑛𝑘−Γ^𝑛_𝑖𝑘𝐿_𝑛𝑗𝐍+_𝑚Γ^𝑚_{𝑖𝑗,𝑘}−Γ^𝑚_{𝑖𝑘,𝑗}+_𝑛Γ^𝑛_𝑖𝑗Γ^𝑚_𝑛𝑘−Γ^𝑛_𝑖𝑘Γ^𝑚_𝑛𝑗+𝑔^𝑚𝑔^𝑛𝐿_𝑖𝑗𝐿_𝑛𝑘−𝐿_𝑖𝑘𝐿_𝑗𝑛𝐱_,𝑚=𝟎, (5.7) where 𝑔¹=𝑔₂/𝑊, 𝑔²=−𝑔₁/𝑊. We can notice that the previous formula differs from its analogue in the Galilean space [10] in the sign of the term 𝑔^𝑚𝑔^𝑛(𝐿_𝑖𝑗𝐿_𝑛𝑘−𝐿_𝑖𝑘𝐿_𝑗𝑛). This is a consequence of the formulas for 𝐍_𝑖 in Proposition 3.4 (with the opposite sign than in the Galilean space).
The component by 𝐱_,𝑚 gives the Gauss (integrability) equation Γ^𝑚_{𝑖𝑗,𝑘}−Γ^𝑚_{𝑖𝑘,𝑗}+_𝑛Γ^𝑛_𝑖𝑗Γ^𝑚_𝑛𝑘−Γ^𝑛_𝑖𝑘Γ^𝑚_𝑛𝑗=−_𝑛𝑔^𝑚𝑔^𝑛𝐿_𝑖𝑗𝐿_𝑛𝑘−𝐿_𝑖𝑘𝐿_𝑗𝑛,𝑚=1,2, (5.8) and the component by 𝐍 the Codazzi-Mainardi equation 𝐿_{𝑖𝑗,𝑘}−𝐿_{𝑖𝑘,𝑗}=−_𝑛Γ^𝑛_𝑖𝑗𝐿_𝑛𝑘−Γ^𝑛_𝑖𝑘𝐿_𝑛𝑗. (5.9) Now, let us assume that the Gaussian curvature 𝐾 of a surface is constant and that the parametric curves asymptotic. For parametrization with asymptotic lines we have 𝐿₁₁=𝐿₂₂=0. We consider spacelike surfaces with negative curvature and timelike surfaces with positive curvature, that is, surfaces that have two families of asymptotic lines. In asymptotic coordinates equation (5.9) for 𝑖=𝑗=1,𝑘=2, and 𝑖=𝑘=2,𝑗=1 reduces to 𝐿_12,1=𝐿₁₂Γ¹₁₁−Γ²₁₂,𝐿_12,2=𝐿₁₂Γ²₂₂−Γ¹₁₂. (5.10) Furthermore, the assumption 𝐾=const.=𝜖(1/𝑚²) implies 𝐿²₁₂=𝑊²(1/𝑚²). By derivating this equation with using we get 𝑊_,1=𝑊Γ¹₁₁−Γ²₁₂,𝑊_,2=𝑊Γ²₂₂−Γ¹₁₂. (5.11) On the other hand, partial derivatives of 𝑊, 𝑊²=−𝜖(𝑔₁𝐱_,2−𝑔₂𝐱_,1)² for an arbitrary parametrization of a regular admissible surface are equal to 𝑊_,1=𝑊Γ¹₁₁+Γ²₁₂,𝑊_,2=𝑊Γ²₂₂+Γ¹₁₂. (5.12) They are obtained by derivating the expression 𝑊²=−𝜖(𝑔₁𝐱_,2−𝑔₂𝐱_,1)² and using 𝑔₂𝐱_,1,1𝑊⋅𝜎=𝜖Γ¹₁₁+𝑔_1,1𝐱_,2𝑔⋅𝜎,₁𝐱_,1,2𝑊⋅𝜎=−𝜖Γ²₁₂+𝑔_1,2𝐱_,1⋅𝜎, (5.13) obtained from for 𝑖=𝑗=1 and 𝑖=1, 𝑗=2.
Now from and it follows that Γ¹₁₂=Γ²₁₂=0. (5.14) By using expressions in we can write this system as 𝑔_1,2𝐱_,1−𝑔₁𝐱_,1,2𝑔⋅𝜎=0,₂𝐱_,1,2−𝑔_1,2𝐱_,2⋅𝜎=0. (5.15) It follows that 𝑔_1,2𝑔₂𝐱_,1−𝑔₁𝐱_,2⋅𝜎=0. (5.16) Furthermore (𝑔₂𝐱_,1−𝑔₁𝐱_,2)⋅𝜎=−𝑊𝜎²≠0, and therefore the previous equation implies 𝑔_1,2=0. (5.17) Substituting (5.17) into the system we obtain (since 𝑔_𝑖≠0, 𝑖=1,2) 𝐱_,1,2⋅𝜎=0. (5.18) Let us analyze conditions (5.17), (5.18). Condition (5.17) implies 𝑔₁=𝑔₁(𝑢), 𝑔₂=𝑔₂(𝑣), that is, functions 𝑔_𝑖 are functions of one parameter only.
Condition (5.18) implies 𝐱_,1,21⋅𝜎=𝑊𝐱_,1,2⋅𝑔₁𝐱_,2−𝑔₂𝐱_,1=1𝑔2𝑊¹ℎ_22,1−𝑔₂ℎ_11,2, (5.19) and therefore 𝑔₁ℎ_22,1−𝑔₂ℎ_11,2=0. (5.20) Condition (5.17) enables us to introduce new coordinates 𝑢=𝑔₁d𝑢,𝑣=𝑔₂d𝑣 (5.21) to obtain 𝑔₁=𝑑𝑥/𝑑𝑢=(𝑑𝑥/𝑑𝑢)(𝑑𝑢/𝑑𝑢)=𝑔₁(1/𝑔₁)=1. Analogously we can obtain 𝑔₂=1. For the coefficients ℎ_𝑖𝑗 of the new parametrization we have ℎ₁₁=𝜕𝐱𝜕𝑢²=1𝑔²₁ℎ₁₁,ℎ₂₂=𝜕𝐱𝜕𝑣²=1𝑔²₂ℎ₂₂. (5.22) Therefore, since 𝑔_1,2=0, ℎ_11,2=𝑔₂𝑔²₁ℎ_11,2,ℎ_22,1=𝑔₁𝑔²₂ℎ_22,1. (5.23) Condition (5.20) now implies (𝜕ℎ₁₁/𝜕𝑣)−(𝜕ℎ₂₂/𝜕𝑢)=ℎ_11,2−ℎ_22,1=(1/𝑔₂𝑔²₁)ℎ_11,2−(1/𝑔₁𝑔²₂)ℎ_22,1=(1/𝑔²₁𝑔²₂)(𝑔₂ℎ_11,2−𝑔₁ℎ_22,1)=0, which means that the considered coordinates are Tchebychev.
Conversely, if the asymptotic curves form the Tchebychev net, let us prove that Φ has constant curvature. The assumptions imply 𝐿₁₁=𝐿₂₂=0 and 𝐱_,1,2⋅𝜎=0, (5.24) where 𝜎=(1/𝑤)(𝐱_,2−𝐱_,1). Expressions in imply that in Tchebyshev coordinates Christoffel symbols Γ¹₁₂, Γ²₁₂, are given by Γ¹₁₂=𝜖𝑊𝐱_,1,2Γ⋅𝜎=0,₂₁₂𝜖=−𝑊𝐱_,1,2⋅𝜎=0. (5.25) Therefore Codazzi-Mainardi equations are equal to 𝐿_12,1=𝐿₁₂Γ¹₁₁,𝐿_12,2=𝐿₁₂Γ²₂₂. (5.26) Now by differentiating 𝐿𝐾=𝜖²¹²𝑊², (5.27) partially in the first variable, we obtain 𝐾_,1𝑊³=2𝜖𝐿₁₂𝐿_12,1𝑊−𝐿₁₂𝑊_,1 (5.28) and (5.26) implies 𝐾_,1𝑊³=2𝜖𝐿²₁₂Γ¹₁₁𝑊−𝑊_,1. (5.29) Since from we have 𝑊_,1=Γ¹₁₁𝑊, (5.30) it follows that 𝐾_,1=0. (5.31) In the same way we can conclude 𝐾_,2=0, (5.32) and therefore 𝐾=const, what was claimed.

Let us now determine the angle between curves of the Tchebyshev net on a regular admissible surface. The angle between nonisotropic unit vectors 𝐚=1,𝑎₂,𝑎₃,𝐛=1,𝑏₂,𝑏₃ (5.33) in the pseudo-Galilean space 𝐺¹₃ is determined by using the following expression: 𝜑²=∠(𝐚,𝐛)²=𝑏₂−𝑎₂²−𝑏₃−𝑎₃². (5.34) The defined angle is invariant under the group of motions . By applying expression (5.34) to the tangent vectors of the Tchebyshev parametric curves, we obtain 𝜑²=𝑦₂−𝑦₁²−𝑧₂−𝑧₁²=𝑊², (5.35) and therefore we can write 𝜑²=−𝜖𝑦₂−𝑦₁²−𝑧₂−𝑧₁². (5.36) The function 𝜑 and the function 𝑊 as well are differentiable of class 𝐶^𝑟, 𝑟≥2, if and only if a surface is not lightlike.

For the Tchebyshev coordinates we have 𝑔₁=𝑔₂=1,Γ¹₁₂=Γ²₁₂=0, (5.37) and the Gauss (integrability) equation (5.8) for 𝑖=𝑗=1,𝑘=2,𝑚=1 turns to Γ¹_11,2+Γ²₁₁Γ¹₂₂𝐿=−²¹²𝑊²=−𝜖𝐾. (5.38) Furthermore, from we have Γ¹₁₁=𝑊_,1𝑊,Γ²₂₂=𝑊_,2𝑊, (5.39) which with implies Γ²₁₁𝑊=−^,1𝑊,Γ¹₂₂𝑊=−^,2𝑊. (5.40) Therefore −𝜖𝐾=𝑊𝑊_,1,2−𝑊_,1𝑊_,2𝑊²+−𝑊_,1𝑊−𝑊_,2𝑊=𝑊_,1,2𝑊, (5.41) that is, 𝑊_,1,2=−𝜖𝐾𝑊. Since 𝑊=±𝜑, where 𝜑 is the angle between Tchebyshev curves, then we have 𝜑_,1,2=−𝜖𝐾𝜑. (5.42) If the Gaussian curvature of the surface is constant 𝐾=𝜖(1/𝑚²), 𝑚∈𝐑, the previous equation turns to the Klein-Gordon equation 𝜑_,1,21=−𝑚²𝜑. (5.43) Therefore we have proved the following theorem:

Theorem 5.3. The angle between curves of the Tchebychev net and the function 𝑊 as well on a spacelike surface of constant negative curvature and timelike surface of constant positive curvature in 𝐺¹₃ satisfy the Klein-Gordon equation.

Finally, the functions 𝜅(𝑢,𝑣)=𝐱_,1,1(𝑢,𝑣)⋅𝐱_,1,1(𝑢,𝑣),𝜅(𝑢,𝑣)=𝐱_,2,2(𝑢,𝑣)⋅𝐱_,2,2(𝑢,𝑣) (5.44) satisfy the Klein-Gordon equation as well. The function 𝜅(𝑢,𝑣₀) is the curvature of a parametric curve 𝑣=𝑣₀ of the Tchebychev net and 𝜅(𝑢₀,𝑣) a parametric curve 𝑢=𝑢₀. Notice that these curves are parametrized by the arc-length, since 𝑥₁=𝑥₂=1.

Theorem 5.4. The functions 𝜅, 𝜅 on a spacelike surface of constant negative curvature and timelike surface of constant positive curvature in 𝐺¹₃ satisfy the Klein-Gordon equation.

Proof. We have 𝐱_,1,1=Γ¹₁₁𝐱_,1−𝐱_,2=Γ¹₁₁(−𝑊𝜎)=−𝑊_,1𝜎 (5.45) and therefore 𝐱_,1,1 is a spacelike (timelike) vector for a spacelike (timelike) surface. Hence we can write 𝜅²(𝑢,𝑣)=−𝜖𝐱_,1,1⋅𝐱_,1,1=𝑊²_,1. (5.46) Now we have 𝜅𝜅_,1=𝑊_,1𝑊_,1,1,𝜅𝜅_,2=𝑊_,1𝑊_,1,2. (5.47) Therefore 𝜅𝜅_,1,2=𝜅𝜅_,2,1−𝜅_,1𝜅_,2=𝑊_,1𝑊_,1,2,1−𝑊_,1,1𝑊_,1,2=𝑊_,1𝑊_,1,2,11=−𝑚²𝑊²_,11=−𝑚²𝜅². (5.48) Therefore we have 𝜅_,1,2=−(1/𝑚²)𝜅. Analogously we prove for 𝜅.

Theorem 5.5. The parametric net on a spacelike surface of revolution (4.6) obtained by pseudo-Euclidean rotations forms the Tchebyshev net in the following parametrization of the surface (𝐾=−1/𝑎²): 𝑥()𝑦()=𝑢,𝑣=𝑢+𝑣,𝑢,𝑣𝐴sin𝑢+𝑣𝑎+𝐵cos𝑢+𝑣𝑎sinh𝑢−𝑣𝑎,𝑧(𝑢,𝑣)=𝐴sin𝑢+𝑣𝑎+𝐵cos𝑢+𝑣𝑎cosh𝑢−𝑣𝑎, (5.49) and on a timelike surface of revolution (4.5) (𝐾=1/𝑎², see Figure 7) 𝑥()𝑦()=𝑢,𝑣=𝑢+𝑣,𝑢,𝑣𝐴sin𝑢+𝑣𝑎+𝐵cos𝑢+𝑣𝑎cosh𝑢−𝑣𝑎,𝑧(𝑢,𝑣)=𝐴sin𝑢+𝑣𝑎+𝐵cos𝑢+𝑣𝑎sinh𝑢−𝑣𝑎. (5.50) The parametric net on a surface of revolution (4.14) obtained by isotropic rotations forms the Tchebyshev net in the following parametrization of a spacelike surface with 𝐾=−2𝑎/𝑏<0,𝑎,𝑏∈𝐑⧵{0} (see Figure 8): 𝑥()𝑦()=1𝑢,𝑣=𝑢+𝑣,𝑢,𝑣4𝑎−𝐴+cosh2𝑎𝑏(𝑢−𝑣)²+𝐴2𝑎−𝐴+cosh2𝑎𝑏()1𝑢−𝑣+𝐵+()2𝑏𝑢+𝑣²,𝑧()=1𝑢,𝑣14𝑎2sinh2𝑎𝑏(2𝑢−2𝑣)−2𝑎𝑏(𝑢−𝑣), (5.51) and of a timelike surface with𝐾=(2𝑎/𝑏)>0,𝑎,𝑏∈𝐑⧵{0}𝑥()𝑦()1𝑢,𝑣=𝑢+𝑣,𝑢,𝑣=−4𝑎𝐴−sinh2𝑎𝑏(𝑢−𝑣)²+𝐴2𝑎𝐴−sinh2𝑎𝑏()1𝑢−𝑣+𝐵+()2𝑏𝑢+𝑣²,𝑧()=1𝑢,𝑣14𝑎2sinh2𝑎𝑏(2𝑢−2𝑣)+2𝑎𝑏(𝑢−𝑣). (5.52)

(a)

(b)

Figure 7 

A timelike surface obtained by pseudo-Euclidean rotations parametrized by the Tchebychev net and the projection of a parametric curve onto 𝑦𝑧-plane.

(a)

(b)

Figure 8 

A spacelike surface obtained by isotropic rotations parametrized by the Tchebychev net and the projection of a parametric curve onto 𝑦𝑧-plane.

Remark 5.6. Notice that for the given parametrizations of surfaces obtained by pseudo-Euclidean rotations we have 𝑔₁=𝑔₂=1,ℎ₁₁=ℎ₂₂, (5.53) that is, Tchebyshev curves satisfy condition similar to that one in the Euclidean space (𝐸=𝐺=1). Therefore, the parametric curves satisfy ℎ_11,2=ℎ_22,1=ℎ_11,1 and ℎ_22,1=ℎ_11,2=ℎ_22,2, which describes ℎ₁₁, ℎ₂₂ as the functions of form 𝜓(𝑢+𝑣). The angle between curves of the Tchebychev net for spacelike and timelike surfaces obtained by pseudo-Euclidean rotations is equal to 𝜑()=2𝑢,𝑣𝑎𝐴sin𝑢+𝑣𝑎+𝐵cos𝑢+𝑣𝑎. (5.54) For spacelike surfaces obtained by isotropic rotations we have ℎ₁₁−ℎ₂₂=𝑢+𝑣2𝑎𝑏², (5.55) and the angle is equal to 𝜑(𝑢,𝑣)=2𝑎𝑏sinh2𝑎𝑏(𝑢−𝑣) (5.56) and for timelike surfaces obtained by isotropic rotations to 𝜑(𝑢,𝑣)=2𝑎𝑏cosh2𝑎𝑏(𝑢−𝑣). (5.57) Furthermore, functions 𝜅(𝑢,𝑣), 𝜅(𝑢,𝑣) are given by 𝜅(𝑢,𝑣)=𝜅()=2𝑢,𝑣𝑎²𝐴cos𝑢+𝑣𝑎−𝐵sin𝑢+𝑣𝑎 (5.58) for spacelike and timelike surfaces obtained by pseudo-Euclidean rotations and by 𝜅(𝑢,𝑣)=𝜅()=2𝑢,𝑣𝑏cosh2𝑎𝑏(𝑢−𝑣),𝜅(𝑢,𝑣)=𝜅()=2𝑢,𝑣𝑏sinh2𝑎𝑏(𝑢−𝑣) (5.59) for spacelike, respectively, timelike surfaces obtained by isotropic rotations.

Remark 5.7. Notice that the torsion of the parametric curves in the Tchebyshev parametrization is constant. This is the consequence of the general result which states that the torsion of asymptotic curves on a spacelike surface of negative curvature and timelike surface of positive curvature is equal to 𝜏=±√𝜖𝐾.

Remark 5.8. From a known solution of the Klein-Gordon equation, we can construct a surface. Let us consider a more general particular solution of the Klein-Gordon equation, that is, the solution 𝜑(𝑢,𝑣)=𝑒^{𝜆(𝑢−𝑣)/𝑎}𝜇()𝐴cos𝑢+𝑣𝑎𝜇()+𝐵sin𝑢+𝑣𝑎,𝜆,𝜇,𝐴,𝐵,𝑎∈𝐑, (5.60) which satisfies 𝜑_,1,2=−((𝜆²+𝜇²)/𝑎²)𝜑. Considering (5.60) as the curvature 𝜅 of a family of curves in the Tchebyshev net, we can construct a surface of constant curvature 𝐾 by means of the Frenet frame {𝑡,𝑛,𝑏} of parametric curves. Frenet formulas are 𝑡′=𝜅𝑛, 𝑛′=𝜏𝑏, 𝑏′=𝜏𝑛, where 𝜏=±(𝜆²+𝜇²)/𝑎², and they allow us to construct fields 𝑡=𝐱₁ or 𝑡=𝐱₂ and 𝑛,𝑏 and therefore a parametrization 𝐱=𝐱(𝑢,𝑣). The obtained family of surfaces contains family of surfaces of revolution (𝜆=0 for surfaces obtained by pseudo-Euclidean rotations and 𝜇=0 for surfaces obtained by isotropic rotations).

Example 5.9. A surface with parametrization 𝑥()𝑦()=1𝑢,𝑣=𝑢+𝑣,𝑢,𝑣𝑒10^{2(𝑢−𝑣)}2cos(𝑢+𝑣)+sin(𝑢+𝑣)√5cosh√5(𝑢−𝑣)−2sinh√5(),𝑧()=1𝑢−𝑣𝑢,𝑣𝑒10^{2(𝑢−𝑣)}2cos(𝑢+𝑣)+sin(𝑢+𝑣)−2cosh√5(𝑢−𝑣)+√5sinh√5(𝑢−𝑣) (5.61) is a spacelike surface parametrized by the Tchebyshev net having 𝐾=−5 (see Figure 9).

Figure 9 

A spacelike surface of constant curvature 𝐾=−5 with 𝜆=2, 𝜇=𝐴=𝑎=1, 𝐵=0.