Surfaces of Constant Curvature in the Pseudo-Galilean Space

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.


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 P 3 R various types of Cayley-Klein spaces can be defined, such as elliptic and hyperbolic space, Euclidean and pseudo-Euclidean

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 P 3 R , f the line absolute line in ω, and I the fixed hyperbolic involution of points of f.Homogeneous coordinates in G 1 3 are introduced in such a way that the absolute plane ω is given by x 0 0, the absolute line f by x 0 x 1 0, and the hyperbolic involution by 0 : 0 : x 2 : x 3 → 0 : 0 : x 3 : x 2 .The last condition is equivalent to the requirement that the conic x 2 2 − x 2 3 0 is the absolute conic.Metric relations are introduced with respect to the absolute figure.In affine coordinates defined by x 0 : x 1 : x 2 : x 3 1 : x : y : z , distance between points P i x i , y i , z i , i 1, 2, is defined by The group of motions of G 1 3 is a six-parameter group given in affine coordinates by x a x, y b cx y cosh ϕ z sinh ϕ, z d ex y sinh ϕ z cosh ϕ.

2.2
It leaves invariant the absolute figure as well as the pseudo-Galilean distance 2.1 of points.

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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 |y 2 − z 2 | equal to zero; they are called lightlike vectors.Isotropic vectors satisfying y 2 − z 2 > 0 are said to be spacelike vectors and vectors satisfying y 2 − z 2 < 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 while in the isotropic plane

2.4
The pseudo-Galilean space G 1 3 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 P 3 11|001 and also called the Galilean space of index 1. 1 3 We will treat a C r -surface, r ≥ 2, as a subset Φ ⊂ G 1 3 for which there exists an open subset D of R 2 and C r -mapping x : / 0, 3.1 then such a surface is admissible and can be locally expressed in the form z z x, y .

3.2
Let Φ ⊂ G 1 3 be a regular admissible surface.We define a side tangential vector by

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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 is equal to the pseudo-Galilean norm of the isotropic vector x ,1 x ,2 − x ,2 x ,1 .In particular, in the parametrization 3.2 we have 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 P 0 of a regular admissible surface, there is a unique isotropic direction defined by the condition x ,1 du 1 x ,2 du 2 0. This isotropic line in a tangent plane in a point P 0 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 P 0 and S ⊥ is perpendicular to the tangent plane.Therefore a unit surface normal field is defined by

3.6
We introduce a pseudo-Galilean cross-product in the following way: where e 2 0, 1, 0 , e 3 0, 0, 1 , e 2 is a unit spacelike, and e 3 is a unit timelike vector, a 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 × G b is the isotropic vector defined by the relation for any vector c.By ∼ above a vector, the projection of a vector on the pseudo-Euclidean yzplane is denoted and by • the pseudo-Euclidean scalar product in the same plane, y 1 , y  are positive definite, while for the timelike surfaces, the form ds 2 1 is positive definite whereas ds 2 2 is negative definite.We have assumed here, without loss of generality, g 1 / 0. In the latter case this means that the matrix of the first fundamental form ds 2 1 0 0 ds 2 2 3.17 is indefinite, analogously to the timelike surfaces in the Minkowski space R 3 1 12 .In particular, for the parametrization 3.2 we have ds 2 dx 2 δ 1 − z 2 ,y dy 2 , 3.18 since dx 0 when δ 1.
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 then the following proposition follows.

Proposition 3.3. One has
Proof.The first coordinate of 3.19 is given by

3.21
Under assumption g 1 / 0 we have and therefore 3.19 turns to The coefficients L ij are obtained by scalar multiplication by N.
In particular, for the parametrization 3.2 we have

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 :

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 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

3.26
International Journal of Mathematics and Mathematical Sciences By using 3.19 it is easy to show that the component by 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 ċ, 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 g 1 du 1 g 2 du 2 0 3.27 (the isotropic direction) and Proof.Let c t x t , y t , z t be a curve on a surface Φ.Since N, Ṅ are isotropic vectors, det ċ, N, Ṅ 0 if and only if ẋ x ,1 du 1 x ,2 du 2 0 which gives 3.27 or det N, ˙ N 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 k 1 for the direction 3.27 is given by and the principal curvature k 2 for the direction 3.28 by 3 .

3.32
The Gaussian curvature of a regular admissible surface is defined by

3.34
The mean curvature of a surface is defined by 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 z 2 − y 2 1 unit timelike sphere y 2 − z 2 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

10
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3.39
By a simple computation we can notice that the following relation holds.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 * k 1 .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 k 1 N.
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 G 1 3 .Since H k 1 /2, the assertion follows.

Surfaces of Revolution
In the pseudo-Galilean space G 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 F 1 , F 2 .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 The invariant b is the radius of the circle.The fixed line of the isotropic rotations 4.2 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 and by rotating a curve u → f u , 0, g u , g > 0, around the x-axis, we obtain a spacelike surface The Gaussian curvature of these surfaces is given by K − 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 International Journal of Mathematics and Mathematical Sciences Therefore, surfaces with constant curvature K 0 are described by the ordinary differential equation Their implicit equation is and the first fundamental form ds 2 du 2 − δg 2 u dv 2 .

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.

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, y 2 − z 2 ±B 2 , see Figures 1 and 4. Cones A / 0, y 2 − z 2 ± Ax B 2 are also surfaces of revolution with vanishing curvature.
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 Let us assume that the rotated curve is parametrized by the arc length International Journal of Mathematics and Mathematical Sciences  that is, the curve is spacelike its tangent vectors are spacelike, −1 or timelike its tangent vectors are timelike, 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: Therefore, the profile curve of a surface with constant curvature is described by the ordinary differential equation f −bK 0 .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 for a spacelike surface and and for a timelike surface, where which implies that the profile curve is a line and obtained surface a parabolic sphere x u, v bv, Au B bv 2 /2, Cu D (Figure 5).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 Therefore the following theorem holds.
The mean curvature H of a surface 4.14 is given by

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.
− , 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 ( −1) and for a timelike surface 1 where h 0 ±2H const, c, c 1 , c 2 const.A surface is obtained by an isotropic rotation of a pseudo-Euclidean circle. 1 3 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

Klein-Gordon Equation and the Tchebyshev Coordinates in G
where E, F, and G are coefficients of the first fundamental form i.e., by Tchebyshev coordinates .Then Theorema Egregium implies ϕ ,u,v −K sin ϕ.

5.2
In particular, for a surface with constant negative curvature K −1/m 2 , the previous equation turns to the sine-Gordon equation for the angle ϕ between parametric curves ϕ ,u,v 1 m 2 sin ϕ.

5.3
Similar results hold in a three-dimensional pseudo-Riemannian manifold of constant curvature e.g., Minkowski space R 1 3 .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, g 1 / 0, g 2 / 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 g 1 g 2 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 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 I 1 3 see 18 .
Theorem 5.2.Asymptotic curves on a C r -surface Φ in G 1 3 , 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 G 1 3 .They are obtained in the following way.Let x x x u 1 , u 2 , y u 1 , u 2 , z u 1 , u 2 be a parametrization of Φ, and let N denote its unit normal field.Multiplying 3.23 by σ and analogous expression obtained when assuming g 2 / 0 , it can be shown that the Christoffel symbols of the second kind defined by 3.19 are given by j x ,1 • σ.

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Under assumption x ,i,j,k x ,i,k,j the following is obtained: where g 1 g 2 /W, g 2 −g 1 /W.We can notice that the previous formula differs from its analogue in the Galilean space 10 in the sign of the term g m g n L ij L nk − L ik L 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 and the component by N the Codazzi-Mainardi equation 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 L 11 L 22 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

International Journal of Mathematics and Mathematical Sciences 19
On the other hand, partial derivatives of W, W 2 − g 1 x ,2 − g 2 x ,1 2 for an arbitrary parametrization of a regular admissible surface are equal to

5.12
They are obtained by derivating the expression W 2 − g 1 x ,2 − g 2 x ,1 2 and using 5.13 obtained from 5.6 for i j 1 and i 1, j 2. Now from 5.11 and 5.12 it follows that By using expressions in 5.6 we can write this system as It follows that Furthermore g 2 x ,1 − g 1 x ,2 • σ −Wσ 2 / 0, and therefore the previous equation implies g 1,2 0. 5.17 Substituting 5.17 into the system 5.15 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 g 1 g 1 u , g 2 g 2 v , that is, functions g i are functions of one parameter only.Condition 5.18 implies Let us now determine the angle between curves of the Tchebyshev net on a regular admissible surface.The angle between nonisotropic unit vectors in the pseudo-Galilean space G 1 3 is determined by using the following expression:

5.34
The defined angle is invariant under the group of motions 2.2 .By applying expression 5.34 to the tangent vectors of the Tchebyshev parametric curves, we obtain and therefore we can write

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 Finally, the functions satisfy the Klein-Gordon equation as well.The function κ u, v 0 is the curvature of a parametric curve v v 0 of the Tchebychev net and κ u 0 , v a parametric curve u u 0 .Notice that these curves are parametrized by the arc-length, since x 1 x 2 1.
Theorem 5.4.The functions κ, κ on a spacelike surface of constant negative curvature and timelike surface of constant positive curvature in G 1 3 satisfy the Klein-Gordon equation.
Proof.We have and therefore x ,1,1 is a spacelike timelike vector for a spacelike timelike surface.Hence we can write

5.46
Now we have
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/a 2 ):

5.49
International Journal of Mathematics and Mathematical Sciences and on a timelike surface of revolution 4.5 (K 1/a 2 , see Figure 7)

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):

5.52
Remark 5.6.Notice that for the given parametrizations of surfaces obtained by pseudo-Euclidean rotations we have

Proposition 3 . 7 .Theorem 3 . 8 .
One has K I − 2H II III 0.Minimal surfaces in a pseudo-Galielan space G 1 3 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.

Figure 2 :
Figure 2: A spacelike timelike surface of revolution with positive negative curvature.

Figure 3 :
Figure 3: A spacelike timelike surface of revolution with negative positive curvature.

Figure 6 :
Figure 6: A surface with constant mean curvature obtained by an isotropic rotation.
,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 1 3 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 u 1 , u 2 , y u 1 , u 2 , z u 1 , u 2 , x ,i ∂x/∂u i , y 1 g 2 1, h 11 h 22 , 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 ,