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.
Abstract
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 G 3 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.
Abstract
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.
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.
Introduction
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
Preliminaries
The group of motions of G 1 3 is a six-parameter group given in affine coordinates by
Preliminaries
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 .
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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
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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.
The Gaussian Curvature of Surfaces in G 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 : 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
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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
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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
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We introduce a pseudo-Galilean cross-product in the following way:
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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 a 1 , a 2 , a 3 
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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
International Journal of Mathematics and Mathematical Sciences
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 2 · z 1 , z 2 y 1 z 1 − y 2 z 2 . Obviously the following proposition holds.
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International Journal of Mathematics and Mathematical Sciences 5 Proposition 3.1. The side tangential vector σ and the normal vector N are unit isotropic vectors that satisfy
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Since the normal vector field satisfies N · N ±1, we distinguish two basic types of admissible surfaces: spacelike surfaces having timelike surface normals −1 and timelike surfaces having spacelike normals 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
3.15
and the surface is spacelike if g 2 1 h 22 − 2g 1 g 2 h 12 g 2 2 h 11 > 0, timelike otherwise. We can notice that if a surface is spacelike, both parts of the first fundamental form, ds 2 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 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
3.15
then the following proposition follows.
3.15
International Journal of Mathematics and Mathematical Sciences 7 Proposition 3.3. One has
3.15
Proof. The first coordinate of 3.19 is given by
3.15
Under assumption g 1 / 0 we have
3.15
and therefore 3.19 turns to
3.15
The coefficients L ij are obtained by scalar multiplication by N.
3.15
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
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 N 2 :
3.25
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
3.25
International Journal of Mathematics and Mathematical Sciences By using 3.19 it is easy to show that the component by N of σ ,i is equal to
3.25
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 .
3.25
Proposition 3.5. The principal directions on a regular admissible surface Φ are given by
3.25
(the isotropic direction) and
3.25
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 ifẋ 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 .
3.25
The principal curvature is given in the next proposition.
3.25
Proposition 3.6. The principal curvature k 1 for the direction 3.27 is given by
3.25
and the principal curvature k 2 for the direction 3.28 by
3.25
3.30
3.25
Proof. 
3.25
International Journal of Mathematics and Mathematical Sciences 9 and for the direction 3.28
3.25
3.32
3.25
The Gaussian curvature of a regular admissible surface is defined by
3.25
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
3.25
3.37
10
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3.38
Particularly, for 3.2 we have N 1/W 0, z ,y , 1 , and
3.38
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.
3.38
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 1 3 there are two types of rotations: pseudo-Euclidean rotations given by the normal form x const.,
Surfaces of Revolution
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
Surfaces of Revolution
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
Surfaces of Revolution
and by rotating a curve u → f u , 0, g u , g > 0, around the x-axis, we obtain a spacelike surface
Surfaces of Revolution
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
Surfaces of Revolution
International Journal of Mathematics and Mathematical Sciences Therefore, surfaces with constant curvature K 0 are described by the ordinary differential equation
Surfaces of Revolution
Their implicit equation is
Surfaces of Revolution
and the first fundamental form
Surfaces of Revolution
The following theorem holds. 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.
Surfaces of Revolution
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 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
Surfaces of Revolution
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
4.18
for a spacelike surface and
4.18
and for a timelike surface, where C ∈ R is a constant.
4.18
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
4.18
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.
4.22
1 If H 0, then f u Au B, g u A 1 u B 1 , A, B, A 1 , B 1 
4.22
− , that is, the surface is generated by an isotropic rotation of an isotropic line (a parabolic sphere). 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 .
5.3
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.
5.3
We proceed with the following definition.
5.3
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 .
5.3
Notice that according to our notation we have h ii,j ∂h ii /∂u j , i, j 1, 2.
5.3
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
5.3
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 . 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.
5.3
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
5.6
18
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Under assumption x ,i,j,k x ,i,k,j the following is obtained:
5.7
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 .
5.7
The component by x ,m gives the Gauss integrability equation 
5.7
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
5.10
Furthermore, the assumption K const. 
5.12
They are obtained by derivating the expression W 2 − g 1 x ,2 − g 2 x ,1 2 and using 
5.15
It follows that g 1,2 g 2 x ,1 − g 1 x ,2 · σ 0.
5.16
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
5.17
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 In the same way we can conclude K ,2 0, 5.32 and therefore K const, what was claimed.
5.18
Let us now determine the angle between curves of the Tchebyshev net on a regular admissible surface. The angle between nonisotropic unit vectors
5.18
in the pseudo-Galilean space G 1 3 is determined by using the following expression:
5.18
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 on a timelike surface of revolution 4.5 (K 1/a 2 , see Figure 7) x u, v u v,
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 ):
5.51
and of a timelike surface with K 2a/b > 0, a, b ∈ R \ {0}
5.51
x u, v u v,
5.52
Remark 5.6. Notice that for the given parametrizations of surfaces obtained by pseudo-Euclidean rotations we have g 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 is a spacelike surface parametrized by the Tchebyshev net having K −5 see Figure 9 .

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