# Surfaces of Constant Curvature in the Pseudo-Galilean Space

Authors: Željka Milin Šipuš; Blaženka Divjak
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

---

## 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.


---

## Body

## 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 spaces𝒫n(R) with an absolute figure, a subset of 𝒫n(R) consisting of a sequence of quadrics and planes [1]. Projectivities of the projective space 𝒫n(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 space𝒫3(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 G31 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-x32=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 Pi=(xi,yi,zi),  i=1,2, is defined by
(2.1)d(P1,P2)={|x2-x1|,ifx1≠x2,|(y2-y1)2-(z2-z1)2|,ifx1=x2.The group of motions ofG31 is a six-parameter group given (in affine coordinates) by
(2.2)x-=a+x,y-=b+cx+ycoshφ+zsinhφ,z-=d+ex+ysinhφ+zcoshφ.
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 Pi,  i=1,2, given by their affine coordinates Pi=(xi,yi) is defined by
(2.3)d(P1,P2)=|(x2-x1)2-(y2-y1)2|,
while in the isotropic plane
(2.4)d(P1,P2)={|x2-x1|,ifx1≠x2,|y2-y1|,ifx1=x2.The pseudo-Galilean spaceG31 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 P11|0013 and also called the Galilean space of index 1.


## 3. The Gaussian Curvature of Surfaces inG31

We will treat aCr-surface, r≥2, as a subset Φ⊂G31 for which there exists an open subset D of R2 and Cr-mapping x:D→G31 satisfying Φ=x(D). A Cr-surface Φ⊂G31 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/∂ui, y,i=∂y/∂ui, z,i=∂z/∂ui, i=1,2. Then a surface is admissible if and only if x,i≠0, for some i=1,2. If we assume
(3.1)|x,1y,1x,2y,2|≠0,
then such a surface is admissible and can be locally expressed in the form
(3.2)z=z(x,y).LetΦ⊂G31 be a regular admissible surface. We define a side tangential vector by
(3.3)σ=1W(x,1x,2-x,2x,1).
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
(3.4)W=|(x,1y,2-x,2y,1)2-(x,1z,2-x,2z,1)2|,
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
(3.5)σ=1W(0,1,z,y),W=|1-z,y2|.
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 pointP0 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
(3.6)N=1W(0,x,1z,2-x,2z,1,x,1y,2-x,2y,1).
We introduce a pseudo-Galilean cross-product in the following way:
(3.7)a×Gb=|0-e2e3a1a2a3b1b2b3|,
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
(3.8)N=1Wx1×Gx2
which for (3.2) turns to N=(1/W)(0,zy,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
(3.9)(a×Gb)·c=-det(a,b,c~),
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 vectorσ and the normal vector N are unit isotropic vectors that satisfy
(3.10)σ·N=0,N2:=ϵ=±1,σ2=-ϵ.Since the normal vector field satisfiesN·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-z,y2>0.The first fundamental form of a surface is induced from the metric of the ambient spaceG31(3.11)ds2=(x,1du1+x,2du2)2+δ(x~,1du1+x~,2du2)2,
where
(3.12)δ={0,ifdirectiondu1:du2isnonisotropic,1,ifdirectiondu1:du2isisotropic.
We introduce the fundamental coefficients
(3.13)gi=x,i,hij=x~,i·x~,j,i,j=1,2,
by which the first fundamental form can be written as
(3.14)ds2=(g1du1+g2du2)2+δ(h11du12+2h12du1du2+h22du22).
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 hij=hji, gi,j=gj,i, i=1,2.Now the functionW has the form
(3.15)W2=-ϵ(g1x,2-g2x,1)2=-ϵ(g12h22-2g1g2h12+g22h11)>0,
and the surface is spacelike if g12h22-2g1g2h12+g22h11>0, timelike otherwise. We can notice that if a surface is spacelike, both parts of the first fundamental form, ds12=(g1du1+g2du2)2 and
(3.16)ds22=h11du12+2h12du1du2+h22du22=-ϵW2g12du22,g1≠0,
are positive definite, while for the timelike surfaces, the form ds12 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
(3.17)(ds1200ds22)
is indefinite, analogously to the timelike surfaces in the Minkowski space R13 [12].In particular, for the parametrization (3.2) we have
(3.18)ds2=dx2+δ(1-z,y2)dy2,
sincedx=0 when δ=1.Example 3.2.
Hyperbolic cylindersz2-y2=R2(y2-z2=R2) are surfaces which are everywhere spacelike (timelike). They are spheres of the space G31, 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 formLij, i,j=1,2, which are the normal components of x,i,j, i=1,2, respectively. If we put
(3.19)x,i,j=∑k=12‍Γijkx,k+LijN,
then the following proposition follows.Proposition 3.3.
One has(3.20)Lij=ϵ1g1(g1x~,i,j-gi,jx~,1)·N=ϵ1g2(g2x~,i,j-gi,jx~,2)·N,Proof.
The first coordinate of (3.19) is given by
(3.21)gi,j=Γij1g1+Γij2g2.
Under assumption g1≠0 we have
(3.22)Γij1=gi,j-Γij2g2g1,
and therefore (3.19) turns to
(3.23)x,i,j=1g1(gi,jx,1+WΓij2σ)+LijN.
From (3.23) it follows that g1x,i,j-gi,jx,1 is an isotropic vector equal to WΓij2σ+Lijg1N. The coefficients Lij are obtained by scalar multiplication by N.In particular, for the parametrization (3.2) we have
(3.24)L11=ϵ1W(0,0,z,x,x)·(0,z,y,1)=-ϵ1Wz,x,x,L12=-ϵ1Wz,x,y,L22=-ϵ1Wz,y,y.FunctionsΓijk 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 vectorσ and the normal vector N are given by
(3.25)σ,i:=∂∂uiσ=1W(g1Li2-g2L1i)N.N,i=1W(g1Li2-g2L1i)σ,i=1,2.Proof.
Vectorsσ,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 Cr-function f, r≥1.
Now, from the definition ofσ it follows that
(3.26)σ,i=(g1W),ix,2-(g2W),ix,1+g1Wx,2,i-g2Wx,1,i.
By using (3.19) it is easy to show that the component by N of σ,i is equal to (1/W)(g1Li2-g2L1i).We will define the Gaussian curvatureK 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,N˙)=0. This property characterizes principal directions in Euclidean space [16].Proposition 3.5.
The principal directions on a regular admissible surfaceΦ are given by
(3.27)g1du1+g2du2=0
(the isotropic direction) and
(3.28)(g2L11-g1L12)du1+(g2L12-g1L22)du2=0.Proof.
Letc(t)=(x(t),y(t),z(t)) be a curve on a surface Φ. Since N, N˙ are isotropic vectors, det(c˙,N,N˙)=0 if and only if x˙=x,1du1+x,2du2=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 curvaturek1 for the direction (3.27) is given by
(3.29)k1=-ϵL11g22-2L12g1g2+L22g12W2
and the principal curvature k2 for the direction (3.28) by
(3.30)k2=L11L22-L122L11g22-2L12g1g2+L22g12.Proof.
Principal curvatures are calculated fromk=II/I, where II=L11du12+2L12du1du2+L22du22 is the second and I=ds2 the first fundamental form. Therefore, for the direction (3.27) we have
(3.31)k1=L11du12+2L12du1du2+L22du22h11du12+2h12du1du2+h22du22=L11g22-2L12g1g2+L22g12h11g22-2h12g1g2+h22g12=-ϵL11g22-2L12g1g2+L22g12W2
and for the direction (3.28)
(3.32)k2=L11du12+2L12du1du2+L22du22(g1du1+g2du2)2=L11L22-L122L11g22-2L12g1g2+L22g12.The Gaussian curvature of a regular admissible surface is defined by(3.33)K=k1k2=-ϵL11L22-L122W2
or for (3.2)
(3.34)K=-ϵz,x,xz,y,y-z,x,y2W4.The mean curvature of a surface is defined by(3.35)H=-ϵ2W2(g22L11-2g1g2L12+g12L22)=k12,
and for (3.2) it turns to
(3.36)H=ϵ2W3z,y,y.
Such definition of H is motivated by Proposition 3.7The third fundamental formIII 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
(3.37)III=dN·dN=(N,1du1+N,2du2)2=N,12du12+2N,1·N,2du1du2+N,22du22,
where
(3.38)N,12=-ϵW2(g22L112-2g1g2L11L12+g12L122),N,1·N,2=-ϵW2(g22L11L12-g1g2(L11L22+L122)+g12L12L22),N,22=-ϵW2(g22L122-2g1g2L12L22+g12L222).
Particularly, for (3.2) we have N=(1/W)(0,z,y,1), and
(3.39)III=-ϵ(d(z,yW)2-d(1W)2)=-ϵ(1W4(z,x,ydx+z,y,ydy)2).
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 spaceG31 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 isκ*=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 G31. Since H=k1/2, the assertion follows.


## 4. Surfaces of Revolution

In the pseudo-Galilean spaceG31 there are two types of rotations: pseudo-Euclidean rotations given by the normal form
(4.1)x-=x,y-=ycosht+zsinht,z-=ysinht+zcosht
and isotropic rotations with the normal form
(4.2)x-(t)=x+bt,y-(t)=y+xt+bt22,z-(t)=z,
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)(4.3)x=const.,y2-z2=R2,R∈R.
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(4.4)z=const.,y=x22b.
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 curveu↦(f(u),g(u),0), g>0, around the x-axis by pseudo-Euclidean rotations, we obtain a timelike surface
(4.5)x(u,v)=(f(u),g(u)coshv,g(u)sinhv),
and by rotating a curve u↦(f(u),0,g(u)), g>0, around the x-axis, we obtain a spacelike surface
(4.6)x(u,v)=(f(u),g(u)sinhv,g(u)coshv).The Gaussian curvature of these surfaces is given byK=-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
(4.7)K=-ϵg′′g.Therefore, surfaces with constant curvatureK0 are described by the ordinary differential equation
(4.8)g′′+ϵK0g=0,K0=const.
Their implicit equation is
(4.9)y2-z2=ϵg2(x)
and the first fundamental form
(4.10)ds2=du2-ϵδg2(u)dv2.
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)
IfK=-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
(4.11)g(u)=Asinh(ua)+Bcosh(ua),A,B,a∈R.(2)
IfK=0, then the general solution of the differential equation (4.8) is
(4.12)g(u)=Au+B,A,B∈R.(3)
IfK=ϵ1/a2, then the general solution of the differential equation (4.8) is
(4.13)g(u)=Asin(ua)+Bcos(ua),A,B,a∈R.Examples of these surfaces are given in Figures2 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 curveu↦(0,f(u),g(u)) about the z-axis by isotropic rotation, we obtain a surface
(4.14)x(u,v)=(v,f(u)+v22b,g(u)),b≠0.
Let us assume that the rotated curve is parametrized by the arc length
(4.15)f′2-g′2=-ϵ,
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:
(4.16)K=-f′′b=K0=const.
Therefore, the profile curve of a surface with constant curvature is described by the ordinary differential equation
(4.17)f′′=-bK0.Theorem 4.2.
The profile curveu↦(0,f(u),g(u)) of a surface with constant curvature obtained by isotropic rotations is given by
(4.18)f(u)=-bK0u22+Au+B,A,B∈R,g(u)=-12bK0((A-bK0u)(A-bK0u)2-1-Arccosh(A-bK0u)+C),
for a spacelike surface and
(4.19)g(u)=-12bK0((A-bK0u)(A-bK0u)2+1+Arcsinh(A-bK0u)+C)
and for a timelike surface, where C∈R is a constant.
IfK0=0, then
(4.20)f(u)=Au+B,g(u)=Cu+D,A,B,C,D∈R,A2-C2=-ϵ
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 curvatureH of the surfaces (4.5), (4.6) is given by
(4.21)H=-ϵ2W2=12g.
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 curvatureH of a surface (4.14) is given by
(4.22)H=-ϵ2f′′g′=-ϵ2f′′f′2+ϵ.Theorem 4.4.
The profile curve of a surface of revolution of constant mean curvature obtained by isotropic rotations (Figure6) in pseudo-Galilean space is as follows. (1)
IfH=0, then f(u)=Au+B, g(u)=A1u+B1,  A,B,A1,B1, A2-A12=-ϵ, that is, the surface is generated by an isotropic rotation of an isotropic line (a parabolic sphere).(2)
IfH≠0, then for a spacelike surface (ϵ=-1)
(4.23)f(u)=1h0sinh(h0u+c)+c1,g(u)=1h0cosh(h0u+c)+c2
and for a timelike surface (ϵ=1)(4.24)f(u)=1h0cosh(h0u+c)+c1,g(u)=1h0sinh(h0u+c)+-2,
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 inG31

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(5.1)E=G=1,F=cosφ,
where E,F,andG are coefficients of the first fundamental form (i.e., by Tchebyshev coordinates). Then Theorema Egregium implies
(5.2)φ,u,v=-Ksinφ.In particular, for a surface with constant negative curvatureK=-1/m2, the previous equation turns to the sine-Gordon equation for the angle φ between parametric curves
(5.3)φ,u,v=1m2sinφ.Similar results hold in a three-dimensional pseudo-Riemannian manifold of constant curvature (e.g., Minkowski spaceR31). 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, h11,2=h22,1.Notice that according to our notation we havehii,j=∂hii/∂uj, 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 byσ=(1/W)(x,2-x,1), where W2=-ϵ(x,2-x,1)2. Furthermore, since
(5.4)h11,2=∂x~,12∂v=2x,1·x,1,2,h22,1=∂x~,22∂u=2x,2·x,1,2,
it follows that
(5.5)σ·x,1,2=1W(x,2-x,1)·x,1,2=12W(h22,1-h11,2)=0.
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 I31 see [18].Theorem 5.2.
Asymptotic curves on aCr-surface Φ in G31, 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 curvesII=0 has two real solutions.
Proof follows from the analogues of the Gauss and Codazzi-Mainardi equations for surfaces in the pseudo-Galilean spaceG31. They are obtained in the following way.
Letx=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
(5.6)Γij1=ϵW(g2x~,i,j-gi,jx~,2)·σ,Γij2=-ϵW(g1x~,i,j-gi,jx~,1)·σ.
Under assumptionx,i,j,k=x,i,k,j the following is obtained:
(5.7)(Lij,k-Lik,j+∑n‍(ΓijnLnk-ΓiknLnj))N+∑m‍(Γijm,k-Γikm,j+∑n‍(ΓijnΓnkm-ΓiknΓnjm+gmgn(LijLnk-LikLjn)))x,m=0,
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 gmgn(LijLnk-LikLjn). This is a consequence of the formulas for Ni in Proposition 3.4 (with the opposite sign than in the Galilean space).
The component byx,m gives the Gauss (integrability) equation
(5.8)Γijm,k-Γikm,j+∑n‍(ΓijnΓnkm-ΓiknΓnjm)=-∑n‍gmgn(LijLnk-LikLjn),m=1,2,
and the component by N the Codazzi-Mainardi equation
(5.9)Lij,k-Lik,j=-∑n‍(ΓijnLnk-ΓiknLnj).
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
(5.10)L12,1=L12(Γ111-Γ122),L12,2=L12(Γ222-Γ121).
Furthermore, the assumption K=const.=ϵ(1/m2) implies L122=W2(1/m2). By derivating this equation with using  we get
(5.11)W,1=W(Γ111-Γ122),W,2=W(Γ222-Γ121).
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
(5.12)W,1=W(Γ111+Γ122),W,2=W(Γ222+Γ121).
They are obtained by derivating the expression W2=-ϵ(g1x,2-g2x,1)2 and using
(5.13)g2x,1,1·σ=WϵΓ111+g1,1x,2·σ,g1x,1,2·σ=-WϵΓ122+g1,2x,1·σ,
obtained from  for i=j=1 and i=1, j=2.
Now from and  it follows that
(5.14)Γ121=Γ122=0.
By using expressions in  we can write this system as
(5.15)(g1,2x,1-g1x,1,2)·σ=0,(g2x,1,2-g1,2x,2)·σ=0.
It follows that
(5.16)g1,2(g2x,1-g1x,2)·σ=0.
Furthermore (g2x,1-g1x,2)·σ=-Wσ2≠0, and therefore the previous equation implies
(5.17)g1,2=0.
Substituting (5.17) into the system  we obtain (since gi≠0, i=1,2)
(5.18)x,1,2·σ=0.
Let us analyze conditions (5.17), (5.18). Condition (5.17) implies g1=g1(u), g2=g2(v), that is, functions gi are functions of one parameter only.
Condition (5.18) implies
(5.19)x,1,2·σ=1Wx,1,2·(g1x,2-g2x,1)=12W(g1h22,1-g2h11,2),
and therefore
(5.20)g1h22,1-g2h11,2=0.
Condition (5.17) enables us to introduce new coordinates
(5.21)u-=∫‍g1du,v-=∫‍g2dv
to obtain g-1=dx/du-=(dx/du)(du/du-)=g1(1/g1)=1. Analogously we can obtain g-2=1. For the coefficients h-ij of the new parametrization we have
(5.22)h-11=(∂x~∂u-)2=1g12h11,h-22=(∂x~∂v-)2=1g22h22.
Therefore, since g1,2=0,
(5.23)h11,2=g2g12h-11,2,h22,1=g1g22h-22,1.
Condition (5.20) now implies (∂h-11/∂v-)-(∂h-22/∂u-)=h-11,2-h-22,1=(1/g2g12)h11,2-(1/g1g22)h22,1=(1/g12g22)(g2h11,2-g1h22,1)=0, which means that the considered coordinates are Tchebychev.
Conversely, if the asymptotic curves form the Tchebychev net, let us prove thatΦ has constant curvature. The assumptions imply L11=L22=0 and
(5.24)x,1,2·σ=0,
where σ=(1/w)(x,2-x,1). Expressions in imply that in Tchebyshev coordinates Christoffel symbols Γ121, Γ122, are given by
(5.25)Γ121=ϵWx,1,2·σ=0,Γ122=-ϵWx,1,2·σ=0.
Therefore Codazzi-Mainardi equations  are equal to
(5.26)L12,1=L12Γ111,L12,2=L12Γ222.
Now by differentiating
(5.27)K=ϵL122W2,
partially in the first variable, we obtain
(5.28)K,1W3=2ϵL12(L12,1W-L12W,1)
and (5.26) implies
(5.29)K,1W3=2ϵL122(Γ111W-W,1).
Since from  we have
(5.30)W,1=Γ111W,
it follows that
(5.31)K,1=0.
In the same way we can conclude
(5.32)K,2=0,
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(5.33)a=(1,a2,a3),b=(1,b2,b3)
in the pseudo-Galilean space G31 is determined by using the following expression:
(5.34)φ2=∠(a,b)2=|(b2-a2)2-(b3-a3)2|.
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
(5.35)φ2=|(y2-y1)2-(z2-z1)2|=W2,
and therefore we can write
(5.36)φ2=-ϵ((y2-y1)2-(z2-z1)2).
The function φ and the function W as well are differentiable of class Cr, r≥2, if and only if a surface is not lightlike.For the Tchebyshev coordinates we have(5.37)g1=g2=1,Γ121=Γ122=0,
and the Gauss (integrability) equation (5.8) for i=j=1,k=2,m=1 turns to
(5.38)Γ11,21+Γ112Γ221=-L122W2=-ϵK.
Furthermore, from  we have
(5.39)Γ111=W,1W,Γ222=W,2W,
which with  implies
(5.40)Γ112=-W,1W,Γ221=-W,2W.
Therefore
(5.41)-ϵK=WW,1,2-W,1W,2W2+(-W,1W)(-W,2W)=W,1,2W,
that is, W,1,2=-ϵKW. Since W=±φ, where φ is the angle between Tchebyshev curves, then we have
(5.42)φ,1,2=-ϵKφ.
If the Gaussian curvature of the surface is constant K=ϵ(1/m2), m∈R, the previous equation turns to the Klein-Gordon equation
(5.43)φ,1,2=-1m2φ.
Therefore we have proved the following theorem:Theorem 5.3.
The angle between curves of the Tchebychev net and the functionW as well on a spacelike surface of constant negative curvature and timelike surface of constant positive curvature in G31 satisfy the Klein-Gordon equation.Finally, the functions(5.44)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)
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 functionsκ, κ- on a spacelike surface of constant negative curvature and timelike surface of constant positive curvature in G31 satisfy the Klein-Gordon equation.Proof.
We have(5.45)x,1,1=Γ111(x,1-x,2)=Γ111(-Wσ)=-W,1σ
and therefore x,1,1 is a spacelike (timelike) vector for a spacelike (timelike) surface. Hence we can write
(5.46)κ2(u,v)=-ϵx,1,1·x,1,1=W,12.
Now we have
(5.47)κκ,1=W,1W,1,1,κκ,2=W,1W,1,2.
Therefore
(5.48)κκ,1,2=(κκ,2),1-κ,1κ,2=(W,1W,1,2),1-W,1,1W,1,2=W,1W,1,2,1=-1m2W,12=-1m2κ2.
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):
(5.49)x(u,v)=u+v,y(u,v)=(Asin(u+va)+Bcos(u+va))sinh(u-va),z(u,v)=(Asin(u+va)+Bcos(u+va))cosh(u-va),
and on a timelike surface of revolution (4.5) (K=1/a2, see Figure 7)
(5.50)x(u,v)=u+v,y(u,v)=(Asin(u+va)+Bcos(u+va))cosh(u-va),z(u,v)=(Asin(u+va)+Bcos(u+va))sinh(u-va).
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)x(u,v)=u+v,y(u,v)=14a(-A+cosh(2ab(u-v)))2+A2a(-A+cosh(2ab(u-v)))+B+12b(u+v)2,z(u,v)=14a(12sinh(2ab(2u-2v))-2ab(u-v)),
and of a timelike surface withK=(2a/b)>0,a,b∈R∖{0}(5.52)x(u,v)=u+v,y(u,v)=-14a(A-sinh(2ab(u-v)))2+A2a(A-sinh(2ab(u-v)))+B+12b(u+v)2,z(u,v)=14a(12sinh(2ab(2u-2v))+2ab(u-v)).A timelike surface obtained by pseudo-Euclidean rotations parametrized by the Tchebychev net and the projection of a parametric curve ontoyz-plane.
(a)
(b)A spacelike surface obtained by isotropic rotations parametrized by the Tchebychev net and the projection of a parametric curve ontoyz-plane.
(a)
(b)Remark 5.6.
Notice that for the given parametrizations of surfaces obtained by pseudo-Euclidean rotations we have(5.53)g1=g2=1,h11=h22,
that is, Tchebyshev curves satisfy condition similar to that one in the Euclidean space (E=G=1). Therefore, the parametric curves satisfy h11,2=h22,1=h11,1 and h22,1=h11,2=h22,2, 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
(5.54)φ(u,v)=2a(Asin(u+va)+Bcos(u+va)).
For spacelike surfaces obtained by isotropic rotations we have
(5.55)h11-h22=u+v2ab2,
and the angle is equal to
(5.56)φ(u,v)=2absinh2ab(u-v)
and for timelike surfaces obtained by isotropic rotations to
(5.57)φ(u,v)=2abcosh2ab(u-v).
Furthermore, functions κ(u,v), κ-(u,v) are given by
(5.58)κ(u,v)=κ-(u,v)=2a2(Acos(u+va)-Bsin(u+va))
for spacelike and timelike surfaces obtained by pseudo-Euclidean rotations and by
(5.59)κ(u,v)=κ-(u,v)=2bcosh2ab(u-v),κ(u,v)=κ-(u,v)=2bsinh2ab(u-v)
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 toτ=±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(5.60)φ(u,v)=eλ(u-v)/a(Acos(μ(u+v)a)+Bsin(μ(u+v)a)),λ,μ,A,B,a∈R,
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(5.61)x(u,v)=u+v,y(u,v)=110e2(u-v)(2cos(u+v)+sin(u+v)(5cosh(5(u-v)-2sinh(5(u-v))))),z(u,v)=110e2(u-v)(2cos(u+v)+sin(u+v)(-2cosh(5(u-v))+5sinh5(u-v)))
is a spacelike surface parametrized by the Tchebyshev net having K=-5 (see Figure 9).Figure 9
A spacelike surface of constant curvatureK=-5 with λ=2, μ=A=a=1, B=0.



---

*Source: 375264-2012-10-15.xml*