# TimelikeW-Surfaces in Minkowski 3-Space ℝ13 and the Sinh-Gordon Equation

Authors: Nadia Alluhaibi; Rashad A. Abdel-Baky
Journal: Journal of Applied Mathematics
 (2022)

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## Abstract

LetM and M∗ be two timelike surfaces in Minkowski 3-space ℝ13. If there exists a spacelike (timelike) Darboux line congruence between each point of M and M∗, then the surfaces are timelike Weingarten surfaces. It turns out their Tschebyscheff angles are solutions of the Sinh-Gordon equation, and the surfaces are related to each other by Backlund’s transformation. Finally, a method to construct new timelike Weingarten surface has been found.


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## Body

## 1. Introduction

Around 1875, Backlund and Bianchi published proofs of several theorems that relate to the transformation of pseudospherical surfaces and that can be used to generate new pseudospherical surfaces from known ones [1, 2]. In particular, surfaces of constant mean or Gaussian curvature in Euclidean 3-space have been studied extensively [3, 4]. With the research and development of the soliton theory, Backlund’s transformation has become an important method to find the solutions of soliton equations. At the same time, the geometricians also play attention to the generalization and development of the geometrical content of the Backlund theorem to the n-dimensional submanifolds with negative constant curvature [5]. Tian [6] has studied Backlund’s transformation on class of surfaces satisfying the relation κ1−mκ2−m=−l2 in Euclidean 3-space E3, where κ1 and κ2 are the principal curvatures and m,l are real constants. Weinuan and Haizhong [7] have studied the same class of surfaces in terms of the so called-Darboux line congruence and improved Tian’s results.On the other hand, the geometry of surfaces of constant curvatures in Minkowski 3-space has been a subject of wide interest. A series of papers are devoted to the construction of surfaces of constant Gaussian curvatures. In 1990, Palmer constructed Backlund’s transformation between spacelike and timelike surfaces of constant negative curvature inℝ13 [8]. At that decade, some researchers gave Backlund’s transformations on Weingarten surfaces [8–11]. The second author presented the Minkowski versions of the Backlund theorem and its application by using the method of moving frames [12]. Gurbuz studied Backlund’s transformations in ℝ1n [13]. Using the same method, Ozdemir and Coken have studied Backlund’s transformations of nonlightlike constant torsion curves in Minkowski 3-space [14]. There are several works about Backlund’s transformations and Sinh-Gordon Equation, for example, [15–19]. The purpose of this paper is to study Backlund’s transformation on class of timelike surfaces satisfying the relation κ1−mκ2+m=−l2, in terms of the so called Darboux line congruence.


## 2. Preliminaries

A line congruence in Euclidean 3-spaceE3 is a two-parameter set of straight lines. Such a congruence has a parameterization in the form [20]:
(1)L:yu,v,λ=pu,v+λξu,v,ξ=1,where pu,v is its base surface (the surface of reference), and ξu,v is the unit vector giving the direction of the straight lines of the congruence, λ being a parameter on each line. The equations
(2)u=ut,v=vt,u′2+v′2≠0,define a ruled surface belonging to the line congruence. The ruled surface is called a developable if and only if
(3)detξt,ξ′t,p′t=0.This is a quadratic equation foru′,v′. If it has two real and distinct roots, then the solutions of this equation define two distinct families of developable ruled surfaces. In generic case, each family consists of the tangent lines to a surface, and these two surfaces M and M∗ are called the focal surfaces of the line congruence. The line congruence gives a mapping f:M⟶M∗ with the property that the line congruence consists of lines which are tangent to both M and M∗ and joining p∈M to fp∈M∗. This simple construction plays a fundamental role in the theory of transformation of surfaces.The classical Backlund theorem studies the transformations of surfaces of constant negative Gaussian curvature in 3-dimensional Euclidean spaceE3 by realizing them as the focal surfaces of a pseudospherical p.s. line congruence. The integrability theorem says that we can construct a new surface in E3 with constant negative Gaussian curvature from a given one.We can rephrase this in more current terminology as follows:Definition 1.
LetL be a line congruence in 3-dimensional Euclidean space E3 with focal surfaces M,M∗, and let f:M⟶M∗ be the function defined above. The line congruence is called a p.s. line congruence if
(i)
The distancepp∗=r is a constant independent of p(ii)
The angle between the two normals atp and p∗ is a constant independent of pTheorem 2 (Backlund 1875).
Suppose thatL is a p.s. line congruence in E3 with the focal surfaces M and M∗, then both M and M∗ have constant negative Gaussian curvature equal to −sin2θ/r2 (such surfaces are called p.s. spherical surfaces).There is also an integrability theorem:Theorem 3.
SupposeM is a surface in E3 of constant negative Gaussian curvature K=−sin2θ/r2, where r>0 and 0<θ<π are constants. Given any unit vector e∈TMp, which is not a principal direction, there exists a unique surface M∗ and a p.s. congruence f:M⟶M∗ such that if p∗=fp, we have pp∗=re, and θ is the angle between the normals at p and p∗.Thus, one can construct one-parameter family of new surface of constant negative Gaussian curvature from a given one, the results, by varyingr.Letℝ13=x1,x2,x3∣x1,x2,x3∈ℝ be the usual oriented 3-dimensional vector space and differential manifold, which is obtained by e1=1,0,0,e2=0,1,0, and e3=0,0,1, and given the Euclidean differential structure. Minkowski 3-space is defined by ℝ13=ℝ13,I2,1, where I2,1=dx12+dx22−dx32. Thus, the metric tensor is given by <x,y>=x1y1+x2y2−x3y3, where x=x1,x2,x3 and y=y1,y2,y3. A vector x in ℝ13 is spacelike, lightlike (null), or timelike if <x,x>>0,<x,x>=0, or <x,x><0, respectively. For any two vectors x=x1,x2,x3, and y=y1,y2,y3 in ℝ13, the vector product of x and y is defined as follows:
(4)x×y=e1e1−e3x1x2x3y1y2y3=x2y3−x3y2,x3y1−x1y3,−x1y2−x2y1.Moreover, forx∈ℝ13, the norm is defined by x=<x,x>, and then the vector x is called a spacelike unit vector if <x,x>=1 and a timelike unit vector if <x,x>=−1. A surface in ℝ13 is called a spacelike or timelike if the induced metric on the surface is a Riemannian or Lorentzian metric, respectively. The hyperbolic and Lorentzian (de Sitter space) unit spheres in Minkowski 3-space ℝ13, respectively, are defined by
(5)h02=x∈ℝ13∣−x12+x22+x32=−1,s12=x∈ℝ13∣−x12+x22+x32=1.A line congruence inℝ13 is called spacelike or timelike according to its direction vector being spacelike or timelike unit. When the congruence is a spacelike (timelike) congruence, then its end points fill a domain on s12h02.


## 3. TimelikeW-Surfaces

LetM be a timelike surface in ℝ13. We choose a local field of orthonormal frame e1,e2,e3 with origin p is a point of M, and the vectors e1,e2 are tangent to M at p, with e1 is timelike. Let ω1,ω2,ω3 be the dual forms to the frame e1,e2,e3 [12]. We can write
(6)dp=Σαωαeα,deα=Σβωαβeβ.Here and through this paper, we shall agree on the index ranges:(7)1≤i,j,k≤2,1≤α,β,ν≤3.Note that(8)ω23+ω32=0,ω21=ω12,ω31=ω13.The structure equations ofℝ13 are
(9)dωα=Σβωβ∧ωβα,dωαβ=Σγωαν∧ωνβ.Restricting these forms to the frame defined above, we have(10)ω3=0,and hence,
(11)0=dω3=Σiωi∧ωi3.This equation implies that unique functionsh11, h12=h21, and h22 exist on M such that
(12)ωi3=Σjhijωj,hij=hji.This known as Cartan’s lemma. Note that(13)hij=ω2iej=<Dejei,e2>=−<Deje2,ei>=<Sej,ei>.So,
(14)K=detS=−<Se1,e1><Se2,e2>+<Se2,e1><Se1,e2>implies that the Gaussian and mean curvatures, respectively, are
(15)K=−h11h22−h12h21,H=12traceS=12h22−h11.Here,D and S are the usual flat connection on ℝ13, and shape operator on TM, respectively. The first equation of (9) gives
(16)dωi=Σjωj∧ωji,where ω12 is the Levi-Civita connection form on M which is uniquely determined by these two equations.The Gauss equation is(17)dω12=ω13∧ω32=Kω1∧ω2.And the Codazzi equations are(18)dω23=ω21∧ω13,dω31=ω32∧ω21.A surface is called a Weingarten surface orW-surface if the two principal curvatures κ1 and κ2 are not independent of one another or, equivalently, if a certain relation Wκ1,κ2=0 is identically satisfied on the surface. We consider a timelike W-surface in ℝ13 satisfying the relation
(19)K+2mH=l2−m2,in which l and m are real constants such that l2−m2>0. Our main result is as follows:Theorem 4.
LetM∗ and M are two timelike surfaces in ℝ13 with a one-to-one correspondence between p∈M and p∗∈M such that
(1)
Lines joining corresponding points are isoclinic withM∗ and M; that is, the angles of lines with M∗ and M are the same constant, e. g., φ(2)
The distance between corresponding pointsp∈M and p∗∈M∗ is a constant r>0(3)
The angle between normal lines at corresponding points ofM and M∗ is a nonzero constant θ
Then, bothM∗ and M are W-surfaces satisfying the same relation as in (19), in which
(i)
When the congruence is spacelike,(20)m=−1+coshθcotφrsinφ,l2=1+coth2θ2cot2φsinh2θr2sin2φ.(ii)
When the congruence is timelike,(21)m=1+cosθtanhφrcoshφ,l2=1+cot2θ2tanh2φsin2θr2cosh2φ.Such a congruence is called Darboux line congruence.Proof.
Case 1. Letξ be a spacelike unit vector along the Darboux line congruence between M and M∗, and then there is an orthonormal moving frame eα∗ adapted to M∗ on a neighborhood of p∗∈M∗. So that
(22)<e3∗,e3>=coshθ,<ξ,e3>=−<ξ,e3∗>=cosφ,where both θ and φ are constants geodesic distances. Now, the vector ξ can be expressed as
(23)ξ=sinφt+cosφe3,where
(24)t=sinhγe1+coshγe2,and γ is the geodesic distance between e1 and the orthogonal projection of ξ on the tangent space of M. With frames chosen above, the immersions p:M⟶ℝ13 and p∗:M⟶ℝ13 are related by the equation
(25)p∗=p+rξ,where r>0 is constant. The normal vector of M∗ can also be written by
(26)e3∗=x1e1+x2e2+coshθe3,<e3∗,e3∗>=1.
Indeed, equations (22) and (26) give
(27)x1=−1+coshθcotφsinhγ+1+coshθ2cot2φ+sinh2θcoshγx2=−1+coshθcotφcoshγ+1+coshθ2cot2φ+sinh2θsinhγ.
By taking differentiation of (25), and using the structure equations, we get
(28)dp∗=ω1+rdγ+ω12sinφcoshγ+ω31cosφe1+ω2+rdγ+ω12sinφsinhγ+ω32cosφe2+rω13sinhγ+ω23coshγsinφe3.
From<dp∗,e3∗>=0, we have
(29)dγ+ω12=−x1ω1+x2ω2+ry1ω13+y2ω23r1+coshθ2cot2φ+sinh2θsinφ,in which
(30)y1=−x1cosφ+sinφcoshθsinhγy2=−x2cosφ+sinφcoshθcoshγ.
It is obvious that(31)dx1=x2dγ,dx2=x1dγ,dy1=y2dγ,dy2=y1dγ.
Taking differentiating of (29), and using the structure equations, we get
(32)dω12=Kω1∧ω2=x2ω1−x1ω2−ry2ω13+y1ω23r1+coshθ2cot2φ+sinh2θsinφ∧dγ+ω12,from which and (29) it follows that
(33)K−2H1+coshθcotφrsinφ=sinh2θr2sin2φ,as claimed, and this means that M is a W-surface. Comparing with (19), the result is clear.
Since (25) can be written as p=p∗+r−ξ, the same calculations are valid for M∗. Then, we would obtain
(34)K∗−2H∗1+coshθcotφrsinφ=sinh2θr2sin2φ,as well.
Case 2. This time, the Darboux line congruence is timelike. As stated in the above case, we can choose the normal vectore3∗ of M∗, so that
(35)<e3∗,e3>=cosθ,<ξ,e3>=−<ξ,e3∗>=sinhφ.
The direction vector along the congruence can be expressed as(36)ξ=coshγe1+sinhγe2coshφ+sinhφe3.
So, the position vector ofM∗is
(37)p∗=p+rξ,where r>0 is constant. The normal vector of M∗is
(38)e3∗=x1e1+x2e2+cosθe3,<e3∗,e3∗>=1.
By a similar procedure as in case 1, we have(39)x1=1+coshθtanhφcoshγ+1+cosθ2tanh2φ+sin2θsinhγ,x2=1+cosθtanhφsinhγ+1+cosθ2tanh2φ+sin2θcoshγ,(40)dγ+ω12=−x1ω1+x2ω2+ry1ω13+y2ω23r1+cosθ2tanh2φ+sin2θcoshφ,where
(41)y1=−x1sinhφ+coshφcosθcoshγ,y2=−x2sinhφ+coshφcosθsinhγ.
It is obvious that(42)dx1=x2dγ,dx2=x1dγ,dy1=−y2dγ,dy2=−y1dγ.
As in the Case 1, an analogous arguments show that(43)K+2H1+cosθtanhφrcoshφ=sin2θr2cosh2φ,K∗+2H∗1+cosθtanhφrcoshφ=sin2θr2cosh2φ,as well. This completes the proof of the theorem.As a special case of the above theorem, the Minkowski version of the Backlund Theorem (1) can be stated as the following [12]:Theorem 5 (Backlund).
Suppose that there is a spacelike (timelike) p.s. line congruence inℝ13 with timelike focal surfaces M and M∗, then both M and M∗ have constant positive Gaussian curvature equal to sinh2θ/r2sin2θ/r2.


## 4. Sinh-Gordon Equation

Now, we consider that the timelikeW-surface M has no umbilical points, that is, we can take the lines of curvature as its parametric curves. The first and second fundamental forms of M are
(44)I=−A2du2+B2dv2,II=−κ1A2du2+κ2B2dv2,where h12=h21=0 and h11=κ1,h22=κ2 are the principal curvatures of M. Then, we have
(45)ω13=κ1Adu,ω23=κ2Bdv,ω1=Adu,ω2=Bdv.Since the differential formsω1 and ω2 are linearly independent at each point of M, the form ω12 can be written uniquely as
(46)ω12=λ1ω1+λ2ω2.We shall calculate the functionsλi by means of equations (16) and (46)
(47)dω1=−λ1ω1∧ω2,dω2=λ2ω1∧ω2.Indeed, we have(48)dω1=∂A∂vdu∧dv,dω2=∂B∂udu∧dv,ω1∧ω2=ABdu∧dv,in view of (45). Hence, substituting in (47),
(49)λ1=1AB∂A∂v,λ2=1AB∂B∂u.Then, (46) gives
(50)ω21=ω12=1B∂A∂vdu+1A∂B∂udv.Thus, Codazzi equation (18) becomes
(51)κ+κ2Av+k1vA=0,κ+κ2Bu+k2uB=0.Letκ1=κ, and then from (19), we get
(52)κ2=−m−l2k−m,κ1+κ2=κ−m2−l2κ−m.Putting (52) into (51), we obtain:
(53)∂∂vlinAκ−m2−l2=0,∂∂ulinBκ−m2−l2κ−m2=0.This equations mean that we can choose two positive valued functionsau and bv such that
(54)A=aul2−m2.lκ−m2−l2,B=bvl2−m2.κ−mκ−m2−l2.Let(55)sinhϑ2=lκ−m2−l2,coshϑ2=κ−mκ−m2−l2.Via (52), then we have
(56)κ1=l2−m2sinhϑ/2coshϑ+ϑ02,κ2=−l2−m2coshϑ/2sinhϑ+ϑ02,where
(57)sinhϑ02=ml2−m2,coshϑ02=ll2−m2.Therefore, we can introduce new parameters locally onM, denoted still by u,v such that its fundamental forms are
(58)I=1l2−m2−sinh2ϑ2du2+cosh2ϑ2dv2,II=1l2−m2coshϑ+ϑ02sinhϑ2du2−sinhϑ+ϑ02coshϑ2dv2.Hence, the local parametersu,v of the timelike W-surface M are called the Tschebyscheff coordinates, and ϑ is called the Tschebyscheff angle. It follows that
(59)ω1=1l2−m2sinhϑ2du,ω2=1l2−m2coshϑ2dv,ω13=coshϑ+ϑ02du,ω23=−sinhϑ+ϑ02dv.The connection formω21 reads
(60)ω21=ω12=12ϑvdu+ϑudv.It follows that(61)ϑuu−ϑvv=sinhϑ+ϑ0.Obviously, there is a one-to-one correspondence between local solutionsϑ>0 of the Sinh-Gordon equation and local timelike W-surfaces in ℝ13 satisfying the condition (2.13) up to rigid motion.We record the following theorem:Theorem 6.
SupposeM is a timelike W-surface without umiblics in ℝ13 satisfying the relation (19), then its Tschebyscheff angle ϑ is a solution of the Sinh-Gordon equation (61).
Conversely, ifϑ~ is a solution of the Sinh-Gordon equation, then
(62)ϑ~uu−ϑ~vv=sinhϑ~+ϑ0.
l, and m are constants such that l2−m2>0, and then there exists locally a timelike W-surface M satisfying the relation (19) in ℝ13 such that ϑ=ϑ~−ϑ0 is its Tschebyscheff angle, where ϑ0 is given by (57).


### 4.1. Backlund’s Transformation

Suppose that a spacelike Darboux line congruence is associated with timelike surfaces,M and M∗ in ℝ13, so M and M∗ are W-surfaces that satisfy the relation (19), and l and m are given by (20). Let u,v be the Tschebyscheff coordinates on M. Then,
(63)ω1=rsinφsinhθsinhϑ2du,ω2=rsinφsinhθcoshϑ2dvω13=coshϑ+ϑ02du,ω23=−sinhϑ+ϑ02dv,where ϑ0 is given by
(64)sinhϑ02=−1+coshθsinhθcotφ,coshϑ02=1+coshθ2cot2φ+sinh2θsinhθ.Then, from (27), we get
(65)x1=sinhθcoshγ+ϑ02,x2=sinhθsinhγ+ϑ02.Letγ=−ϑ∗/2, and then equations (41) and (65) can be expressed as
(66)x1=sinhθcoshϑ∗−ϑ02,x2=−sinhθsinhϑ∗−ϑ02,(67)y1=sinφ−coshθcoshϑ02sinhϑϑ0∗2−sinhϑ02coshϑ∗−ϑ02y2=sinφcoshθcoshϑ02coshϑ∗−ϑ02+sinhϑ02sinhϑ∗−ϑ02.The differential form (equation (29)) is Backlund’s transformation. We can write it as a system of partial differential equations. Then, (29) reads
(68)r−12dϑ∗+ω12sinhθsinφcoshϑ02=−x1ω1+x2ω2+ry1ω13+y2ω23.By substituting (63), (66), and (67) into (68), we get
(69)ϑu∗−ϑv2sinhθ=sinhϑ+ϑ02coshϑ∗−ϑ02+coshθcoshϑ+ϑ02sinhϑ∗−ϑ02ϑv∗−ϑu2sinhθ=coshϑ+ϑ02sinhϑ∗−ϑ02+coshθsinhϑ+ϑ02coshϑ∗−ϑ02.Equation (69) is Backlund’s transformation of solutions of the Sinh-Gordon equation (61). In fact, we have the following:Proposition 7.
Ifϑ is a solution of the Sinh-Gordon equation (61) and θ>0, then equation (69) on ϑ is completely integrable, and ϑ∗ satisfies the equation:
(70)ϑuu∗−ϑvv∗=sinhϑ∗−ϑ0.Proof.
Since (69),
(71)ϑuv∗−ϑvv2.sinhθ=ϑv2coshϑ+ϑ02coshϑ∗−ϑ02+coshθsinhϑ+ϑ02sinhϑ∗−ϑ02+ϑv∗2sinhϑ+ϑ02sinhϑ∗−ϑ02+coshθcoshϑ+ϑ02coshϑ∗−ϑ02,ϑvu∗−ϑuu2.sinhθ=ϑu2sinhϑ+ϑ02sinhϑ∗−ϑ02+coshθcoshϑ+ϑ02coshϑ∗−ϑ02+ϑu∗2coshϑ+ϑ02coshϑ∗−ϑ02+coshθsinhϑ+ϑ02sinhϑ∗−ϑ02.
From the last two equations we can have(72)ϑuu−ϑvv2.sinhθ=ϑv∗−ϑu2sinhϑ+ϑ02sinhϑ∗−ϑ02+coshθcoshϑ+ϑ02coshϑ∗−ϑ02−ϑu∗−ϑv2coshϑ+ϑ02coshϑ∗−ϑ02+coshθsinhϑ+ϑ02sinhϑ∗−ϑ02.
Via (69), (72) becomes
(73)ϑuu−ϑvv2.sinh2θ=sinhϑ+ϑ02sinhϑ∗−ϑ02+coshθcoshϑ+ϑ02coshϑ∗−ϑ02×coshϑ+ϑ02sinhϑ∗−ϑ02+coshθsinhϑ+ϑ02coshϑ∗−ϑ02−coshϑ+ϑ02coshϑ∗−ϑ02+coshθsinhϑ+ϑ02sinhϑ∗−ϑ02×sinhϑ+ϑ02coshϑ∗−ϑ02+coshθcoshϑ+ϑ02sinhϑ∗−ϑ02,(74)ϑuu−ϑvv=sinhϑ+ϑ0.By similar argument, we can also have the following Sinh-Gordon equation:(75)ϑuu∗−ϑvv∗=sinhϑ∗−ϑ0.Therefore, (72) give Backlund’s transformation of solutions of the last two Sinh-Gordon equations. Furthermore, if the congruence is a timelike Darboux line congruence associated with timelike surfaces M and M∗ in ℝ13, then from (40), we can find Backlund’s transformation. Therefore, for the spacelike Darboux line congruence, we record the following theorem and other case is similar:Theorem 8.
Suppose that we have a spacelike Darboux line congruence associated with timelike surfacesM and M∗ in ℝ13, then both M and M∗ are W-surfaces satisfying the same relation (19), and their Tschebyscheff angles ϑ of M and ϑ∗ of M∗ are both solutions of the Sinh-Gordon equation, and these surfaces are related to each other by the Backlund’s transformations (69).Proof.
From equation (26), we have
(76)e3×e3∗=sinhθsinhγ+ϑ02e1+coshγ+ϑ02e2=sinhθsinhϑ02t+coshϑ02sinhγe1+coshγe2,e=sinhϑ02t+coshϑ02sinhγe1+coshγe2=coshϑ02t⊥+sinhϑ02t=sinhγ+ϑ02e1+coshγ+ϑ02e2.
Then, the vectore is tangent to both M and M∗, andϑ0/2 is the angle between t⊥ and e:
(77)t⊥=sinhγe1+coshγe2,t=coshγe1+sinhγe2.
For a spacelike Darboux line congruence, as we stated near a nonumbilical point onM, we have a local frame field in which e1 and e2 are along the principal directions. In addition, we have two other local frame fields p;t,t⊥,e3 and p;e,e⊥,e3 on M, where
(78)e⊥≔e×e3=−coshγ+ϑ02e1−sinhγ+ϑ02e2=−coshϑ02t−sinhϑ02t⊥,
and a local frame fieldp∗;e∗,e∗⊥,e3∗ on M∗, where
(79)e∗≔−e=−sinhγ+ϑ02e1−coshγ+ϑ02e2=sinhϑ∗−ϑ02e1−coshϑ∗−ϑ02e2,,e∗⊥≔e∗×e3∗=coshθcoshϑ∗−ϑ02e1−sinhϑ∗−ϑ02e2+sinhθe3=coshθcoshϑ02t+sinhϑ02t⊥+sinhθe3..
Denoteη1∗,η2∗ the coframe dual to p∗;e∗,e∗⊥,e3∗ on M∗. By (28), we have
(80)dp∗=1coshϑO/2sinhϑ∗2ω!+coshϑ∗2ω2−rcosφ−sinhϑ∗2ω31+coshϑ∗2ω23e+rsinφsinhθ.coshθcoshϑ0/2−sinhϑ∗2ω3!+coshϑ∗2ω23t+rsinφ−sinhϑ∗2ω3!+coshϑ∗2ω23e3.
So, we have(81)η1∗≔<dp∗,e∗>=rsinφsinhθ−sinhϑ∗2sinhϑ+ϑ02du+coshϑ∗2coshϑ+ϑ02dv,(82)η2∗≔−<dp∗,e∗⊥>=rsinφsinhθ−sinhϑ∗2coshϑ+ϑ02du+coshϑ∗2sinhϑ+ϑ02dv.
To find outη31∗ and η32∗, we take the differentiation of (28) and get
(83)de3∗=1coshϑ0/2sinhθrsinφ−coshϑ∗−ϑ02ω!−sinhϑ∗−ϑ02ω2−cotφsinhθcoshϑ∗−ϑ02ω31−sinhϑ∗−ϑ02ω23e+coshθcoshϑ0/2coshϑ∗−ϑ02ω3!+sinhϑ∗−ϑ02ω32t+sinhθcoshϑ∗−ϑ02ω3!−sinhϑ∗−ϑ02ω23e3.
Therefore, we obtain(84)η31∗≔<de3∗,e∗>=coshϑ∗−ϑ02sinhϑ+ϑ02du+sinhϑ∗−ϑ02coshϑ+ϑ02dv,η32∗≔−<de3∗,e∗⊥>=coshϑ∗−ϑ02coshϑ+ϑ02du+sinhϑ∗−ϑ02sinhϑ+ϑ02dv.
Consider local frame fieldsp∗;e1∗,e2∗,e3∗ on M∗ such that
(85)e∗=−sinhϑ+ϑ02e1∗+coshϑ+ϑ02e2∗e⊥∗=−coshϑ+ϑ02e1∗+sinhϑ+ϑ02e2∗and denote its dual by ω1∗,ω2∗. Then, we get
(86)ω1∗=sinhϑ+ϑ02η1∗−coshϑ+ϑ02η2∗=rsinφsinhθsinhϑ∗2du,ω2∗=coshϑ+ϑ02η1∗−sinhϑ+ϑ02η2∗=rsinφsinhθcoshϑ∗2dv,,ω31∗=sinhϑ+ϑ02η31∗−coshϑ+ϑ02η32∗=coshϑ∗−ϑ02du,ω32∗=coshϑ+ϑ02η31∗−sinhϑ+ϑ02η32∗=sinhϑ∗−ϑ02dv.
It follows thate1∗ and e2∗ are the principal directions on the surface M∗, and its fundamentals forms become
(87)I∗=1l2−m2sinh2ϑ∗2du2−cosh2ϑ∗2dv2,II∗=1l2−m2−sinhϑ∗2coshϑ∗−ϑ02du2+coshϑ∗2sinhϑ∗−ϑ02dv2.
This means thatu and v are Tschebyscheff coordinates on M∗, and ϑ∗ is its Tschebyscheff angle.As a Minkowski version of integrability theorem, we may therefore state the following theorem:Theorem 9.
SupposeM is a timelike surface satisfying the relation (19) in ℝ13, for any given real number θ>0, we can construct a spacelike Darboux line congruence such that the solution of the completely integrable equation (69) is the Tschebyscheff angle of the corresponding surface M∗.Proof.
Letu,v be the Tschebyscheff coordinates on M. Then, equation (69) is completely integrable by Proposition (44). Let ϑ∗ be the solution of (69) such that ϑ∗u0,v0=ϑu0,v0. Putting
(88)r=1l2−m2sinhθsinφ,then (57) and (63) hold. Let
(89)ξ=sinφ−sinhϑ∗2e1+coshϑ∗2e2+cosφe3,(90)p∗=p+rξ.
We want to prove thatp∗ is a timelike surface, and that the above formula gives a spacelike Darboux line congruence in ℝ13 associated with M and M∗.
Let(91)e3∗=sinhθcoshϑ∗−ϑ02e1−sinhϑ∗−ϑ02e2+coshθe3.
By differentiation of (90), we get
(92)dp∗=ω1e1+ω2e2+rsinφ−dϑ∗2+ω12coshϑ∗2e1−sinhϑ∗2e2+rcosφω31e1+ω32e2+rsinφ−sinhϑ∗2ω31+coshϑ∗2ω23e3.
By using of (91), we find
(93)de3∗=<dp∗,e3∗>=−sinhθcoshϑ∗−ϑ02ω1+sinhϑ∗−ϑ02ω2+rsinhθcosφcoshϑ02−dϑ∗2+ω12+rcoshθsinφ−sinhϑ∗2ω31+coshϑ∗2ω23−sinhθcosφcoshϑ∗−ϑ02ω31−sinhϑ∗−ϑ02ω23=0,in view of (69). Thus, e3∗ is normal vector of M∗. From (91), we have, <e3∗,e3∗>=1. Then, M∗ is a timelike surface.
From the definition ofξ, and e3∗, we have
(94)<ξ,e3>=−<ξ,e3∗>=cosφ=const.,<e3,e3∗>=coshθ=const.,<ξ,ξ>=1.Hence, the line congruence given by (69) is a spacelike Darboux line congruence. From Theorem 8, ϑ∗ is the Tschebyscheff angle of M∗.In similar arguments, we can give the corresponding theorems for Case 2 in Theorem3, and we omit the details here.


## 4.1. Backlund’s Transformation

Suppose that a spacelike Darboux line congruence is associated with timelike surfaces,M and M∗ in ℝ13, so M and M∗ are W-surfaces that satisfy the relation (19), and l and m are given by (20). Let u,v be the Tschebyscheff coordinates on M. Then,
(63)ω1=rsinφsinhθsinhϑ2du,ω2=rsinφsinhθcoshϑ2dvω13=coshϑ+ϑ02du,ω23=−sinhϑ+ϑ02dv,where ϑ0 is given by
(64)sinhϑ02=−1+coshθsinhθcotφ,coshϑ02=1+coshθ2cot2φ+sinh2θsinhθ.Then, from (27), we get
(65)x1=sinhθcoshγ+ϑ02,x2=sinhθsinhγ+ϑ02.Letγ=−ϑ∗/2, and then equations (41) and (65) can be expressed as
(66)x1=sinhθcoshϑ∗−ϑ02,x2=−sinhθsinhϑ∗−ϑ02,(67)y1=sinφ−coshθcoshϑ02sinhϑϑ0∗2−sinhϑ02coshϑ∗−ϑ02y2=sinφcoshθcoshϑ02coshϑ∗−ϑ02+sinhϑ02sinhϑ∗−ϑ02.The differential form (equation (29)) is Backlund’s transformation. We can write it as a system of partial differential equations. Then, (29) reads
(68)r−12dϑ∗+ω12sinhθsinφcoshϑ02=−x1ω1+x2ω2+ry1ω13+y2ω23.By substituting (63), (66), and (67) into (68), we get
(69)ϑu∗−ϑv2sinhθ=sinhϑ+ϑ02coshϑ∗−ϑ02+coshθcoshϑ+ϑ02sinhϑ∗−ϑ02ϑv∗−ϑu2sinhθ=coshϑ+ϑ02sinhϑ∗−ϑ02+coshθsinhϑ+ϑ02coshϑ∗−ϑ02.Equation (69) is Backlund’s transformation of solutions of the Sinh-Gordon equation (61). In fact, we have the following:Proposition 7.
Ifϑ is a solution of the Sinh-Gordon equation (61) and θ>0, then equation (69) on ϑ is completely integrable, and ϑ∗ satisfies the equation:
(70)ϑuu∗−ϑvv∗=sinhϑ∗−ϑ0.Proof.
Since (69),
(71)ϑuv∗−ϑvv2.sinhθ=ϑv2coshϑ+ϑ02coshϑ∗−ϑ02+coshθsinhϑ+ϑ02sinhϑ∗−ϑ02+ϑv∗2sinhϑ+ϑ02sinhϑ∗−ϑ02+coshθcoshϑ+ϑ02coshϑ∗−ϑ02,ϑvu∗−ϑuu2.sinhθ=ϑu2sinhϑ+ϑ02sinhϑ∗−ϑ02+coshθcoshϑ+ϑ02coshϑ∗−ϑ02+ϑu∗2coshϑ+ϑ02coshϑ∗−ϑ02+coshθsinhϑ+ϑ02sinhϑ∗−ϑ02.
From the last two equations we can have(72)ϑuu−ϑvv2.sinhθ=ϑv∗−ϑu2sinhϑ+ϑ02sinhϑ∗−ϑ02+coshθcoshϑ+ϑ02coshϑ∗−ϑ02−ϑu∗−ϑv2coshϑ+ϑ02coshϑ∗−ϑ02+coshθsinhϑ+ϑ02sinhϑ∗−ϑ02.
Via (69), (72) becomes
(73)ϑuu−ϑvv2.sinh2θ=sinhϑ+ϑ02sinhϑ∗−ϑ02+coshθcoshϑ+ϑ02coshϑ∗−ϑ02×coshϑ+ϑ02sinhϑ∗−ϑ02+coshθsinhϑ+ϑ02coshϑ∗−ϑ02−coshϑ+ϑ02coshϑ∗−ϑ02+coshθsinhϑ+ϑ02sinhϑ∗−ϑ02×sinhϑ+ϑ02coshϑ∗−ϑ02+coshθcoshϑ+ϑ02sinhϑ∗−ϑ02,(74)ϑuu−ϑvv=sinhϑ+ϑ0.By similar argument, we can also have the following Sinh-Gordon equation:(75)ϑuu∗−ϑvv∗=sinhϑ∗−ϑ0.Therefore, (72) give Backlund’s transformation of solutions of the last two Sinh-Gordon equations. Furthermore, if the congruence is a timelike Darboux line congruence associated with timelike surfaces M and M∗ in ℝ13, then from (40), we can find Backlund’s transformation. Therefore, for the spacelike Darboux line congruence, we record the following theorem and other case is similar:Theorem 8.
Suppose that we have a spacelike Darboux line congruence associated with timelike surfacesM and M∗ in ℝ13, then both M and M∗ are W-surfaces satisfying the same relation (19), and their Tschebyscheff angles ϑ of M and ϑ∗ of M∗ are both solutions of the Sinh-Gordon equation, and these surfaces are related to each other by the Backlund’s transformations (69).Proof.
From equation (26), we have
(76)e3×e3∗=sinhθsinhγ+ϑ02e1+coshγ+ϑ02e2=sinhθsinhϑ02t+coshϑ02sinhγe1+coshγe2,e=sinhϑ02t+coshϑ02sinhγe1+coshγe2=coshϑ02t⊥+sinhϑ02t=sinhγ+ϑ02e1+coshγ+ϑ02e2.
Then, the vectore is tangent to both M and M∗, andϑ0/2 is the angle between t⊥ and e:
(77)t⊥=sinhγe1+coshγe2,t=coshγe1+sinhγe2.
For a spacelike Darboux line congruence, as we stated near a nonumbilical point onM, we have a local frame field in which e1 and e2 are along the principal directions. In addition, we have two other local frame fields p;t,t⊥,e3 and p;e,e⊥,e3 on M, where
(78)e⊥≔e×e3=−coshγ+ϑ02e1−sinhγ+ϑ02e2=−coshϑ02t−sinhϑ02t⊥,
and a local frame fieldp∗;e∗,e∗⊥,e3∗ on M∗, where
(79)e∗≔−e=−sinhγ+ϑ02e1−coshγ+ϑ02e2=sinhϑ∗−ϑ02e1−coshϑ∗−ϑ02e2,,e∗⊥≔e∗×e3∗=coshθcoshϑ∗−ϑ02e1−sinhϑ∗−ϑ02e2+sinhθe3=coshθcoshϑ02t+sinhϑ02t⊥+sinhθe3..
Denoteη1∗,η2∗ the coframe dual to p∗;e∗,e∗⊥,e3∗ on M∗. By (28), we have
(80)dp∗=1coshϑO/2sinhϑ∗2ω!+coshϑ∗2ω2−rcosφ−sinhϑ∗2ω31+coshϑ∗2ω23e+rsinφsinhθ.coshθcoshϑ0/2−sinhϑ∗2ω3!+coshϑ∗2ω23t+rsinφ−sinhϑ∗2ω3!+coshϑ∗2ω23e3.
So, we have(81)η1∗≔<dp∗,e∗>=rsinφsinhθ−sinhϑ∗2sinhϑ+ϑ02du+coshϑ∗2coshϑ+ϑ02dv,(82)η2∗≔−<dp∗,e∗⊥>=rsinφsinhθ−sinhϑ∗2coshϑ+ϑ02du+coshϑ∗2sinhϑ+ϑ02dv.
To find outη31∗ and η32∗, we take the differentiation of (28) and get
(83)de3∗=1coshϑ0/2sinhθrsinφ−coshϑ∗−ϑ02ω!−sinhϑ∗−ϑ02ω2−cotφsinhθcoshϑ∗−ϑ02ω31−sinhϑ∗−ϑ02ω23e+coshθcoshϑ0/2coshϑ∗−ϑ02ω3!+sinhϑ∗−ϑ02ω32t+sinhθcoshϑ∗−ϑ02ω3!−sinhϑ∗−ϑ02ω23e3.
Therefore, we obtain(84)η31∗≔<de3∗,e∗>=coshϑ∗−ϑ02sinhϑ+ϑ02du+sinhϑ∗−ϑ02coshϑ+ϑ02dv,η32∗≔−<de3∗,e∗⊥>=coshϑ∗−ϑ02coshϑ+ϑ02du+sinhϑ∗−ϑ02sinhϑ+ϑ02dv.
Consider local frame fieldsp∗;e1∗,e2∗,e3∗ on M∗ such that
(85)e∗=−sinhϑ+ϑ02e1∗+coshϑ+ϑ02e2∗e⊥∗=−coshϑ+ϑ02e1∗+sinhϑ+ϑ02e2∗and denote its dual by ω1∗,ω2∗. Then, we get
(86)ω1∗=sinhϑ+ϑ02η1∗−coshϑ+ϑ02η2∗=rsinφsinhθsinhϑ∗2du,ω2∗=coshϑ+ϑ02η1∗−sinhϑ+ϑ02η2∗=rsinφsinhθcoshϑ∗2dv,,ω31∗=sinhϑ+ϑ02η31∗−coshϑ+ϑ02η32∗=coshϑ∗−ϑ02du,ω32∗=coshϑ+ϑ02η31∗−sinhϑ+ϑ02η32∗=sinhϑ∗−ϑ02dv.
It follows thate1∗ and e2∗ are the principal directions on the surface M∗, and its fundamentals forms become
(87)I∗=1l2−m2sinh2ϑ∗2du2−cosh2ϑ∗2dv2,II∗=1l2−m2−sinhϑ∗2coshϑ∗−ϑ02du2+coshϑ∗2sinhϑ∗−ϑ02dv2.
This means thatu and v are Tschebyscheff coordinates on M∗, and ϑ∗ is its Tschebyscheff angle.As a Minkowski version of integrability theorem, we may therefore state the following theorem:Theorem 9.
SupposeM is a timelike surface satisfying the relation (19) in ℝ13, for any given real number θ>0, we can construct a spacelike Darboux line congruence such that the solution of the completely integrable equation (69) is the Tschebyscheff angle of the corresponding surface M∗.Proof.
Letu,v be the Tschebyscheff coordinates on M. Then, equation (69) is completely integrable by Proposition (44). Let ϑ∗ be the solution of (69) such that ϑ∗u0,v0=ϑu0,v0. Putting
(88)r=1l2−m2sinhθsinφ,then (57) and (63) hold. Let
(89)ξ=sinφ−sinhϑ∗2e1+coshϑ∗2e2+cosφe3,(90)p∗=p+rξ.
We want to prove thatp∗ is a timelike surface, and that the above formula gives a spacelike Darboux line congruence in ℝ13 associated with M and M∗.
Let(91)e3∗=sinhθcoshϑ∗−ϑ02e1−sinhϑ∗−ϑ02e2+coshθe3.
By differentiation of (90), we get
(92)dp∗=ω1e1+ω2e2+rsinφ−dϑ∗2+ω12coshϑ∗2e1−sinhϑ∗2e2+rcosφω31e1+ω32e2+rsinφ−sinhϑ∗2ω31+coshϑ∗2ω23e3.
By using of (91), we find
(93)de3∗=<dp∗,e3∗>=−sinhθcoshϑ∗−ϑ02ω1+sinhϑ∗−ϑ02ω2+rsinhθcosφcoshϑ02−dϑ∗2+ω12+rcoshθsinφ−sinhϑ∗2ω31+coshϑ∗2ω23−sinhθcosφcoshϑ∗−ϑ02ω31−sinhϑ∗−ϑ02ω23=0,in view of (69). Thus, e3∗ is normal vector of M∗. From (91), we have, <e3∗,e3∗>=1. Then, M∗ is a timelike surface.
From the definition ofξ, and e3∗, we have
(94)<ξ,e3>=−<ξ,e3∗>=cosφ=const.,<e3,e3∗>=coshθ=const.,<ξ,ξ>=1.Hence, the line congruence given by (69) is a spacelike Darboux line congruence. From Theorem 8, ϑ∗ is the Tschebyscheff angle of M∗.In similar arguments, we can give the corresponding theorems for Case 2 in Theorem3, and we omit the details here.


## 5. Conclusion

Mathematical techniques based on the method of moving frames have been shown to be suitable for study of geometry of the timelikeW-surfaces in Minkowski 3-Space ℝ13 and the Sinh-Gordon equation. We believe that the study of Backlund’s transformations of W-surfaces via the method of moving frames may shed some light on current research problems and perhaps suggest new ones.



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