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The Backlund’s theorem in Minkowski 3-space R31
Applied Mathematics and Computation 160 (2005) 41–50 www.elsevier.com/locate/amc
The BacklundÕs theorem in Minkowski 3-space R31 Rashad A. Abdel-Baky Department of Mathematics, Faculty of Science, University of Assiut, 71516 Assiut, Egypt
Abstract In this paper we present the Minkowski versions of the BacklundÕs theorem and its application by using the method of moving frames. Ó 2003 Published by Elsevier Inc. Keywords: Line congruence; Backlund theorem; Sinh-Gordon equation
1. Introduction The classical Backlund theorem studies the transformations of surfaces of constant negative curvature in the Euclidean space E3 by realizing them as the focal surfaces of a pseudo-spherical (p.s.) line congruence. The integrability theorem says that we can construct a new surface in E3 with constant negative curvature from a given one. A congruence of lines is an immersed surface in the Grassmann manifold of all lines in E3 . Locally we can suppose the lines be oriented, with their points given by yðu; v; kÞ ¼ pðu; vÞ þ kfðu; vÞ;
kfk ¼ 1;
k being a parameter on each line [2]. The equations u ¼ uðtÞ;
v ¼ vðtÞ;
u02 þ v02 6¼ 0;
define a ruled surface belonging to the congruence. It is a developable if and only if the determinate ðf; dp; dfÞ ¼ 0: 0096-3003/$ - see front matter Ó 2003 Published by Elsevier Inc. doi:10.1016/S0096-3003(03)00716-1
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R.A. Abdel-Baky / Appl. Math. Comput. 160 (2005) 41–50
This is a quadratic equation in du, dv. Suppose that it has two real and distinct roots, then there are two families of developables surfaces, each of which (in generic case) consists of the tangent lines of two surfaces M and M such that the congruence consists of the lines joining p 2 M to f ðpÞ 2 M . This construction plays a fundamental role in the theory of transformation of surfaces. Definition 1.1. Consider a line congruence with focal surfaces M, and M such that its lines are the common tangents at p 2 M to p ¼ f ðpÞ 2 M . The congruence is called a pseudo-spherical (p.s.) if i(i) kpp k ¼ r, which is a constant independent of p, (ii) The angle between the two normals np and np is equal to a constant h independent of p. We can now state the classical BacklundÕs theorem: Theorem 1.1. Suppose there is a p.s. congruence in E3 with the focal surfaces M and M such that the distance r between corresponding points and the angle h between corresponding normals are constants. Then both M and M have constant negative Gaussian curvature equal to sin2 h=r2 . There is also an integrability theorem: Theorem 1.2. Suppose M is a surface in E3 of constant negative Gaussian curvature K ¼ sin2 h=r2 , where r > 0 and 0 < h < p are constants. Given any unit vector e 2 T ðMp Þ, which is not a principal direction, there exists a unique surface M and a p.s. congruence f : M ! M such that if p ¼ f ðpÞ, we have pp ¼ re and h 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 varying r.
2. The Backlunds theorem in Minkowski 3-space R31 In affine geometry Chern and Terng [1] have give a modern proof of the classical Backlund theorem. Mc-Nertny in [4], treated the same problem by choosing a moving frame on a timelike surface M, such that one vector in the frame points in the direction of the congruence. She proved that: for the timelike surface the Gaussian curvature must be positive for the existence of a p.s. congruence.
R.A. Abdel-Baky / Appl. Math. Comput. 160 (2005) 41–50
43
Our approach here is to consider the Minkowski versions of the same problem, in which the results of Mc-Nertny is special case and to give the Backlund transformations. Let M be a spacelike surface in R31 . 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. Let x1 ; x2 ; x3 be the dual forms to the frame e1 ; e2 ; e3 [3]. We can write X X dp ¼ xa ea ; dea ¼ xab eb : ð2:1Þ a
b
Here and through this paper we shall agree on the index ranges 1 6 i; j; k 6 2;
1 6 a; b; m 6 3:
Note that: x12 þ x21 ¼ 0;
x31 ¼ x13 ;
x32 ¼ x23 :
ð2:2Þ
R31
are: The structure equations of X X dxa ¼ xb ^ xba ; dxab ¼ xam ^ xmb :
ð2:3Þ
c
b
Restricting these forms to the frame defined above, we have x3 ¼ 0
ð2:4Þ
and hence 0 ¼ dx3 ¼
X
xi ^ xi3 :
ð2:5Þ
i
By CartanÕs lemma we may write X xi3 ¼ hij xj ; hij ¼ hji :
ð2:6Þ
j
The first equation of (2.3) gives X dxi ¼ xj ^ xji ;
ð2:7Þ
j
where x12 is the Live-Civita connection form on M which is uniquely determined by these two equations. The Gauss equation is X dxij ¼ xik ^ xki þ Xij ; ð2:8Þ k
where X12 ¼ x13 ^ x32 ¼ x13 ^ x23 ¼ Kx1 ^ x2 ;
ð2:9Þ
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R.A. Abdel-Baky / Appl. Math. Comput. 160 (2005) 41–50
K ¼ detðhij Þ is the Gaussian curvature of the surface M. The Codazzi equations are: X dxi3 ¼ xij ^ xj3 :
ð2:10Þ
j
In the following theorems, we can now state the Minkowski versions of the BacklundÕs theorem: Theorem 2.1. Suppose there is a p.s. spacelike congruence f in R31 between space like focal surfaces M, and M such that the distance r between corresponding points and the angle h between corresponding normals are constant. Then both M, and M have constant negative Gaussian curvature equal to
sinh2 h=r2 . Proof. Since a p.s. congruence exists between M, and M then there is an orthonormal moving frame ea adapted to M on a neighborhood of p ¼ f ðpÞ. Choose the direction vector along the congruence as e1 ¼ cos ce1 þ sin ce2 :
ð2:11Þ
We can take the normal vector of M so that he3 ; e3 i ¼ cosh h;
he3 ; e3 i ¼ 1;
ð2:12Þ
where both h and c are constants. Now, the normal vector of M can be expressed as e3 ¼ x1 e1 þ x2 e2 cosh he3 :
ð2:13Þ
Substituting (2.13) into (2.12), then x1 ¼ sinh h sin c;
x2 ¼ sinh h cos c:
By the vector product in R31 , we have 2 3 2 e1 cos c sin c 6 7 6 4 e2 5 ¼ 4 sin c cosh h cos c cosh h e3 sin c sinh h cos c sinh h
ð2:14Þ
0
32
e1
3
76 7
sinh h 54 e2 5:
cosh h
ð2:15Þ
e3
Suppose locally M is given by an immersion p : U ! R31 , where U is an open subset of the u; v plane, then M is given by p ¼ p þ re1 :
ð2:16Þ
Taking differentiation of (2.16) and using the structure equations, we get dp ¼ ðx1 þ rx21 sin cÞe1 þ ðx2 þ rx12 cos cÞe2 þ rðx13 cos c þ x23 sin cÞe3 : ð2:17Þ
R.A. Abdel-Baky / Appl. Math. Comput. 160 (2005) 41–50
45
On the other hand, from hdp ; e3 i ¼ 0, we have rx12 ¼ x1 sin c x2 cos c þ r coth hðx13 cos c þ x23 sin cÞ:
ð2:18Þ
Again, taking differentiating of (2.18), and using the structure equations, we get rx13 ^ x23 ¼ x12 ^ ðx1 cos c þ x2 sin c þ r coth hðx13 cos c þ x23 sin cÞÞ: ð2:19Þ Thus, by using (2.19) and (2.18), we obtain r2 x13 ^ x23 ¼ ðx1 sin c þ x2 cos c þ r coth hðx13 cos c þ x23 sin cÞÞ ^ ðx1 cos c þ x2 sin c þ r coth hðx13 cos c þ x23 sin cÞÞ ð2:20Þ or by means of (2.7)–(2.10), it reduces to ð1 þ r2 K coth2 hÞx1 ^ x2 ¼ r2 Kx1 ^ x2 : 2
ð2:21Þ
2
Thus, we have K ¼ sinh h=r , as claimed. Since (2.7) can be written as p ¼ p þ rð e1 Þ, the same calculations are valid for M , then we would obtain K ¼ sinh2 h=r2 as well. This completes the proof of the theorem. Now that we have, for spacelike surfaces, shown constant negative Gaussian curvature to be a necessary condition for the existence of a p.s. spacelike congruence. Let M be a timelike surface in R31 . We choose a local field of orthonormal frame ea with origin p is a point of M and the vectors e1 ; e2 are tangent to M at p with e1 is timelike. Then, the same equations (2.1)–(2.11) are exists, except that (2.2) is replaced by x12 ¼ x21 ;
x31 ¼ x13 ;
x32 þ x23 ¼ 0
ð2:22Þ
and the Gaussian curvature K ¼ detðhij Þ. Theorem 2.2. Suppose there is a p.s. congruence f in R31 between timelike focal surfaces M, and M such that the distance r between corresponding points and the angle h between corresponding normals are constant. Then (A) if kpp k ¼ r2 ; i:e: the congruences is timelike, and he3 ; e3 i ¼ cos h; 0 < h < p, both M, and M have constant positive Gaussian curvature equal to sin2 h=r2 , (B) if kpp k ¼ r2 ; i:e: the congruences is spacelike, and he3 ; e3 i ¼ cosh h; h 6¼ 0, both M, and M have constant positive Gaussian curvature equal to sinh2 h=r2 .
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Proof. Case (A) Since a p.s. congruence exists between M, and M then there is an orthonormal moving frame ea adapted to M on a neighborhood of p ¼ f ðpÞ. Choose the direction vector along the congruence as e1 ¼ cosh ce1 þ sinh ce2 :
ð2:23Þ
We can take the normal vector of M so that he3 ; e3 i ¼ cos h;
he3 ; e3 i ¼ 1:
ð2:24Þ
The normal vector of M can be expressed as e3 ¼ x1 e1 þ x2 e2 þ cos he3 :
ð2:25Þ
By a similar procedure as in Theorem 2.1, we get x1 ¼ sin h sinh c;
x2 ¼ sin h cosh c;
then we have 2 3 2 e1 cosh c 4 e2 5 ¼ 4 sinh c cos h e3
sinh c sin h
sinh c cosh c cos h
cosh c sin h
ð2:26Þ 32 3 0 e1 sin h 54 e2 5: cos h e3
ð2:27Þ
Letting p and p denotes the position vectors for M and M in local coordinates, respectively, then p ¼ p þ re1 :
ð2:28Þ
Taking differentiation of (2.28) and using the structure equations, we get dp ¼ ðx1 þ rx21 sinh cÞe1 þ ðx2 þ rx21 cosh cÞe2 þ rðx13 cosh c þ x23 sinh cÞe3 : ð2:29Þ
And from hdp
; e3 i
¼ 0, we have
rx21 ¼ x1 sinh c x2 cosh c þ r cot hðx13 cosh c þ x23 sinh cÞ:
ð2:30Þ
Again, taking differentiating of (2.30), and using the structure equations, we get rx23 ^ x31 ¼ x12 ^ ðx1 cosh c x2 sinh c þ r cot hðx23 cosh c þ x13 sinh cÞÞ: ð2:31Þ
Thus, by using (2.30) and (2.31), we obtain r2 x23 ^ x31 ¼ ð x2 cosh c þ x1 sinh c þ r cot hðx13 cosh c þ x23 sinh cÞÞ ^ ðx1 cosh c x2 sinh c þ r cot hðx23 cosh c þ x13 sinh cÞÞ ð2:32Þ or by means of (2.7)–(2.10), and (2.22) it reduces to ð1 r2 K cot2 hÞx1 ^ x2 ¼ r2 Kx1 ^ x2 :
ð2:33Þ
R.A. Abdel-Baky / Appl. Math. Comput. 160 (2005) 41–50
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Thus, we have K ¼ sin2 h=r2 , as claimed. Also, we can get for M , that K ¼ sin2 h=r2 as well. This completes the proof of the theorem. Case (B): This time the line congruence is spacelike. As stated in Case (A), the existence of a p.s. congruence between M and M yields that the direction vector along the congruence can be written as e3 ¼ x1 e1 þ x2 e2 þ cosh he3 : As in the last case, we get 2 3 2 e1 cosh h cosh c cosh h sinh c 4 e2 5 ¼ 4 sinh c cosh c e3 cosh c sinh h sinh c sinh h
ð2:34Þ 32 3 e1 sinh h 0 54 e2 5: cosh h e3
ð2:35Þ
Also, we can have dp ¼ ðx1 þ rx21 cosh cÞe1 þ ðx2 þ rx12 sinh cÞe2 þ rðx13 sinh c þ x23 cosh cÞe3 :
ð2:36Þ
Alternatively, hdp ; e3 i ¼ 0, then rx21 ¼ x2 sinh c x1 cosh c þ r coth hðx23 cosh c þ x13 sinh cÞ:
ð2:37Þ
By similar arguments as in the Case (A), we have that K ¼ sin h2 h=r2 , as claimed. Also, we can get for M , that K ¼ sinh2 h=r2 as well. This completes the proof of the theorem.
3. The Sinh-Gordon equation Suppose that M is a spacelike surface, and has no umbilical point. Then we can take the lines of curvature as its parametric curves, and write its fundamental forms as I ¼ A2 du2 þ B2 dv2 ; ð3:1Þ II ¼ k1 A2 du2 þ k2 B2 dv2 ; where k1 , k2 are the principal curvatures of M. According to Eq. (2.7), we get that x13 ¼ k1 A du; x23 ¼ k2 B dv; ð3:2Þ x1 ¼ A du; x2 ¼ B dv; where hii ¼ ki and h12 ¼ 0. Then the Codazzi equations can be written by ðk1 k2 ÞAv þ ðk1 Þv A ¼ 0; ð3:3Þ ðk2 k1 ÞBu þ ðk2 Þu B ¼ 0:
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R.A. Abdel-Baky / Appl. Math. Comput. 160 (2005) 41–50
The Live-Civita connection form will be 1 1
x21 ¼ x12 ¼ Av du þ Bu dv: B A
ð3:4Þ
Now suppose M has constant negative Gaussian curvature K 1; then k1 k2 ¼ 1. By (3.3), we get 2
2
k1 1 k2 1 ð3:5Þ Av þ ðk1 Þv A ¼ 0; Bu þ ðk2 Þu B ¼ 0: k1 k2 Or 1 o o linðk12 1Þ ¼ lin A; 2 ov ov
1 o o 2 linðk21
1Þ ¼ lin B: 2 ou ou
ð3:6Þ
Eqs. (3.6) means that, we can choose two positive valued functions aðuÞ, bðvÞ such that k12 1 ¼
aðuÞ ; A2
k22 1 ¼
bðvÞ : B2
ð3:7Þ
By making change in the coordinate system, we may assume that a ¼ b ¼ 1. Let k1 ¼ coth u, k2 ¼ tanh u, then A ¼ sinh u, B ¼ cosh u, so that the fundamental forms are: I ¼ sinh2 u du2 þ cosh2 u dv2 ;
II ¼ sinh u cosh uðdu2 þ dv2 Þ;
ð3:8Þ
so 2u is the angle between the asymptotic direction on M. It follows from (3.2) that x13 ¼ cosh u du;
x23 ¼ sinh u dv;
x1 ¼ sinh u du;
x2 ¼ cosh u dv: ð3:9Þ
Thus we get dx12 ¼ x13 ^ x32 ¼ sinh u cosh u du ^ dv; in view of the Gauss equation. And from Eq. (3.4), we have 2
o u o2 u dx12 ¼ þ du ^ dv: ou2 ov2
ð3:10Þ
ð3:11Þ
From Eqs. (3.10) and (3.11) it follows that o2 u o2 u þ ¼ sinh u cosh u: ou2 ov2
ð3:12Þ
Let U ¼ 2u is the angle between the two asymptotic directions, then o2 U o2 U þ 2 ¼ sinh U: ou2 ov
ð3:13Þ
R.A. Abdel-Baky / Appl. Math. Comput. 160 (2005) 41–50
49
We call (3.13) the Sinh-Gordon equation in analogy with the classical SinhGordon equation in the Euclidean case [2]. The converse is also true by the existence and uniqueness theorem of surfaces theory. We record the following theorem: Theorem 3.1. Let M is a spacelike surface in R31 . For any positive real number U there is a one-to-one correspondence between solutions U of the Sinh-Gordon equation and the spacelike surfaces of constant negative Gaussian curvature K 1 in R31 up to Lorentzian rigid motion. Since we have K 1, then r ¼ sinh h. So Eqs. (2.18), (3.4), and (3.9) gives
oU U U U sinh h ¼ 2 sinh sin c þ cosh
þ h þ cosh
h cos c; ov 2 2 2 ð3:14Þ oU U sinh h ¼ 2 cosh cos c þ ou 2
sinh
U þ h sinh 2
U
h 2
sin c: ð3:15Þ
We call Eqs. (3.14) and (3.15) the Backlund transformations. Thus if a spacelike p.s. congruence associated with spacelike surfaces M and M , then we obtain the same Backlund transformations. In fact the following theorem is proved: Theorem 3.2. Suppose a p.s. spacelike congruence associated with spacelike focal surfaces M and M in R31 . Then both M and M have Gauassian curvature K 1, and the angles U and U between their asymptotic directions are both solutions of the Sinh-Gordon equation (3.13), and they are related to each other by the Backlund transformations (3.14) and (3.15). Now, let M is a timelike surface and has no umbilical points. By a similar procedure as in the above case, we get the Sinh-Gordon equation o2 U o2 U
2 ¼ sinh U: ou2 ov
ð3:16Þ
And the Backlund transformations, which are correspondence to Case (A) and Case (B) in Theorem 2.2, respectively, are:
oU U U U sin h ¼ 2 sinh sinh c þ cosh þ c þ cosh
c cos h; ov 2 2 2 ð3:17Þ
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R.A. Abdel-Baky / Appl. Math. Comput. 160 (2005) 41–50
oU U sin h ¼ 2 cosh cosh c þ ou 2
cosh
U U þ c cosh
c cos h: 2 2 ð3:18Þ
And oU U sinh h ¼ 2 sinh cosh c þ ov 2 oU U sinh h ¼ 2 cos cosh c þ ou 2
sinh
sinh
U U þ c sinh
c cosh h; 2 2
U þ c þ sinh 2
U
c 2
ð3:19Þ cosh h: ð3:20Þ
Thus we can give the following theorem: Theorem 3.3. Suppose a p.s. timelike (spacelike) congruence associated with timelike focal surfaces M and M in R31 . Then both M and M have Gaussian curvature K 1, and the angles U and U between their asymptotic directions are both solutions of the Sinh-Gordon equation (3.16), and they are related to each other by the Backlund transformations (3.17)–(3.20).
References [1] S.S. Chern, C.L. Terng, An analogue of BacklundÕs theorem in affine geometry, Rocky Mt. J. Math. 10 (I) (1980). [2] L.P. Eisenhart, A treatise in differential geometry of curves and surfaces, Ginn Camp, New York, 1969. [3] S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, I, II, New York, 1963, 1969. [4] L.V. Mc-Nertney, One-parameter families of surfaces with constant curvature in Lorentz 3-Space, Ph.D., Brown University, 1980.
The BacklundÕs theorem in Minkowski 3-space R31 Rashad A. Abdel-Baky Department of Mathematics, Faculty of Science, University of Assiut, 71516 Assiut, Egypt
Abstract In this paper we present the Minkowski versions of the BacklundÕs theorem and its application by using the method of moving frames. Ó 2003 Published by Elsevier Inc. Keywords: Line congruence; Backlund theorem; Sinh-Gordon equation
1. Introduction The classical Backlund theorem studies the transformations of surfaces of constant negative curvature in the Euclidean space E3 by realizing them as the focal surfaces of a pseudo-spherical (p.s.) line congruence. The integrability theorem says that we can construct a new surface in E3 with constant negative curvature from a given one. A congruence of lines is an immersed surface in the Grassmann manifold of all lines in E3 . Locally we can suppose the lines be oriented, with their points given by yðu; v; kÞ ¼ pðu; vÞ þ kfðu; vÞ;
kfk ¼ 1;
k being a parameter on each line [2]. The equations u ¼ uðtÞ;
v ¼ vðtÞ;
u02 þ v02 6¼ 0;
define a ruled surface belonging to the congruence. It is a developable if and only if the determinate ðf; dp; dfÞ ¼ 0: 0096-3003/$ - see front matter Ó 2003 Published by Elsevier Inc. doi:10.1016/S0096-3003(03)00716-1
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R.A. Abdel-Baky / Appl. Math. Comput. 160 (2005) 41–50
This is a quadratic equation in du, dv. Suppose that it has two real and distinct roots, then there are two families of developables surfaces, each of which (in generic case) consists of the tangent lines of two surfaces M and M such that the congruence consists of the lines joining p 2 M to f ðpÞ 2 M . This construction plays a fundamental role in the theory of transformation of surfaces. Definition 1.1. Consider a line congruence with focal surfaces M, and M such that its lines are the common tangents at p 2 M to p ¼ f ðpÞ 2 M . The congruence is called a pseudo-spherical (p.s.) if i(i) kpp k ¼ r, which is a constant independent of p, (ii) The angle between the two normals np and np is equal to a constant h independent of p. We can now state the classical BacklundÕs theorem: Theorem 1.1. Suppose there is a p.s. congruence in E3 with the focal surfaces M and M such that the distance r between corresponding points and the angle h between corresponding normals are constants. Then both M and M have constant negative Gaussian curvature equal to sin2 h=r2 . There is also an integrability theorem: Theorem 1.2. Suppose M is a surface in E3 of constant negative Gaussian curvature K ¼ sin2 h=r2 , where r > 0 and 0 < h < p are constants. Given any unit vector e 2 T ðMp Þ, which is not a principal direction, there exists a unique surface M and a p.s. congruence f : M ! M such that if p ¼ f ðpÞ, we have pp ¼ re and h 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 varying r.
2. The Backlunds theorem in Minkowski 3-space R31 In affine geometry Chern and Terng [1] have give a modern proof of the classical Backlund theorem. Mc-Nertny in [4], treated the same problem by choosing a moving frame on a timelike surface M, such that one vector in the frame points in the direction of the congruence. She proved that: for the timelike surface the Gaussian curvature must be positive for the existence of a p.s. congruence.
R.A. Abdel-Baky / Appl. Math. Comput. 160 (2005) 41–50
43
Our approach here is to consider the Minkowski versions of the same problem, in which the results of Mc-Nertny is special case and to give the Backlund transformations. Let M be a spacelike surface in R31 . 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. Let x1 ; x2 ; x3 be the dual forms to the frame e1 ; e2 ; e3 [3]. We can write X X dp ¼ xa ea ; dea ¼ xab eb : ð2:1Þ a
b
Here and through this paper we shall agree on the index ranges 1 6 i; j; k 6 2;
1 6 a; b; m 6 3:
Note that: x12 þ x21 ¼ 0;
x31 ¼ x13 ;
x32 ¼ x23 :
ð2:2Þ
R31
are: The structure equations of X X dxa ¼ xb ^ xba ; dxab ¼ xam ^ xmb :
ð2:3Þ
c
b
Restricting these forms to the frame defined above, we have x3 ¼ 0
ð2:4Þ
and hence 0 ¼ dx3 ¼
X
xi ^ xi3 :
ð2:5Þ
i
By CartanÕs lemma we may write X xi3 ¼ hij xj ; hij ¼ hji :
ð2:6Þ
j
The first equation of (2.3) gives X dxi ¼ xj ^ xji ;
ð2:7Þ
j
where x12 is the Live-Civita connection form on M which is uniquely determined by these two equations. The Gauss equation is X dxij ¼ xik ^ xki þ Xij ; ð2:8Þ k
where X12 ¼ x13 ^ x32 ¼ x13 ^ x23 ¼ Kx1 ^ x2 ;
ð2:9Þ
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R.A. Abdel-Baky / Appl. Math. Comput. 160 (2005) 41–50
K ¼ detðhij Þ is the Gaussian curvature of the surface M. The Codazzi equations are: X dxi3 ¼ xij ^ xj3 :
ð2:10Þ
j
In the following theorems, we can now state the Minkowski versions of the BacklundÕs theorem: Theorem 2.1. Suppose there is a p.s. spacelike congruence f in R31 between space like focal surfaces M, and M such that the distance r between corresponding points and the angle h between corresponding normals are constant. Then both M, and M have constant negative Gaussian curvature equal to
sinh2 h=r2 . Proof. Since a p.s. congruence exists between M, and M then there is an orthonormal moving frame ea adapted to M on a neighborhood of p ¼ f ðpÞ. Choose the direction vector along the congruence as e1 ¼ cos ce1 þ sin ce2 :
ð2:11Þ
We can take the normal vector of M so that he3 ; e3 i ¼ cosh h;
he3 ; e3 i ¼ 1;
ð2:12Þ
where both h and c are constants. Now, the normal vector of M can be expressed as e3 ¼ x1 e1 þ x2 e2 cosh he3 :
ð2:13Þ
Substituting (2.13) into (2.12), then x1 ¼ sinh h sin c;
x2 ¼ sinh h cos c:
By the vector product in R31 , we have 2 3 2 e1 cos c sin c 6 7 6 4 e2 5 ¼ 4 sin c cosh h cos c cosh h e3 sin c sinh h cos c sinh h
ð2:14Þ
0
32
e1
3
76 7
sinh h 54 e2 5:
cosh h
ð2:15Þ
e3
Suppose locally M is given by an immersion p : U ! R31 , where U is an open subset of the u; v plane, then M is given by p ¼ p þ re1 :
ð2:16Þ
Taking differentiation of (2.16) and using the structure equations, we get dp ¼ ðx1 þ rx21 sin cÞe1 þ ðx2 þ rx12 cos cÞe2 þ rðx13 cos c þ x23 sin cÞe3 : ð2:17Þ
R.A. Abdel-Baky / Appl. Math. Comput. 160 (2005) 41–50
45
On the other hand, from hdp ; e3 i ¼ 0, we have rx12 ¼ x1 sin c x2 cos c þ r coth hðx13 cos c þ x23 sin cÞ:
ð2:18Þ
Again, taking differentiating of (2.18), and using the structure equations, we get rx13 ^ x23 ¼ x12 ^ ðx1 cos c þ x2 sin c þ r coth hðx13 cos c þ x23 sin cÞÞ: ð2:19Þ Thus, by using (2.19) and (2.18), we obtain r2 x13 ^ x23 ¼ ðx1 sin c þ x2 cos c þ r coth hðx13 cos c þ x23 sin cÞÞ ^ ðx1 cos c þ x2 sin c þ r coth hðx13 cos c þ x23 sin cÞÞ ð2:20Þ or by means of (2.7)–(2.10), it reduces to ð1 þ r2 K coth2 hÞx1 ^ x2 ¼ r2 Kx1 ^ x2 : 2
ð2:21Þ
2
Thus, we have K ¼ sinh h=r , as claimed. Since (2.7) can be written as p ¼ p þ rð e1 Þ, the same calculations are valid for M , then we would obtain K ¼ sinh2 h=r2 as well. This completes the proof of the theorem. Now that we have, for spacelike surfaces, shown constant negative Gaussian curvature to be a necessary condition for the existence of a p.s. spacelike congruence. Let M be a timelike surface in R31 . We choose a local field of orthonormal frame ea with origin p is a point of M and the vectors e1 ; e2 are tangent to M at p with e1 is timelike. Then, the same equations (2.1)–(2.11) are exists, except that (2.2) is replaced by x12 ¼ x21 ;
x31 ¼ x13 ;
x32 þ x23 ¼ 0
ð2:22Þ
and the Gaussian curvature K ¼ detðhij Þ. Theorem 2.2. Suppose there is a p.s. congruence f in R31 between timelike focal surfaces M, and M such that the distance r between corresponding points and the angle h between corresponding normals are constant. Then (A) if kpp k ¼ r2 ; i:e: the congruences is timelike, and he3 ; e3 i ¼ cos h; 0 < h < p, both M, and M have constant positive Gaussian curvature equal to sin2 h=r2 , (B) if kpp k ¼ r2 ; i:e: the congruences is spacelike, and he3 ; e3 i ¼ cosh h; h 6¼ 0, both M, and M have constant positive Gaussian curvature equal to sinh2 h=r2 .
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R.A. Abdel-Baky / Appl. Math. Comput. 160 (2005) 41–50
Proof. Case (A) Since a p.s. congruence exists between M, and M then there is an orthonormal moving frame ea adapted to M on a neighborhood of p ¼ f ðpÞ. Choose the direction vector along the congruence as e1 ¼ cosh ce1 þ sinh ce2 :
ð2:23Þ
We can take the normal vector of M so that he3 ; e3 i ¼ cos h;
he3 ; e3 i ¼ 1:
ð2:24Þ
The normal vector of M can be expressed as e3 ¼ x1 e1 þ x2 e2 þ cos he3 :
ð2:25Þ
By a similar procedure as in Theorem 2.1, we get x1 ¼ sin h sinh c;
x2 ¼ sin h cosh c;
then we have 2 3 2 e1 cosh c 4 e2 5 ¼ 4 sinh c cos h e3
sinh c sin h
sinh c cosh c cos h
cosh c sin h
ð2:26Þ 32 3 0 e1 sin h 54 e2 5: cos h e3
ð2:27Þ
Letting p and p denotes the position vectors for M and M in local coordinates, respectively, then p ¼ p þ re1 :
ð2:28Þ
Taking differentiation of (2.28) and using the structure equations, we get dp ¼ ðx1 þ rx21 sinh cÞe1 þ ðx2 þ rx21 cosh cÞe2 þ rðx13 cosh c þ x23 sinh cÞe3 : ð2:29Þ
And from hdp
; e3 i
¼ 0, we have
rx21 ¼ x1 sinh c x2 cosh c þ r cot hðx13 cosh c þ x23 sinh cÞ:
ð2:30Þ
Again, taking differentiating of (2.30), and using the structure equations, we get rx23 ^ x31 ¼ x12 ^ ðx1 cosh c x2 sinh c þ r cot hðx23 cosh c þ x13 sinh cÞÞ: ð2:31Þ
Thus, by using (2.30) and (2.31), we obtain r2 x23 ^ x31 ¼ ð x2 cosh c þ x1 sinh c þ r cot hðx13 cosh c þ x23 sinh cÞÞ ^ ðx1 cosh c x2 sinh c þ r cot hðx23 cosh c þ x13 sinh cÞÞ ð2:32Þ or by means of (2.7)–(2.10), and (2.22) it reduces to ð1 r2 K cot2 hÞx1 ^ x2 ¼ r2 Kx1 ^ x2 :
ð2:33Þ
R.A. Abdel-Baky / Appl. Math. Comput. 160 (2005) 41–50
47
Thus, we have K ¼ sin2 h=r2 , as claimed. Also, we can get for M , that K ¼ sin2 h=r2 as well. This completes the proof of the theorem. Case (B): This time the line congruence is spacelike. As stated in Case (A), the existence of a p.s. congruence between M and M yields that the direction vector along the congruence can be written as e3 ¼ x1 e1 þ x2 e2 þ cosh he3 : As in the last case, we get 2 3 2 e1 cosh h cosh c cosh h sinh c 4 e2 5 ¼ 4 sinh c cosh c e3 cosh c sinh h sinh c sinh h
ð2:34Þ 32 3 e1 sinh h 0 54 e2 5: cosh h e3
ð2:35Þ
Also, we can have dp ¼ ðx1 þ rx21 cosh cÞe1 þ ðx2 þ rx12 sinh cÞe2 þ rðx13 sinh c þ x23 cosh cÞe3 :
ð2:36Þ
Alternatively, hdp ; e3 i ¼ 0, then rx21 ¼ x2 sinh c x1 cosh c þ r coth hðx23 cosh c þ x13 sinh cÞ:
ð2:37Þ
By similar arguments as in the Case (A), we have that K ¼ sin h2 h=r2 , as claimed. Also, we can get for M , that K ¼ sinh2 h=r2 as well. This completes the proof of the theorem.
3. The Sinh-Gordon equation Suppose that M is a spacelike surface, and has no umbilical point. Then we can take the lines of curvature as its parametric curves, and write its fundamental forms as I ¼ A2 du2 þ B2 dv2 ; ð3:1Þ II ¼ k1 A2 du2 þ k2 B2 dv2 ; where k1 , k2 are the principal curvatures of M. According to Eq. (2.7), we get that x13 ¼ k1 A du; x23 ¼ k2 B dv; ð3:2Þ x1 ¼ A du; x2 ¼ B dv; where hii ¼ ki and h12 ¼ 0. Then the Codazzi equations can be written by ðk1 k2 ÞAv þ ðk1 Þv A ¼ 0; ð3:3Þ ðk2 k1 ÞBu þ ðk2 Þu B ¼ 0:
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R.A. Abdel-Baky / Appl. Math. Comput. 160 (2005) 41–50
The Live-Civita connection form will be 1 1
x21 ¼ x12 ¼ Av du þ Bu dv: B A
ð3:4Þ
Now suppose M has constant negative Gaussian curvature K 1; then k1 k2 ¼ 1. By (3.3), we get 2
2
k1 1 k2 1 ð3:5Þ Av þ ðk1 Þv A ¼ 0; Bu þ ðk2 Þu B ¼ 0: k1 k2 Or 1 o o linðk12 1Þ ¼ lin A; 2 ov ov
1 o o 2 linðk21
1Þ ¼ lin B: 2 ou ou
ð3:6Þ
Eqs. (3.6) means that, we can choose two positive valued functions aðuÞ, bðvÞ such that k12 1 ¼
aðuÞ ; A2
k22 1 ¼
bðvÞ : B2
ð3:7Þ
By making change in the coordinate system, we may assume that a ¼ b ¼ 1. Let k1 ¼ coth u, k2 ¼ tanh u, then A ¼ sinh u, B ¼ cosh u, so that the fundamental forms are: I ¼ sinh2 u du2 þ cosh2 u dv2 ;
II ¼ sinh u cosh uðdu2 þ dv2 Þ;
ð3:8Þ
so 2u is the angle between the asymptotic direction on M. It follows from (3.2) that x13 ¼ cosh u du;
x23 ¼ sinh u dv;
x1 ¼ sinh u du;
x2 ¼ cosh u dv: ð3:9Þ
Thus we get dx12 ¼ x13 ^ x32 ¼ sinh u cosh u du ^ dv; in view of the Gauss equation. And from Eq. (3.4), we have 2
o u o2 u dx12 ¼ þ du ^ dv: ou2 ov2
ð3:10Þ
ð3:11Þ
From Eqs. (3.10) and (3.11) it follows that o2 u o2 u þ ¼ sinh u cosh u: ou2 ov2
ð3:12Þ
Let U ¼ 2u is the angle between the two asymptotic directions, then o2 U o2 U þ 2 ¼ sinh U: ou2 ov
ð3:13Þ
R.A. Abdel-Baky / Appl. Math. Comput. 160 (2005) 41–50
49
We call (3.13) the Sinh-Gordon equation in analogy with the classical SinhGordon equation in the Euclidean case [2]. The converse is also true by the existence and uniqueness theorem of surfaces theory. We record the following theorem: Theorem 3.1. Let M is a spacelike surface in R31 . For any positive real number U there is a one-to-one correspondence between solutions U of the Sinh-Gordon equation and the spacelike surfaces of constant negative Gaussian curvature K 1 in R31 up to Lorentzian rigid motion. Since we have K 1, then r ¼ sinh h. So Eqs. (2.18), (3.4), and (3.9) gives
oU U U U sinh h ¼ 2 sinh sin c þ cosh
þ h þ cosh
h cos c; ov 2 2 2 ð3:14Þ oU U sinh h ¼ 2 cosh cos c þ ou 2
sinh
U þ h sinh 2
U
h 2
sin c: ð3:15Þ
We call Eqs. (3.14) and (3.15) the Backlund transformations. Thus if a spacelike p.s. congruence associated with spacelike surfaces M and M , then we obtain the same Backlund transformations. In fact the following theorem is proved: Theorem 3.2. Suppose a p.s. spacelike congruence associated with spacelike focal surfaces M and M in R31 . Then both M and M have Gauassian curvature K 1, and the angles U and U between their asymptotic directions are both solutions of the Sinh-Gordon equation (3.13), and they are related to each other by the Backlund transformations (3.14) and (3.15). Now, let M is a timelike surface and has no umbilical points. By a similar procedure as in the above case, we get the Sinh-Gordon equation o2 U o2 U
2 ¼ sinh U: ou2 ov
ð3:16Þ
And the Backlund transformations, which are correspondence to Case (A) and Case (B) in Theorem 2.2, respectively, are:
oU U U U sin h ¼ 2 sinh sinh c þ cosh þ c þ cosh
c cos h; ov 2 2 2 ð3:17Þ
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R.A. Abdel-Baky / Appl. Math. Comput. 160 (2005) 41–50
oU U sin h ¼ 2 cosh cosh c þ ou 2
cosh
U U þ c cosh
c cos h: 2 2 ð3:18Þ
And oU U sinh h ¼ 2 sinh cosh c þ ov 2 oU U sinh h ¼ 2 cos cosh c þ ou 2
sinh
sinh
U U þ c sinh
c cosh h; 2 2
U þ c þ sinh 2
U
c 2
ð3:19Þ cosh h: ð3:20Þ
Thus we can give the following theorem: Theorem 3.3. Suppose a p.s. timelike (spacelike) congruence associated with timelike focal surfaces M and M in R31 . Then both M and M have Gaussian curvature K 1, and the angles U and U between their asymptotic directions are both solutions of the Sinh-Gordon equation (3.16), and they are related to each other by the Backlund transformations (3.17)–(3.20).
References [1] S.S. Chern, C.L. Terng, An analogue of BacklundÕs theorem in affine geometry, Rocky Mt. J. Math. 10 (I) (1980). [2] L.P. Eisenhart, A treatise in differential geometry of curves and surfaces, Ginn Camp, New York, 1969. [3] S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, I, II, New York, 1963, 1969. [4] L.V. Mc-Nertney, One-parameter families of surfaces with constant curvature in Lorentz 3-Space, Ph.D., Brown University, 1980.