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arXiv:1105.1867v1 [math.DG] 10 May 2011

Backlund transformations of curves in the Galilean and

pseudo-Galilean spaces

S¨uleyman Cengiza,1,, Nevin G¨urb¨uzb,1

aKaratekin University, Mathematics Department, C¸ ankırı

bEski¸sehir Osmangazi University, Mathematics Department, Eski¸sehir

Abstract

Backlund transformations of admissible curves in the Galilean 3-space and

pseudo-Galilean 3-space and also spatial Backlund transformations of space

curves in Galilean 4-space preserve the torsions under certain assumptions.

Keywords: Backlund transformations, pseudo-Galilean space, Galilean space

2010 MSC: 53A35

1. Introduction

In the 1890s Bianchi, Lie, and ﬁnally Backlund looked at what are now called

Backlund transformations of surfaces. In modern parlance, they begin with two

surfaces in Euclidean space in a line congruence: there is a mapping between

the surfaces M1and M2such that the line through any two corresponding

points is tangent to both surfaces. Backlund proved that if a line congruence

satisﬁed two additional conditions, that the line segment joining corresponding

points has constant length, and that the normals at corresponding points form

a constant angle, then the two surfaces are necessarily surfaces of constant

negative curvature. He was also able to show that a Backlund transformation

is integrable, in the sense that given a point on a surface of constant negative

curvature and a tangent line segment at that point, a new surface of constant

negative curvature can be found, containing the endpoint of the line segment,

that is a Backlund transform of the original surface.

The classical Backlund theorem studies the transformation of surfaces of

constant negative curvature in R3by realizing them as the focal surfaces of a

pseudo-spherical line congruence. The integrability theorem says that we can

construct a new surface in R3with constant negative curvature from a given

one. In [1] Tenenblat and Terng established a high dimension generalization of

Backlund’s theorem which is very interesting both for physical and mathematical

reasons. After that Chern and Terng customized Backlund theorem for aﬃne

Email addresses: suleymancengiz@karatekin.edu.tr (S¨uleyman Cengiz),

toprak400@gmail.com (Nevin G¨urb¨uz)

Preprint submitted to Elsevier May 11, 2011

surfaces [2]. By the same year this transformation was reduced to corresponding

asymptotical lines by Terng [3] and following years Tenenblat expanded the

Backlund transformation of two surfaces in R3

1to space forms[4]. In 1990 Palmer

constructed a Backlund transformation between spacelike and timelike surfaces

of constant negative curvature in E3

1[5]. At that decade some researchers gave

Backlund transformations on Weingarten surfaces [6–9].

In 1998 Calini and Ivey [10] proposed a geometric realization of the Back-

lund Transformation for the sine-Gordon equation in the context of curves of

constant torsion. Since the asymptotic lines on a pseudospherical surface have

constant torsion, the Backlund transformation can be restricted to get a trans-

formation that carries constant torsion curves to constant torsion curves. Later

the converse of the idea was proved and generalized for the n-dimensional case

by Nemeth [11]. In [12] Nemeth studied a similar concept for constant torsion

curves in the 3-dimensional constant curvature spaces. Shief and Rogers used

an analogue of the classical Backlund transformation for the generation of soli-

ton surfaces [13]. In [14] Chou, Kouhua and Yongbo obtained the Backlund

transformation on timelike surfaces with constant mean curvature in R2

1.Zuo,

Chen, Cheng studied Backlund theorems in three dimensional de Sitter space

and anti-de Sitter space [15]. Abdel-Baky presented the Minkowski versions

of the Backlund theorem and its application by using the method of moving

frames [16]. G¨urb¨uz studied Backlund transformations in Rn

1[17]. Using the

same method ¨

Ozdemir and C¸ ¨oken have studied Backlund tra nsformations of

non-lightlike constant torsion curves in Minkowski 3-space[18].

In this paper we show that a restriction of Backlund theorem on space curves

satisfying the given three conditions preserves the torsions of the curves in

Galilean and pseudo-Galilean spaces. For the necessary deﬁnitions and the-

orems of Galilean and pseudo-Galilean spaces we refered [20–23]

2. Preliminaries

The Galilean space G3is the three dimensional real aﬃne space with the

absolute ﬁgure {w, f , I}, where wis the ideal plane, fis a line in wand Iis

the ﬁxed elliptic involution of points of f.

The scalar product of two vectors X= (a1, a2, a3) and Y= (b1, b2, b3) in G3

is deﬁned by

< X, Y >G=a1.b1a16= 0 or b16= 0

a2·b2+a3·b3, a1= 0 and b1= 0

An admissible curve α:I⊂R→G3of the class Cr(r ≥3) in the Galilean

space G3is deﬁned by the parametrization

α(s) = (s, x(s), y(s))

where s is the arc length of αwith the diﬀerential form ds =dx. The curvature

κ(s) and the torsion τ(s) of an admissible curve in G3are given by κ(s) =

2

py′′2(s)−z′′2(s) and τ(s) = (det(α′(s), α′′(s), α′′′ (s))/κ2(s) respectively. The

associated moving trihedron is given by

E1=α′(s) = (1, x′(s), y′(s))

E2=(0, x′′(s), y′′(s))

px′′2(s) + y′′2(s)

E3=(0,−y′′(s), x′′ (s))

px′′2(s) + y′′2(s)

Then the Frenet formulas in the Galilean space G3becomes:

E′

1=κE2

E′

2=τE3(1)

E′

3=−τE2

3. Backlund transformations of admissible curves in the Galilean space

G3

Theorem 1. Suppose that ψis a transformation between two admissible curves

αand eαin the Galilean space G3with eα=ψ(α)such that in the corresponding

points we have:

i. The line segment [eα(s)α(s)] at the intersection of the osculating planes of

the curves has constant length r

ii. The distance vector eα(s)−α(s)has the same angle γ6=π

2with the tangent

vectors of the curves

iii. The binormals of the curves have the same constant angle φ6= 0.

Then these curves are congruent with the curvatures and torsions

˜κ=κ=−2dγ

ds

eτ=τ=sin φ

r

and the transformation of the curves is given by

eα=α+2C

τ2+C2(cos γE1+ sin γE2)

where C=τtan φ

2is a constant and γis a solution of the diﬀerential equation

dγ

ds =τsin γtan φ

2

3

Proof. Denote by (E1,E2,E3) and (f

E1,f

E2,f

E3) the Frenet frames of the curves

αand eαin the Galilean space G3.Let f

E3be a unit binormal of eα.

If we denote by W1the unit vector of eα−α, then we can complete W1,E3

and W1,f

E3to the positively oriented orthonormal frames (W1, W2, W3) and

(W1,f

W2,f

W3) where W3=E3,f

W3=f

E3and γis the angle between W1and

E1.The frames (W1, W2, W3) and (W1,f

W2,f

W3) can be obtained by rotating

the frames (E1,E2,E3) and (f

E1,f

E2,f

E3) around E3and f

E3with an angle γ

respectively. So we can write

W1

W2

W3

=

cos γsin γ0

−sin γcos γ0

0 0 1

E1

E2

E3

and

W1

f

W2

f

W3

=

cos γsin γ0

−sin γcos γ0

0 0 1

f

E1

f

E2

f

E3

.

Similarly for a rotation around W1by the angle φ

f

W2= cos φW2−sin φW3

f

W3= sin φW2+ cos φW3

From the above equations we write

f

E1= (cos2γ+ sin2γcos φ)E1+ cos γsin γ(1 −cos φ)E2

+ sin γsin φE3

f

E2= cos γsin γ(1 −cos φ)E1+ (sin2γ+ cos2γcosh φ)E2(2)

−cos γsin φE3

f

E3=−sin γsin φE1+ sin φcos γE2+ cos φE3

Using (1) and (2) for f

E3

df

E3

ds =−eτf

E2

= (−eτcos γsin γ(1 −cos φ))E1

+(−eτ(sin2γ+ cos2γcos φ))E2

+(eτsinh φcos γ)E3

and taking derivative of f

E3in (2) with respect to s

df

E3

ds = (−sin φcos γdγ

ds )E1

+(−τcos φ−sin γsin φ(κ+dγ

ds ))E2

+(τsin φcos γ)E3

4

then equating the two statements above we obtain

eτ=τ

dγ

ds =τsin γtanh φ

2

Similarly, diﬀerentiating f

E1and f

E2from (2) and using (1)

˜κ=κ=−2dγ

ds

Now αis a unit speed curve. Diﬀerentiating

r2= (˜α−α)2

and substituting the distance vector

eα−α=r(cos γE1+ sin γE2) (3)

we ﬁnd that ˜αis also a unit speed curve.

Next taking the derivative of (3) we obtain:

f

E1= (1 −rsin γdγ

ds )E1+rcos γ(κ+dγ

ds )E2+τr sin γE3

From this equation and the Frenet frames (2)

eτ=τ=sin φ

r

Then rearranging this equality we get

r=

2τtan φ

2

τ21 + tan2φ

2

Finally with the aid of (3) , naming the constant C=τtan φ

2, the Backlund

transformation of the curves is

eα=α+2C

τ2+C2(cos γE1+ sin γE2).

4. Backlund Transformations of admissible curves in the pseudo-Galilean

space G3

1

The pseudo-Galilean space G3

1is the three dimensional real aﬃne space with

the absolute ﬁgure {w, f , I}, where wis the ideal plane, fis a line in wand I

is the ﬁxed hyperbolic involution of the points of f.

The scalar product of two vectors X= (a1, a2, a3) and Y= (b1, b2, b3) in G3

1

is deﬁned by

5

< X, Y >G=a1.b1a16= 0 or b16= 0

a2·b2−a3·b3, a1= 0 and b1= 0

The curvature κ(s) and the torsion τ(s) of an admissible curve α(s) = (s, x(s), y(s))

in G3

1are given by

κ(s) = p|x′′2(s)−y′′2(s)|and τ(s) = (det(α′(s), α′′ (s), α′′′(s))/κ2(s)

respectively. The associated moving trihedron is given by

E1=a′(s) = (1, x′(s), y′(s))

E2=(0, x′′(s), y′′(s))

p|x′′2(s)−y′′2(s)|

E3=(0, εy′′ (s), εz′′ (s))

p|x′′2(s)−y′′2(s)|

where ε=∓1. The Frenet formulas in the pseudo-Galilean space G3

1have the

following form:

E′

1=κE2

E′

2=τE3(4)

E′

3=τE2

4.1. Backlund transformations of admissible curves which have time-

like binormals in the pseudo-Galilean space G3

1:

Theorem 2. Suppose that ψis a transformation between two admissible curves

αand eαin the pseudo-Galilean space G3

1with eα=ψ(α)such that in the corre-

sponding points we have:

i. The line segment [eα(s)α(s)] at the intersection of the osculating planes of

the curves has constant length r

ii. The distance vector eα−αhas the same angle γ6=π

2with the tangent

vectors of the curves

iii. The timelike binormals of the curves have the same constant angle φ6= 0.

Then these curves have equal torsions

eτ=τ=−sinh φ

r

and the Backlund transformation of the curves is

eα=α+2C

C2−τ2(cos γE1+ sin γE2)

where C=τtanh φ

2is a constant and γis a solution of the diﬀerential

equation

dγ

ds =τsin γtanh φ

2.

6

Proof. Denote by (E1,E2,E3) and (f

E1,f

E2,f

E3) the Frenet frames of the curves

αand eαin the pseudo-Galilean space G3

1respectively. Let f

E3be a unit timelike

binormal of eαsuch that Df

E3,f

E3E=−1.For the rotations of frames we can

write

W1

W2

W3

=

cos γsin γ0

−sin γcos γ0

0 0 1

E1

E2

E3

,

W1

f

W2

f

W3

=

cos γsin γ0

−sin γcos γ0

0 0 1

f

E1

f

E2

f

E3

and

f

W2= cosh φW2+ sinh φW3

f

W3= sinh φW2+ cosh φW3

From the equations above we can write

f

E1= (cos2γ+ sin2γcosh φ)E1+ cos γsin γ(1 −cosh φ)E2

−sin γsinh φE3

f

E2= cos γsin γ(1 −cosh φ)E1+ (sin2γ+ cos2γcosh φ)E2(5)

+ cos γsinh φE3

f

E3=−sin γsinh φE1+ sinh φcos γE2+ cosh φE3

Diﬀerentiating f

E3with respect to the arc length s and using the Frenet equations

(4) for f

E3we ﬁnd

eτ=τ

dγ

ds =τsin γtanh φ

2

Next taking the derivative of the distance vector

eα−α=r(cos γE1+ sin γE2)

and by (5) we get

eτ=τ=−sinh φ

r

Then rearranging the equality above

r=

2τtanh φ

2

τ2tanh2φ

2−1

Finally with the aid of distance vector, naming the constant C=τtanh φ

2,

the Backlund transformation is obtained as

eα=α+2C

C2−τ2(cos γE1+ sin γE2)

7

4.2. Backlund transformations of admissible curves which have time-

like normals in the pseudo-Galilean space G3

1:

Theorem 3. Suppose that ψis a transformation between two admissible curves

αand eαin the pseudo-Galilean space G3

1with eα=ψ(α)such that in the corre-

sponding points we have:

i. The line segment [eα(s)α(s)] at the intersection of the osculating planes of

the curves has constant length r

ii. The distance vector eα−αhas the same angle γ6= 0 with the tangent

vectors of the curves

iii. The timelike normals of the curves have the same constant angle φ6= 0.

Then these curves have the relation between their torsions

eτ=−τ=−sinh φ

r

and the Backlund transformation of the curves is given by

eα=α+2C

C2−τ2(cosh γE1+ sinh γE2)

where C=τtanh φ

2is a constant and γis a solution of the diﬀerential

equation

dγ

ds =−τsinh γtanh φ

2

Proof. Denote by (E1,E2,E3) and (f

E1,f

E2,f

E3) the Frenet frames of the curves

αand eαin the pseudo-Galilean space G3

1respectively. Let f

E2be a unit timelike

normal of eαsuch that Df

E2,f

E2E=−1.

Again by the notation of previous proof it can be writen

W1

W2

W3

=

cosh γsinh γ0

sinh γcosh γ0

001

E1

E2

E3

,

W1

f

W2

f

W3

=

cosh γsinh γ0

sinh γcosh γ0

0 0 1

f

E1

f

E2

f

E3

and

f

W2= cosh φW2+ sinh φW3

f

W3=−sinh φW2+ cosh φW3

From the above equations we write

f

E1= (cosh2γ−sinh2γcosh φ)E1+ cosh γsinh γ(1 −cosh φ)E2

−sinh γsinh φE3

f

E2= cosh γsinh γ(−1 + cosh φ)E1+ (−sinh2γ+ cosh2γcosh φ)E2(6)

+ cosh γsinh φE3

f

E3=−sinh γsinh φE1−sinh φcosh γE2+ cosh φE3

8

Diﬀerentiating f

E3with respect to the arc length s and using Frenet equation

for f

E3we ﬁnd

eτ=−τ

dγ

ds =−τsinh γtanh φ

2

Next taking the derivative of the distance vector

eα−α=r(cosh γE1+ sinh γE2)

and from (6) it can be found

eτ=−τ=sinh φ

r

Then rearranging the equality above we get

r=

2τtanh φ

2

τ2tanh2φ

2−1

Finally with the aid of distance vector, naming the constant C=τtanh φ

2,

the transformation is obtained as

eα=α+2C

C2−τ2(cosh γE1+ sinh γE2)

5. Spatial Backlund transformations of curves in Galilean space G4

The Galilean space G4consists of a four dimensional real aﬃne space en-

dowed with global absolute time and Euclidean metric structure E over the si-

multaneity hyperplanes deﬁned as the three-dimensional real aﬃne spaces with

underlying vector space Ker(t) of the absolute time functional which is a non

zero linear functional t : V →R on the underlying vector space V of E.

The scalar product of two vectors X= (a1, a2, a3, a4) and Y= (b1, b2, b3, b4)

in G4is deﬁned by

< X, Y >G=a4.b4a46= 0 or b46= 0

a1·b1+a2·b2+a3·b3, a4= 0 and b4= 0

Let α(s) = (x(s), y(s), z(s), t(s)) be the position vector of a curve. Then the

condition α′=E1,|E1|= 1 is equivalent to the condition t(s) = s. Thus natural

equations of a curve α(s) = (x(s), y(s), z(s), s) in G4are

κ(s) = px′′2(s) + y′′2(s) + z′′2(s)

τ(s) = (det(α′(s), α′′ (s), α′′′(s))/κ2(s).

9

The unit tangent vector, the unit normal vector, the unit binormal vector and

the temporal vector (of the time axis) of the curve are shown by E1, E2, E3, E4

respectively. Thus

E1=α′(s) = (x′(s), y′(s), z′(s),1)

E2=E′

1

κ(s)

E3=E′

2

τ(s)

E4=µ E1∧E2∧E3

where µis chosen as ∓1 for det(E1, E2, E3, E4) to be 1.

The Frenet equations in the Galilean 4-space with the spatial Frenet vectors

E1, E2, E3and the temporal vector E4are given by

E′

1=κE2

E′

2=−κE1+τE3(7)

E′

3=−τE2

E′

4=−σE3

Galilean geometry is the study of properties of ﬁgures that are invariant

under the Galilean transformations. In general a Galilean transformation in n

spatial dimensions takes the (n+ 1)-vector (u, t) to the (n+ 1)-vector (Ru+vt+

a, t +a0) where R∈SO(n), v ∈Rnand a∈Rn. Particularly in G4, a spatial

rotation of reference frame happens for the plane spanned by two spatial axes

holding the other plane stationary.

5.1. Spatial Backlund transformations of curves in the Galilean 4-

space:

Theorem 4. Suppose that ψis a transformation between two curves αand eα

in the Galilean space G4with eα=ψ(α).We have:

i. The line segment [eα(s)α(s)] at the intersection of the osculating planes of

the curves has constant length r

ii. The distance vector eα−αhas the same Euclidean angle γ6=π

2with the

tangent vectors of the curves

iii. The binormals of the curves have the same constant Euclidean angle φ6= 0.

Then curvatures, torsions and the spatial Backlund transformation of the curves

are given by

˜κ=−κ−2dγ

ds

eτ=τ=sin φ

r

eα=α+2C

τ2+C2(cos γE1+ sin γE2)

10

where the Backlund parameter is C=τtan φ

2and γis a solution of the

diﬀerential equation dγ

ds =τsin γtan φ

2−κ.

Proof. Denote by (E1,E2,E3,E4) and (f

E1,f

E2,f

E3,f

E4) the Frenet frame of

the curves αand eαin the Galilean space G4respectively. Let f

E3be the unit

binormal of eα.

If we denote by W1the unit vector of eα−α, then we can complete W1,E3,E4

and W1,f

E3,f

E4to the positively oriented orthonormal frames (W1, W2, W3, W4)

and (W1,f

W2,f

W3,f

W4) where W3=E3,f

W3=f

E3, W4=E4,f

W4=f

E4. For a

spatial rotation of the E1E2plane holding the other plane constant we can write

W1

W2

W3

W4

=

cos γsin γ0 0

−sin γcos γ0 0

0 0 1 0

0 0 0 1

E1

E2

E3

E4

and similarly for f

E1f

E2plane

W1

f

W2

f

W3

f

W4

=

cos γsin γ0 0

−sin γcos γ0 0

0 0 1 0

0 0 0 1

f

E1

f

E2

f

E3

f

E4

.

Also we can rotate spatially the f

W2f

W3plane by the transformation

W1

f

W2

f

W3

f

W4

=

1 0 0 0

0 cos φ−sin φ0

0 sin φcos φ0

0 0 0 1

W1

W2

W3

W4

From the above equations we ﬁnd the Frenet vectors

f

E1= (cos2γ+ sin2γcos φ)E1+ cos γsin γ(1 −cos φ)E2

+ sin γsin φE3

f

E2= cos γsin γ(1 −cos φ)E1+ (sin2γ+ cos2γcos φ)E2(8)

−cos γsin φE3

f

E3=−sin γsin φE1+ sin φcos γE2+ cos φE3

f

E4=E4

Since αis a unit speed curve, diﬀerentiating the below

||˜α−α||2=<˜α−α, ˜α−α >=r2

and substituting the distance vector

eα−α=r(cos γE1+ sin γE2)

11

we ﬁnd that ˜αis also a unit speed curve. By the derivative of above equation

and of f

E1with respect to the arclength s we ﬁnd

τ=sin φ

r

dγ

ds =τsin γtan φ

2−κ

Also from the derivative of f

E3and use of Frenet equations gives

˜τ=τ

A similar approach for f

E1results in the equality

˜κ=−κ−2dγ

ds

Hence the transformation of the curves becomes

eα=α+2C

τ2+C2(cos γE1+ sin γE2)

with C=τtan(φ

2).

6. References

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12

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