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Backlund transformations of curves in the Galilean and pseudo-Galilean spaces

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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.
arXiv:1105.1867v1 [math.DG] 10 May 2011
Backlund transformations of curves in the Galilean and
pseudo-Galilean spaces
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]. 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:IRG3of 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))2(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=2
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
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 eαα, 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 φW2sin 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 γ
ds )E1
+(τcos φsin γsin φ(κ+
ds ))E2
+(τsin φcos γ)E3
4
then equating the two statements above we obtain
eτ=τ
ds =τsin γtanh φ
2
Similarly, differentiating f
E1and f
E2from (2) and using (1)
˜κ=κ=2
ds
Now αis a unit speed curve. Differentiating
r2= (˜αα)2
and substituting the distance vector
eαα=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 γ
ds )E1+rcos γ(κ+
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·b2a3·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))2(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 eααhas 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
C2τ2(cos γE1+ sin γE2)
where C=τtanh φ
2is a constant and γis a solution of the differential
equation
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
ds =τsin γtanh φ
2
Next taking the derivative of the distance vector
eαα=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
21
Finally with the aid of distance vector, naming the constant C=τtanh φ
2,
the Backlund transformation is obtained as
eα=α+2C
C2τ2(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 eααhas 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
C2τ2(cosh γE1+ sinh γE2)
where C=τtanh φ
2is a constant and γis a solution of the differential
equation
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= (cosh2γsinh2γcosh 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 φE1sinh φcosh γE2+ cosh φE3
8
Differentiating f
E3with respect to the arc length s and using Frenet equation
for f
E3we find
eτ=τ
ds =τsinh γtanh φ
2
Next taking the derivative of the distance vector
eαα=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
21
Finally with the aid of distance vector, naming the constant C=τtanh φ
2,
the transformation is obtained as
eα=α+2C
C2τ2(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))2(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 E1E2E3
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 RSO(n), v Rnand aRn. 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 eααhas 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=κ2
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
ds =τsin γtan φ
2κ.
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 eαα, 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 φsin 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
||˜αα||2=<˜αα, ˜αα >=r2
and substituting the distance vector
eαα=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
ds =τsin γtan φ
2κ
Also from the derivative of f
E3and use of Frenet equations gives
˜τ=τ
A similar approach for f
E1results in the equality
˜κ=κ2
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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13
... 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 (see [4]). In recent years, Gürbüz [7] studied Bäcklund transformations in R n 1 . ...
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In this study we have defined Bäcklund transformations of curves according to Bishop frame preserving the natural curvatures under certain assumptions in Minkowski 3-space.
... This section is taken from [4]. Introduction Let α : I ⊂ R → E 3 be an arbitrary curve in Euclidean 3-space E 3 . ...
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We define Bäcklund transformations of curves according to the Bishop frame which preserve the natural curvatures under certain assumptions in Euclidean 3-space.
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K. Tenenblat expanded the Bäcklund transformation of two-dimensional surfaces in ℝ 3 to space forms. This transformation was reduced to corresponding asymptotic lines by Terng. Surfaces in three-dimensional constant curvature have constant Gaussian curvature and their asymptotic lines have constant torsion. This case was presented by A. Calim and T. Ivey. Recently S. Nemeth generalized their theorem for constant torsion curves in n-dimensional constant curvature. In this paper we also extend this to the n-dimensional Lorentz space.
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The Bäcklund transformation of two surfaces of ℝ 3 with the same constant negative Gaussian curvature transforms an asymptotic line of one surface into an asymptotic line of the other. Since the asymptotic lines of such a surface have constant torsion, it is natural to restrict the Bäcklund transformations to such curves. This idea was developed by A. Calini and T. Ivey in [J. Knot Theory Ramifications 7, 719-746 (1998; Zbl 0912.53002)]. We prove the converse of their theorem and generalize the transformation to the n-dimensional case.
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