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Integrable motions of curves in projective geometries
Journal of Geometry and Physics 60 (2010) 972–985
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Integrable motions of curves in projective geometries Yanyan Li a,b,∗ , Changzheng Qu b,c , Shichang Shu a a
Institute of Mathematics and Information Science, Xianyang Normal University, Xianyang 712000, PR China
b
Center for Nonlinear Studies, Northwest University, Xi’an 710069, PR China
c
Department of Mathematics, Northwest University, Xi’an 710069, PR China
article
info
Article history: Received 23 June 2009 Accepted 3 March 2010 Available online 12 March 2010 Keywords: Motion of curve Integrable system Projective geometry Kaup–Kupershmidt equation KdV equation
abstract In this paper, integrable systems satisfied by the curvatures of curves under inextensible motions in projective geometries are identified. It is shown that motions of curves in two-, three- and four-dimensional projective geometries are respectively related to the Kaup–Kupershmidt hierarchy, a coupled Kaup–Kupershmidt–KdV system and their extension. Crown Copyright © 2010 Published by Elsevier B.V. All rights reserved.
1. Introduction It has been known that integrable systems are closely related to the motions of curves or surfaces. There have been many works devoted to this subject. About the relationship between integrable systems and motions of curves in the plane, Goldstein and Petrich [1] in an intriguing paper showed that the celebrated mKdV equation naturally arises from inextensible motion of curves in Euclidean geometry, where ‘‘naturally’’ means that the normal velocity in the curve flow is the derivative of curvature of the curve with respect to the arc-length. It was of great interest that Nakayama, Segur and Wadati [2] set up a correspondence between mKdV hierarchy and inextensible motions of plane curves in Euclidean geometry. Later, it was also shown that the KdV hierarchy [3,4], Burgers hierarchy [5,6] and Sawada–Kotera hierarchy [7] naturally arise from inextensible motions of plane curves in centro-affine geometry, similarity geometry and affine geometry, respectively. They can be regarded as a centro-affine version, similarity version and affine version of the mKdV hierarchy. As compared to the integrable motions of plane curves, it is of great interest to study motions of space curves in various geometric settings. The pioneering work concerning the relationship between integrable systems and invariant curve motions is due to Hasimoto [8]. He showed that the vortex filament flow is related to the non-linear Schrödinger equation via the so-called Hasimoto transformation. Indeed, the Schrödinger flow is an inextensible motion of space curves in Euclidean geometry. Lamb [9] used the Hasimoto transformation to connect other motions of curves to the mKdV and sine-Gordon equations. Lakshmanan [10] interpreted the dynamics of a nonlinear string of fixed length in R3 through consideration of the motion of an arbitrary rigid body along it, deriving the AKNS spectral problem without spectral parameter. Langer and Perline [11] showed that the dynamics of a non-stretching vortex filament in R3 gives the NLS hierarchy. Motions of curves on S 2 and S 3 were considered by Doliwa and Santini [12]. Langer et al. [13] obtained the system of mKdV equations from
∗
Corresponding author at: Institute of Mathematics and Information Science, Xianyang Normal University, Xianyang 712000, PR China. E-mail address: [email protected] (Y. Li).
0393-0440/$ – see front matter Crown Copyright © 2010 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.geomphys.2010.03.001
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inextensible motions of space curves in Euclidean geometries. Nakayama [14,15] showed that the defocusing nonlinear Schrödinger equation and a coupled system of KdV equations and their hyperbolic type arise from motions of curves in hyperboloids in the Minkowski space. In [6], integrable motions of space curves in similarity geometries were also studied, a coupled system of mKdV–Burgers hierarchy and their higher-dimensional extension were obtained. The well-known Boussinesq equation can be obtained from the inextensible motion of space curves in affine geometry and centro-affine geometries [7]. It is also of great interest to study motions of curves in various Manifolds and Homogeneous spaces. A number of integrable systems were obtained in [16–32]. More interestingly, the Hamiltonian structure can be identified clearly from the invariant geometric flows, which provides a nice geometric interpretation for Hamiltonian structure to integrable systems. To obtain higher-dimensional integrable systems from invariant curves or surface motions in various geometries, Lakshmanan, Myrzakulov et al. [33–38] studied motions of curves in Euclidean geometries by adding an extra space variable; many higher-dimensional integrable systems can be obtained in this way. This provides a geometric interpretation for higher-dimensional integrable systems, although it is not natural. It has been known that there are three integrable fifth-order evolution equations of the form ut = u5 + auu3 + bu1 u2 + cu2 u1 where a, b and c are constants, u(t , x) is a function of t and x, which are the fifth-order KdV equation, the Sawada–Kotera equation and the Kaup–Kupershmidt equation. The fifth-order KdV equation and the Sawada–Kotera equation have been shown to associate with inextensible motions of curves respectively in centro-affine geometry and affine geometry. So a question arises: Does the Kaup–Kupershmidt equation arise from a motion of curves in certain geometries? The purpose of this paper is to investigate the relationship between integrable systems and inextensible motions of curves in two-, three- and four-dimensional projective geometries. The paper is outlined as follows: In Section 2, we give a brief account of projective differential geometry of curves. In Section 3, we discuss the inextensible motion of plane curves in the two-dimensional projective geometry P2 . It will be shown that the Kaup–Kupershmidt hierarchy naturally arises from such motion flow. In Section 4, we consider motion of curves in the three-dimensional projective geometry P3 , a coupled Kaup–Kupershmidt and KdV system is obtained. Motion of curves in the four-dimensional projective geometry P4 is studied in Section 5. Finally, Section 6 gives concluding remarks on this work. 2. Projective differential geometry of curves Projective differential geometry is an old subject; earlier work on this topic can be traced back to [39]. From Klein’s Erlanger program, it is the study of the differential geometry which is invariant with respect to the projective group, i.e. homography. Let us first give an account of projective differential geometry P2 of plane curves [39–42]. Let A be a point of γ parametrized by s with projective coordinates x0 (s), x1 (s) and x2 (s). Denote it by A = [x0 (s), x1 (s), x2 (s)]T . The coordinates xi of A satisfy the third order differential equation
000 z 000 A
z 00 A00
z0 A0
z = 0. A
It can be written as z 000 + pz 00 + qz 0 + rz = 0
(2.1)
where p=−
|A000 A0 A| , |A00 A0 A|
q=−
|A000 A00 A| , |A00 A0 A|
r =−
|A000 A00 A0 | . |A00 A0 A|
It is necessary to assume that |A00 A0 A| 6= 0, which is equivalent to that the curve has no inflection points. Eq. (2.1) defines a class of curves projectively equal to each other and to the curve γ , but it does not necessarily give all curves projectively ¯ = λ(s)A, we obtain a new equation (2.1) which from the same curve γ , also defines equal to γ . In fact, if we change A to A curves that are projectively equal to it. This also applies to the change of parameter s → s¯ = f (s). This allows us to simplify Eq. (2.1). We look for two functions λ(s) and f (s) in order to eliminate the terms in z 0 and z 00 in the equation. We shall see that there are infinitely many possible ways, as s¯ is only defined up to homography. Now, we want to transform the equation A000 + pA00 + qA0 + rA = 0
(2.2)
to the equation of the form
¯ 000 + r¯ A¯ = 0 A
(2.3)
by the transformations
¯ = λ(s)A, A
s¯ = f (s),
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provided f (s) and λ(s) satisfy the system
λ0 f 00 3 − 0 λ f
3λ00
= p,
λ
−
6λ0 f 00
λ f0
−
f 000 f0
+
3f 00
2
f 02
= q,
(2.4)
where r¯ satisfies 2 λ000 3λ00 f 00 λ0 f 000 3λ0 f 00 3 − − + + r¯ f 0 = r . λ λ f0 λ f0 λ f 02
(2.5)
From the above expression, we also have 1 f 000 2
f0
3 f 00
−
4
2
f 02
=−
1 12
p2 −
1 0 1 p + q. 4 4
(2.6)
The left hand side of (2.6) is the Schwarzian of s¯ with respect to s. Once s¯ is known we deduce λ by the first one of (2.4). It is easily to see that s¯ is defined only up to homography since the Schwarzian is invariant under homography. If we take into account the expression of s¯ and of λ(s) in (2.5), this reduces to 3
r¯ f 0 = r −
1 3
pq +
2 27
p3 −
1 0 1 1 q + pp0 + p00 ≡ H (s), 3 6
2
(2.7)
which can be written as r¯ ds¯3 = Hds3 . Thus we have 1
1
dσ = r¯ 3 ds¯ = H 3 ds. Notice that H involves derivatives of the fourth order; it is easy to check that the arc-length is invariant under the projective transformation group. Using the projective arc-length σ to parametrize the curve γ , and noting the expression (2.4) for p, Eq. (2.2) has no terms in d2 A/dσ 2 , so Eq. (2.2) can be written in the form d3 A dσ 3
+ 2κ
dA dσ
+ hA = 0.
(2.8)
The quantities h and κ are invariant with respect to a projective transformation, but they are not independent, since for any parameter s, we must have 1
dσ = H 3 ds. In particular, if we have dσ = ds, that is to say H = 1 which, taking into account Eq. (2.6), enables us to write h = κ 0 + 1. Hence if the curve is parametrized by arc-length, Eq. (2.8) becomes d3 A dσ 3
+ 2κ
dA dσ
+ (κ 0 + 1)A = 0.
(2.9)
This leaves only one invariant κ in the equation, and it is the projective curvature introduced previously. The Schwarzian {¯s}σ of s¯ with respect to σ is given by
{¯s}σ =
1 2
κ
and we can deduce
κ = 2{¯s}σ . To determine κ we need the following result. Proposition 2.1. Given two functions x(s) and y(s) of the same variable s, we have
{y}x dx2 = [{y}s − {x}s ]ds2 .
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Applying this proposition to x(s) = σ (s) and y(s) = s¯(s), we obtain
{¯s}σ dσ 2 = [{¯s}s − {σ }s ]ds2 . Hence, taking into account the expression of κ , we have
{¯s}s − {σ }s . ( ddsσ )2
κ=2
Since {¯s}s is known, we need to compute {σ }s . In fact, we have dσ ds
1
= H3.
Deriving logarithmically
σ 00 1 H0 = σ0 3 H and a further derivative yields 2 σ 000 σ 00 2 1 H0 1 H 00 − − . = σ0 3 H 3 H2 σ 02
Thus we have
{σ }s =
1 σ 000 2 σ0
−
3 σ 00
2
4 σ 02
=
1 H 00 6 H
−
7 H0
2
36 H 2
.
Plugging them into the expression for κ , we get
" κ=H
2
1 1 1 1 H 00 7 H0 − p0 − p2 + q − + 2 6 2 3 H 18 H 2
− 32
# .
(2.10)
It is readily checked that the projective curvature is related to the affine curvature of the curve γ via
00 2 # − 23 " φ 1 φ 000 7 φ φ − − + κ= , 2 2 3 φ0 18 φ 0 where the 0 denotes the derivative with respect to the affine arc-length. The Frenet frame of the curve is the moving frame of P2 denoted by A, A(1) , A(2) , defined by the following formulas A(1) = A(2) =
dA
,
dσ d2 A dσ 2
+ κ A.
It gives a moving frame of P2 , where the point A(1) is the tangent to the curve in A and the line hA, A(2) i is the projective normal. Notice that det(A, A(1) , A(2) ) = 1. The following Frenet formulas are an immediate consequence of the definition of the Frenet frame and the third-order differential equation (2.8):
A A(1) = A(2) σ
0
κ −1
!
A 0 1 A(1) . 0 A(2)
1 0
κ
(2.11)
Now the Frenet frame is given by A(1) (s) = A(2) (s) =
dA dσ d2 A dσ 2
1
= H− 3
dA ds
, 2
+ κA = H− 3
d2 A ds2
1
5
− H− 3 H0 3
dA ds
+ κ A.
It is easy to verify the fundamental theorem for curves in P2 . Theorem 2.1. Given a continuous function κ(σ ) on a closed interval a ≤ σ ≤ b. Then there exists a unique plane curve up to the homography, its arc-length is σ and curvature is κ(σ ). It follows from the fundamental Theorem 2.1 that the curves with constant projective curvature fall in three categories:
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1. The general parabola: y = xm , m 6∈ {2, 21 , −1}; 2. The logarithmic spiral: ρ = emθ , m 6= 0; 3. The exponential: y = ex . Similarly, we can establish the theory of the curves in n-dimensional projective geometry Pn [42]. In the threedimensional case, we obtain the Serrent–Frenet formulas for curves in P3 [42] 0 A 3κ A(1) (2) = − 16 5 A (3) A σ −τ
1 0
0 1
4κ
0
4
3κ
−
5
0 A 0 A(1) 1 A(2) , A(3) 0
(2.12)
where κ and τ are the projective curvatures of the curves in P3 . In the four-dimensional case, the Serrent–Frenet formulas read as [42]
A 0 (1) A 4κ1 (2) A = 264 A(3) 104κ 2 (4) 24κ3 A
1 0 6κ1 108 8κ2
σ
0 1 0 6κ1 48
0 0 1 0 4κ1
A 0 (1) 0 A (2) 0 A , 1 A(3) 0 A(4)
(2.13)
where κ1 , κ2 and κ3 are the projective curvatures of the curves in P4 . The curves are determined uniquely by the curvatures κ1 , κ2 and κ3 up to the projective transformation group. 3. Motion of curves in two-dimensional projective geometry P2 In this section, we consider inextensible motion of curves in two-dimensional projective geometry P2 by using the previous frame structure to obtain the Kaup–Kupershmidt hierarchy. For convenience, instead of A by γ in (2.11), we have the Serret–Frenet formulas in P2
γ 0 ( 1 ) γ = κ −1 γ (2) σ
γ 0 1 γ (1) , 0 γ (2)
!
1 0
κ
(3.1)
where σ (p, t ) is the projective arc-length and κ is the projective curvature of the curve γ (t , σ ), p is a free parameter, independent of time t. One may parametrize the curve γ by
[γ , γ (1) , γ (2) ] = 1.
(3.2) 2
The projective curve flow in P is governed by
γt = F γ + Gγ (1) + H γ (2) ,
(3.3)
where F , G and H depend on the projective curvature κ and its derivatives. Let L be the perimeter for a closed curve, then
I L=
gdp,
where g = dσ /dp. Normally, we require that the curve is not inextensible and the arc-length does not depend on time t. In view of the formula for the projective arc-length, this is the case if d/dt [γ , γ (1) , γ (2) ] = 0. At the same time, using the projective Serret–Frenet formula (3.1), one can readily find gt g
1
1
3
3
= F + Gσ + Hσ σ + κ H .
Using it, one obtains the time variation of the perimeter dL dt
I =
F + Gσ +
1 3
Hσ σ +
1 3
κ H dσ .
So the inextensibility condition means
I F+
1 3
κ H dσ = 0,
F = −Gσ −
1 3
Hσ σ −
1 3
(3.4)
κH.
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The first equation in (3.4) means that the perimeter for a closed curve is invariant under the motion (3.3), and the second one means that the arc-length σ commutes with time t. In view of Eqs. (3.1) and (3.3), one obtains the time evolution of the frame in two-dimensional projective geometry
γ F γ (1) = F1 F2 γ (2) t
G G1 G2
H H1 H2
! γ γ (1) , γ (2)
(3.5)
where F , G, H, Fi , Gi and Hi , i = 1, 2 are functions of curvature κ to be determined. Eqs. (3.1) and (3.5) are compatible, that is to say
γ γ ( 1 ) ( 1 ) γ = γ , γ (2) t σ γ (2) σ t
one has F1 = Fσ + κ G − H , G1 = F + Gσ + κ H , H1 = G + Hσ , F2 = −κt − κ F + F1σ + κ G1 − H1 , G2 = F1 − κ G + G1σ + κ H1 , H2 = G1 − κ H + H1σ , and
−F + κ F1 − F2σ − κ G2 + H2 = 0, κt = F2 + G − κ G1 + G2σ + κ H2 , G2 + H − κ H1 + H2σ = 0.
(3.6)
Clearing up and simplifying the above equations, one arrives at the following equations F = −Gσ − G=
1 18
1 3
Hσ σ −
Hσ σ σ σ −
5 9
1 3
κH,
κ Hσ σ −
5 18
κσ +
1
2
Hσ −
1 9
1
(κσ σ − 4κ 2 )H − ∂σ−1 [(κσ σ − 4κ 2 )Hσ ], 9
F1 = Fσ + κ G − H , G1 = F + Gσ + κ H , H1 = G + Hσ ,
(3.7)
1 3 1 1 F2 = − Fσ σ − Gσ σ σ − G − κ Hσ σ − (3κσ + 1)Hσ − 2 2 2 2
1 2
κσ σ − κ 2 H ,
G2 = 2Fσ + Gσ σ + κ G + 2κ Hσ + (κσ − 1)H , H2 = F + 2Gσ + Hσ σ ,
κt =
3 2
Fσ σ +
1 2
Gσ σ σ + 2κ Gσ + κσ G +
3 2
3
1
2
2
κ Hσ σ + (κσ − 1)Hσ + κσ σ H .
Substituting F and G into the last equation of system (3.7), we obtain the evolution equation satisfied by the curvature κ
κt = −18Ω Hσ ,
(3.8)
where
Ω = ∂σ6 − 12κ∂σ4 − 36κσ ∂σ3 − 49κσ σ ∂σ2 + 36κ 2 ∂σ2 − 35κσ σ σ ∂σ + 120κκσ ∂σ − 13κσ σ σ σ
+ 82κκσ σ + 69κσ2 − 32κ 3 + 27 − (2κσ σ σ σ σ − 20κκσ σ σ − 50κσ κσ σ + 40κ 2 κσ )∂σ−1 + 2κσ ∂σ−1 (κσ σ − 4κ 2 ). Indeed, taking κ = − 12 κ˜ , we arrive at
Ω = ∂σ6 + 6κ∂ ˜ σ4 + 18κ˜ σ ∂σ3 +
+
41 2
κ˜ κ˜ σ σ
49
35
13
κ˜ σ σ ∂σ2 + 9κ˜ 2 ∂σ2 + κ˜ σ σ σ ∂σ + 30κ˜ κ˜ σ ∂σ + κ˜ σ σ σ σ 2 2 2 69 2 25 1 3 + κ˜ σ + 4κ˜ + 27 + κ˜ σ σ σ σ σ + 5κ˜ κ˜ σ σ σ + κ˜ σ κ˜ σ σ + 5κ˜ 2 κ˜ σ ∂σ−1 + κ˜ σ ∂σ−1 (κ˜ σ σ + 2κ˜ 2 ). 4
2
2
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It is the recursion operator of the Kaup–Kupershmidt equation [43]
κt = κσ σ σ σ σ − 10κκσ σ σ − 25κσ κσ σ + 20κ 2 κσ .
(3.9)
Its integrability has been explored extensively. Letting Hσ = −(1/18)Ω n−1 κσ , we get the Kaup–Kupershmidt hierarchy
κt = Ω n κσ ,
n ≥ 1.
Taking H = −18κ , we arrive at the seventh-order Kaup–Kupershmidt equation
κt = κσ σ σ σ σ σ σ − 14κκσ σ σ σ σ − 49κσ κσ σ σ σ − 84κσ σ κσ σ σ + 56κ 2 κσ σ σ + 252κκσ κσ σ + 70κσ3 −
224 3
κ 3 κσ + 27κσ .
4. Motion of curves in three-dimensional projective geometry P3 Now we study the motion of space curves in three-dimensional projective geometry P3 . In three-dimensional projective geometry, the Serret–Frenet formulas for curves read as [42]
0 γ 3κ ( 1 ) γ (2) = − 16 5 γ γ (3) σ −τ
1 0
0 1
4κ
0
−
4 5
0 γ 0
γ (1)
1 γ (2) .
3κ
0
γ
(4.1)
(3)
The three-dimensional projective curve flow is governed by
γt = E γ + F γ (1) + Gγ (2) + H γ (3) ,
(4.2)
where E, F , G and H are velocities of the curve motion, depending on the projective curvatures κ , τ and their derivatives with respect to the projective arc-length σ . In terms of (4.1) and (4.2), the time evolution of the frame in three-dimensional projective geometry is given by
γ E ( 1 ) γ E1 (2) = E γ 2 (3) E3 γ t
F F1 F2 F3
G G1 G2 G3
γ H γ (1) H1 , H2 γ (2) H3 γ (3)
where E, F , G, H, Ei , Fi , Gi and Hi , i = 1, 2, 3 are functions of the curvatures, to be determined. The commutativity condition ∂σ ∂t = ∂t ∂σ leads to the following equations E1 = Eσ + 3 κ F −
16 5
G − τ H,
F1 = E + Fσ + 4κ G − G1 = F + Gσ + 3κ H ,
4 5
H,
H1 = G + Hσ , E2 = −3κt − 3κ E + E1σ + 3κ F1 − F2 = E1 − 3κ F + F1σ + 4κ G1 − G2 = F1 − 3κ G + G1σ + 3κ H1 ,
4 5
16 5
G1 − τ H1 ,
H1 ,
H2 = G1 − 3κ H + H1σ , 16 16 E − 4κ E1 + E2σ + 3κ F2 − G2 − τ H2 , E3 = 5 5 16 4 F3 = E2 − 4κt + F − 4κ F1 + F2σ + 4κ G2 − H2 , 5 5 16 G3 = F2 + G − 4κ G1 + G2σ + 3κ H2 , 5 16 H3 = G2 + H − 4κ H1 + H2σ , 5
(4.3)
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and 4 4 F1 − 3κ F2 + F3σ + 4κ G3 − H3 = 0, 5 5 4 G3 + τ H + H1 − 3κ H2 + H3σ = 0, 5 4 1 κt = (F3 + τ G + G1 − 3κ G2 + G3σ + 3κ H3 ), 3 5 4 16 τt = −τ E − E1 + 3κ E2 − E3σ − 3κ F3 + G3 + τ H3 . 5 5 Simplifying the above equations, one has the following equations E3 + τ F +
1 3 1 3 E = − Fσ − Gσ σ − 2κ G − Hσ σ σ − 2κ Hσ − κσ H − H , 2 4 2 5 5 5 1 8 2 1 1 8 2 1 1 −1 Gσ σ σ σ − κ Gσ σ − κσ + 1 Gσ − κσ σ − κ + τ G − ∂σ κσ σ − κ + τ Gσ F = 12 3 6 6 3 6 3 3 3 1 5 5 1 1 4 2 1 + Hσ σ σ σ σ − κ Hσ σ σ + − κσ + Hσ σ − κσ σ − κ + τ Hσ − 3κ H 24 6 6 6 3 12 12 4 1 5 1 − ∂σ−1 κσ σ − κ 2 + τ Hσ σ + κ Hσ , 6 3 12 3 16 E1 = Eσ + 3κ F − G − τ H, 5 4 F1 = E + Fσ + 4κ G − H , 5 G1 = F + Gσ + 3κ H ,
H1 = G + Hσ , 7 21 3 33 24 62 3 3 3 7 6 9 E2 = E1σ − κ E − F2σ + κ F1 − F − G1 − κ G2 − G3σ − τ G − τ H1 + H2 − κ H3 , 10 10 10 10 25 25 10 10 10 10 25 10 4 F2 = E1 − 3κ F + F1σ + 4κ G1 − H1 , 5 G2 = F1 − 3κ G + G1σ + 3κ H1 , H2 = G1 − 3κ H + H1σ , 16 16 E3 = E − 4κ E1 + E2σ + 3κ F2 − G2 − τ H2 , 5 5 3 9 3 3 24 38 33 7 7 3 6 21 F3 = E1σ − κ E + F2σ − κ F1 + F − G1 + κ G2 − τ G − G3σ − τ H1 − H2 − κ H3 , 10 10 10 10 25 25 10 10 10 10 25 10 16 G − 4κ G1 + G2σ + 3κ H2 , G3 = F2 + 5 16 H3 = G2 + H − 4κ H1 + H2σ , 5 1 1 6 3 3 3 κt = − Fσ σ σ + 2κ Fσ + κσ F − Gσ σ σ σ + 2κ Gσ σ + κσ − Gσ − Hσ σ σ σ σ − κσ + Hσ σ 2 5 2 5 2 20 9 12 2 3 18 3 − κσ σ − κ 2 + τ Hσ − κσ σ σ − κκσ + τσ H , 5 5 5 5 5 10
1 2 τt = 6Fσ σ + 4τ Fσ + τσ F − Gσ σ σ σ σ σ + 10κ Gσ σ σ σ + 25κσ + Gσ σ σ + (24κσ σ − 32κ 2 + 6τ )Gσ σ 2 5 208 1 + 11κσ σ σ − 96κκσ + κ + 4τσ Gσ + (2κσ σ σ σ − 32κκσ σ − 32κσ2 + 20κσ + τσ σ )G − Hσ σ σ σ σ σ σ 5 5 11 11 19 18 116 2 14 + κ Hσ σ σ σ σ + κσ − Hσ σ σ σ + κσ σ + κ + τ Hσ σ σ 2 5 5 5 5 5 1 348 244 27 272 4 576 3 + − κσ σ σ + κκσ + κ + τσ Hσ σ + −κσ σ σ σ + κκσ σ + 56κσ + 43κσ2 − κτ − κ 10 5 5 10 5 5 5 78 177 3 − 12 + τσ σ Hσ + − κσ σ σ σ σ + κκσ σ σ + κσ κσ σ + 18κσ σ + 2κσ τ 10 5 5 864 2 8 1 − κ κσ − κτσ + τσ σ σ H . 5
5
10
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Y. Li et al. / Journal of Geometry and Physics 60 (2010) 972–985
Substituting the expression for F into the equations for the projective curvatures κ and τ , one obtains the following equations
κ A = 1 A2 τ t
B1 B2
Gσ Hσ
,
(4.4)
where 1 3
τσ σ +
5
1
2
1
1
τσ ∂σ + τ ∂σ2 − κτ − κσ ∂σ−1 τ + Ω2 (τσ ∂σ−1 ), 6 3 6 12 1 7 4 1 8 2 29 53 104 2 9 27 B1 = Ω1 + τ Ω2 ∂σ − ∂σ + κ∂σ + κσ ∂σ + κσ σ − κ + κσ σ σ − κκσ ∂σ−1 2 12 30 3 15 10 5 6 15 5 5 2 1 1 1 3 −1 1 2 1 −1 − κσ ∂σ κ + τσ σ σ + τσ σ ∂σ + τσ ∂σ − κ − ∂σ + κσ − τ − κσ ∂σ−1 τ ∂σ , 3 24 6 24 6 10 12 5 12 5 1 20 28 5 64 208 1 τσ ∂σ3 − κτ + ∂σ2 − 10κσ τ + κτσ ∂σ − 6κσ σ τ + κσ τσ − κ 2 τ − κ A2 = τ ∂σ4 + 3 12 3 5 3 6 3 5 4 4 1 64 8 5 1 + τ2 − κσ σ σ τ + κσ σ τσ − κκσ τ − 20κσ − κ 2 τσ + τ τσ ∂σ−1 − τσ ∂σ−1 (2κσ σ − 16κ 2 + τ ), 3 3 3 3 3 6 6 1 14 196 2 5 1 6 1 14 10 42 ∂σ + τ ∂σ5 − κ − τσ ∂σ4 − 7κσ + κτ + ∂σ3 − κσ σ + 5κσ τ − κ + κτσ B2 = 20 24 3 5 5 5 6 6 5 37 588 5 32 2 104 41 2 2 28 2 − τ ∂σ − κσ σ σ + 3κσ σ τ − κκσ + κσ τσ − κ τ − κ − τσ + τ ∂σ − 2κσ σ σ σ A1 = Ω1 +
15 2
12
5
352
5 32
1
12
3
5
576
30
4
3 292
1
κκσ σ − κσ σ τσ + κκσ τ + 59κσ + 10κσ − κ + κ τσ − κτ − 12 + τσ σ − κσ σ σ τ + 3 6 3 5 3 15 2 5 5 3 78 177 864 2 23 1 −1 − τ τσ − κσ σ σ σ σ − κκσ σ σ − κσ κσ σ + κ κσ + 10κσ τ + κτσ − τσ σ σ ∂σ −
12 1 12
10
5
τσ ∂σ−1 ((2κσ σ
2
5
3
5
2
5
10
− 16κ + τ )∂σ + 20κ), 2
where
Ω1 = −
+
1
13
35
49
41
23
∂σ6 + κ∂σ4 + 3κσ ∂σ3 + κσ σ ∂σ2 − 6κ 2 ∂σ2 + κσ σ σ ∂σ − 20κκσ ∂σ + κσ σ σ σ − κκσ σ − κσ2 12 12 12 3 2 6 1 10 25 40 1 κ3 − + κσ σ σ σ σ − κκσ σ σ − κσ κσ σ + κ 2 κσ ∂σ−1 − κσ ∂σ−1 (κσ σ − 8κ 2 )
24 32 3
5
6
3
3
3
3
is the recursion operator of the Kaup–Kupershmidt equation, and
Ω2 = ∂σ2 − 4κ − 2κσ ∂σ−1 is the recursion operator of the KdV equation. Letting G = 6, H = 0, one gets the following system 1
κt = κσ σ σ σ σ − 20κκσ σ σ − 50κσ κσ σ + 80κ 2 κσ + τσ σ σ − 2κτσ − κσ τ , 2 τt = −8κσ σ σ τ − 2κσ σ τσ + 120κσ + 128κκσ τ − 5τ τσ + 16κ 2 τσ ,
which can be written as 1
κt = κσ σ σ σ σ − 20κκσ σ σ − 50κσ κσ σ + 80κ 2 κσ + (∂σ2 − 4κ − 2κσ ∂σ−1 )τσ , 2
τt = −(2κσ σ + 5τ − 16κ 2 )τσ − 8τ (∂σ2 − 4κ − 2κσ ∂σ−1 )κσ + (80κτ + 120)κσ . Note that
κt = κσ σ σ σ σ − 20κκσ σ σ − 50κσ κσ σ + 80κ 2 κσ is the fifth-order Kaup–Kupershmidt equation and ∂σ2 − 4κ − 2κσ ∂σ−1 is the recursion operator of the KdV equation
κt = κσ σ σ − 6κκσ . So the system (4.5) can be regarded as a coupled Kaup–Kupershmidt–KdV system.
(4.5)
Y. Li et al. / Journal of Geometry and Physics 60 (2010) 972–985
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5. Motion of curves in four-dimensional projective geometry P4
In four-dimensional projective geometry, the Serret–Frenet formulas for curves read as [42]
γ 0 (1) 4κ1 γ (2) γ = 264 γ (3) 104κ2 (4) γ 24κ3 σ
1
0
0
0
1
0
6κ1
0
1
108
6κ1
0
8κ2
48
4κ1
γ 0 γ (1) (2) 0 γ . (3) 1 γ γ (4) 0 0
(5.1)
The four-dimensional projective curve flow is governed by
γt = E γ + F γ (1) + Gγ (2) + H γ (3) + Lγ (4) ,
(5.2)
where E, F ,G, H and L are velocities, depending on the projective curvatures κi , i = 1, 2, 3. In terms of (5.1) and (5.2), the time evolution in four-dimensional projective geometry is given by
γ E (1) E1 γ (2) γ = E2 E γ (3) 3 (4) E4 γ t
F F1 F2 F3 F4
G G1 G2 G3 G4
H H1 H2 H3 H4
γ L (1) γ L1 (2) L2 γ L3 γ (3) L4 γ (4)
where E, F , G, H, L, Ei , Fi , Gi , Hi and Li , i = 1, 2, 3, 4 are functions of κi , i = 1, 2, 3, to be determined. The commutativity condition ∂σ ∂t = ∂t ∂σ leads to the following equations E1 = Eσ + 4κ1 F + 264G + 104κ2 H + 24κ3 L, F1 = E + Fσ + 6κ1 G + 108H + 8κ2 L, G1 = F + Gσ + 6κ1 H + 48L, H1 = G + Hσ + 4κ1 L, L1 = H + Lσ , E2 = −4κ1t − 4κ1 E + E1σ + 4κ1 F1 + 264G1 + 104κ2 H1 + 24κ3 L1 , F2 = E1 − 4κ1 F + F1σ + 6κ1 G1 + 108H1 + 8κ2 L1 , G2 = F1 − 4κ1 G + G1σ + 6κ1 H1 + 48L1 , H2 = G1 − 4κ1 H + H1σ + 4κ1 L1 , L2 = H1 − 4κ1 L + L1σ , E3 = −264E − 6κ1 E1 + E2σ + 4κ1 F2 + 264G2 + 104κ2 H2 + 24κ3 L2 , F3 = −6κ1t + E2 − 264F − 6κ1 F1 + F2σ + 6κ1 G2 + 108H2 + 8κ2 L2 , G3 = F2 − 264G − 6κ1 G1 + G2σ + 6κ1 H2 + 48L2 , H3 = G2 − 264H − 6κ1 H1 + H2σ + 4κ1 L2 , L3 = H2 − 264L − 6κ1 L1 + L2σ , E4 = −104κ2t − 104κ2 E − 108E1 − 6κ1 E2 + E3σ + 4κ1 F3 + 264G3 + 104κ2 H3 + 24κ3 L3 , F4 = E3 − 104κ2 F − 108F1 − 6κ1 F2 + F3σ + 6κ1 G3 + 108H3 + 8κ2 L3 , G4 = −6κ1t + F3 − 104κ2 G − 108G1 − 6κ1 G2 + G3σ + 6κ1 H3 + 48L3 , H4 = G3 − 104κ2 H − 108H1 − 6κ1 H2 + H3σ + 4κ1 L3 , L4 = H3 − 104κ2 L − 108L1 − 6κ1 L2 + L3σ ,
(5.3)
982
Y. Li et al. / Journal of Geometry and Physics 60 (2010) 972–985
and F4 − 24κ3 G − 8κ2 G1 − 48G2 − 4κ1 G3 + G4σ + 6κ1 H4 + 48L4 = 0, H4 − 24κ3 L − 8κ2 L1 − 48L2 − 4κ1 L3 + L4σ = 0, 1
1 66 1 1 26 39 1 E1σ − F− κ1 F1 + F2σ − κ2 G + G1 + G3σ 20 5 10 20 5 5 20 24 1 1 6 2 12 1 − κ3 H + κ2 H1 + 3H2 + κ1 H3 + H4σ + κ3 L1 + κ2 L2 + L3 + κ1 L4 , 5 5 10 20 5 5 5 5 27 3 1 3 1 3 13 E3σ − κ3 F − κ2 F1 − F2 = − κ2 E − E1 − κ1 E2 + 14 28 56 112 14 14 7 1 33 3 13 27 3 1 + F4σ + G3 + κ1 G4 + κ2 H3 + H4 + κ3 L3 + κ2 L4 , 112 14 56 14 28 14 14 1 1 1 1 13 = −κ3 E − κ2 E1 − 2E2 − κ1 E3 + E4σ + κ1 F4 + 11G4 + κ2 H4 + κ3 L4 . 3 6 24 6 3
κ1t = − κ1 E + 5 6
κ2t
κ3t
(5.4)
Simplifying the above equations, one has the following equations 16 5
32
64
32
67
318
429
7338
κ1σ − 1260 Fσ − κ1σ σ F + Gσ σ σ σ σ − κ1 Gσ σ σ − κ1σ − Gσ σ 5 10 5 5 5 219 38 51 286 κ1σ σ − 224κ12 − 448κ2 Gσ − κ1σ σ σ − 224κ1 κ1σ − 224κ2σ G + Hσ σ σ σ σ σ − κ1 Hσ σ σ σ 5 5 10 5 21 1146 1408 Hσ σ σ + κ1σ σ + 224κ12 + κ2 Hσ σ 57κ1σ + 5 5 5 656 48 138 κ1σ σ σ + 224κ1 κ1σ + 13080κ1 + κ2σ + 120κ3 Hσ + κ1σ σ σ σ + 8856κ1σ 5 5 5 112 33 38 144 2004 κ2σ σ + 72κ3σ H + Lσ σ σ σ σ σ σ − κ1 Lσ σ σ σ σ + κ1σ − Lσ σ σ σ 5 25 5 5 5 2346 1208 2 2568 10748 10628 3376 κ1σ σ − κ1 − κ2 Lσ σ σ + κ1σ σ σ − κ1 κ1σ − κ2σ + 7800κ1 25 25 25 25 25 25 108 1278 7848 2 10548 204 κ 3 Lσ σ + κ1σ σ σ σ − κ1σ − κ3σ − κ1 κ1σ σ + 8208κ1σ + 60480 + 2688κ1 κ2 5 25 25 25 25 10428 246 3216 6048 κ2σ σ + 384κ13 Lσ + κ1σ σ σ σ σ − κ1 κ1σ σ σ − κ1σ κ1σ σ + 2520κ1σ σ + 768κ12 κ1σ
Fσ σ σ σ −
5
κ1 Fσ σ −
5
− − + − + + − −
25
3376 25
25
κ2σ σ σ + 1792κ1σ κ2 + 1792κ1 κ2σ −
25
192 5
25
κ3σ σ L = 0,
(5.5)
and
κ1 κ2 κ3
! = t
A1 A2 A3
B1 B2 B3
C1 C2 C3
D1 D2 D3
!
Fσ Gσ H , σ Lσ
(5.6)
where 1 A1 = − Ω , 2 3 B1 = − Ω ∂σ + 63, 4 9 4 9 1 1 C1 = − ∂ σ − κ1σ − 63 ∂σ − (27κ1σ σ − 36κ12 − 112κ2 ) − (9κ1σ σ σ − 54κ1 κ1σ − 8κ2σ )∂σ−1 , 20 2 5 5
1 22 62 ∂σ5 − κ1 ∂σ3 − (5κ1σ − 18)∂σ2 − (36κ1σ σ − 28κ12 − 56κ2 )∂σ − κ1σ σ σ − κ1 κ1σ 10 5 5 5 102 252 48 24 2 − 36κ1 − κ2σ − 6κ3 − κ1σ σ σ σ − 4κ1 κ1σ σ − 2κ1σ − κ1σ − κ2σ σ − κ3σ ∂σ−1 , 5 5 5 5 2 4 2 12 45 12 12 4 12 A2 = ∂σ − κ1 ∂σ2 − κ1σ − ∂σ − κ1σ σ − κ12 − 4κ2 − κ1σ σ σ − κ1 κ1σ − κ2σ ∂σ−1 , D1 = −
35
1
7
35
4
35
35
35
35
Y. Li et al. / Journal of Geometry and Physics 60 (2010) 972–985
99
191
619
747
57
983
58
κ1σ σ − κ12 − 10κ2 ∂σ 1120 280 560 280 70 35 129 15 135 186 3462 − κ1σ σ σ − κ3 − κ2σ − κ1 κ1σ − κ1 560 14 14 35 35 3 58 264 20 3 − κ1σ σ σ σ − κ1 κ1σ σ − 2κ12σ − κ1σ − κ2σ σ − κ3σ ∂σ−1 , 280 35 7 7 7 9 6 79 13 3981 81 52 2 24 4 3 C2 = ∂σ − κ1 ∂ σ + κ1σ − ∂σ + κ1σ σ − κ1 + κ2 ∂σ2 160 280 560 280 70 35 7 927 30 1941 33 15 + κ1σ σ σ − κ1 κ1σ + κ1 + κ2σ + κ3 ∂σ 560 7 10 70 7 166 31 2 1635 288 3 1208 127 15 261 κ1σ σ σ σ − κ1 κ1σ σ − κ1σ + κ1σ + κ1 + κ1 κ2 − κ2σ σ + κ3σ + 1890 + 280 35 14 7 35 35 35 7 27 57 177 1107 432 2 + κ1σ σ σ σ σ − κ1 κ1σ σ σ − κ1σ κ1σ σ + κ1σ σ + κ1 κ1σ 140 35 70 14 35 634 66 9 + 20κ1σ κ2 + κ1 κ2σ − κ2σ σ σ + κ3σ σ ∂σ−1 , 35 35 14 9 7 687 737 2 101 211 423 212 D2 = ∂σ + κ1 ∂σ5 + κ1σ − ∂σ4 + κ1σ σ − κ1 − κ2 ∂σ3 700 1400 280 140 200 350 175 567 6687 363 3667 69 2871 1903 2 + κ1σ σ σ − κ1 κ1σ + κ1 − κ2σ + κ3 ∂σ + κ1σ σ σ σ − κ1 κ1σ σ 200 700 70 700 140 1400 175 1387 2 1248 3 2729 33 1089 627 7356 − κ1σ − κ1σ + κ1 + κ1 κ2 − κ2σ σ − κ3σ + 1485 ∂σ + κ1σ σ σ σ σ B2 =
− +
∂σ5 −
κ1 ∂σ3 −
175 117
70 3903
20 3312
350
κ1 κ1σ σ σ −
7
175
κ3σ σ −
3
20 31
5
25
+ 300κ2σ − κ3σ σ σ −
∂σ2 −
175 3792 2
κ1σ κ1σ σ +
27
κ12 + 444κ2 −
κ1σ +
175 3557
κ1 κ1σ −
κ2σ σ σ
700 213
70
1400
12
1116
κ1σ σ + κ1 κ3 + κ1 κ2σ 25 35 171 837 306 2 + 62κ1σ κ2 + κ1σ σ σ σ σ σ − κ1σ σ σ − κ1σ σ 35
1400
− +
70 848
70 41
35 249
κ1 κ1σ σ −
280
κ1σ σ σ σ σ −
κ12σ −
35
175
35 8612 35
κ1 κ1σ σ σ σ +
2256
70 24
14
175
κ κ
175
+ 24κ1σ σ κ2 + κ1 κ3σ 35 19296 69 2536 1056 2 + 34κ1σ κ2σ + κ1 κ1σ − κ1σ κ1σ σ σ + κ1 κ2σ σ + κ1 κ1σ σ ∂σ−1 , 35 25 175 175 4 5 923 4 2 6 6 48 3 2 A3 = − ∂σ + κ1 ∂σ + κ1σ − ∂σ + κ1σ σ + κ1 − 8κ2 ∂σ 35 35 7 20 35 35 32 12 334 8 4 4 64 + κ1σ σ σ + κ1 κ1σ + κ1 + 5κ3 + κ1σ σ σ σ + κ1 κ1σ σ + κ12σ + κ1σ + κ3σ ∂σ−1 , 35 35 5 35 35 35 5 233 6 103 7543 389 429 773 B3 = − ∂σ + κ1 ∂σ4 + κ1σ − ∂σ3 + κ1σ σ − κ12 − 34κ2 ∂σ2 1120 56 80 280 112 35 2657 2689 585 75 145 479 + κ1σ σ σ − κ1 κ1σ − κ1 − κ2σ + κ3 ∂σ + κ1σ σ σ σ 112 2019
κ2σ σ σ σ −
350 972
14
κ1σ + 16κ13 + 112κ1 κ2 −
κ1 κ1σ σ σ −
3088 35
κ1σ σ −
2 1 1σ
634 35
112
65 2
3
20604
2
5
κ2σ σ − κ3σ +
κ1σ κ1σ σ + 16κ κ
2 1 1σ
169 6 κ1σ κ2 − κ2σ σ σ − κ3σ σ ∂σ−1 , 3 3 21 7 23 7 347 1403 8599 393 197 2 146 5 C3 = − ∂σ + κ1 ∂ σ + κ1σ + ∂σ4 + κ1σ σ − κ1 − κ2 ∂σ3 160 280 560 280 560 35 7 55 741 8989 153 5 − κ1σ σ σ + κ1 κ1σ + κ1 + κ2σ − κ3 ∂σ2 16 70 10 10 7 349 199 3 2 121439 16 2232 293 60 2328 − κ1σ σ σ σ + κ1 κ1σ σ − κ1σ + κ1σ + κ13 − κ1 κ2 + κ2σ σ + κ3σ + ∂σ +
112
κ1 κ2σ +
80
112
70
5
70
35
35
70
7
5
984
Y. Li et al. / Journal of Geometry and Physics 60 (2010) 972–985
17
159
κ1σ σ σ σ σ −
κ1 κ1σ σ σ −
67
89641
5
249
κ1 κ1σ σ +
2032
5344
664
κ1σ σ + κ12 κ1σ + κ1σ κ2 + κ1 κ2σ 8 35 5 70 35 105 105 12012 2 598 13368 99 27 248 − 10κ1 κ3 − κ1 − κ2σ σ σ − κ2 + κ3σ σ − κ1σ σ σ σ σ σ + κ1σ κ2σ −
105
5
80
− −
35 72
175 35356 105 46896
κ κ
2 1 1σ
+
κ1 κ1σ −
κ12 κ1σ σ +
14
70
35 23271
700 1792 2
15 7988
25 106382
7 6032
243
κ1 κ2 +
κ1 κ2σ σ − 10κ1σ κ3 +
1728
7
κ
5 334
κ1σ κ2σ +
25 50563
525 487
2 1σ σ
35
+
28
κ1 κ3σ −
κ2σ σ σ σ −
87302
35 25
175 53691 350
κ1 κ2σ σ −
525 576
κ1 κ12σ −
10188
κ1 κ1σ + 40κ2 κ1σ σ − 2κ1 κ3σ 177 2 97 − κ1 κ1σ σ σ σ + κ1σ σ σ − κ2σ − κ1σ σ + κ3σ σ σ − κ2σ σ σ σ ∂σ−1 , 35 70 5 35 14 35 143 8 83 296 11357 2253 1929 2 66 D3 = − ∂σ − κ1 ∂σ6 − κ1σ − ∂σ5 − κ1σ σ − κ1 + κ2 ∂σ4 4200 1400 175 700 350 350 175 343 4201 5933 37 233 κ1σ σ σ − κ2σ + κ1 − κ1 κ1σ − κ3 ∂σ3 − 20 20 14 140 140 8107 2 2981 412 18608 37724 46637 26637 41 + κ1σ + κ2σ σ − κ1σ σ σ σ − κ1σ − κ1 κ2 + κ1 κ1σ σ − − κ3σ 140 84 35 35 525 700 5 140 28292 3 8212 51372 2 102922 494 152713 14843 − κ1 ∂σ2 + κ2σ σ σ + κ2 + κ1 − κ1σ σ − κ1 κ2σ + κ1 κ3 525 420 25 175 350 525 35 23113 39101 134674 51598 2 967 81 + κ1σ κ1σ σ + κ1 κ1σ σ σ − κ1σ κ2 − κ1 κ1σ + κ3σ σ − κ1σ σ σ σ σ ∂σ −
κ1σ κ1σ σ σ +
864
5
28 73
κ1σ σ σ +
1238
5 6272
20
140 844
κ3σ σ σ +
κ1σ σ κ2 +
35 147587
1056
κ1σ κ3
κ1σ κ1σ σ σ
2100 38376
κ12 κ1σ σ + κ1 κ1σ σ σ σ + 43152κ1 + κ14 + κ22 − κ3 − κ2σ 525 5 15 5 25 2100 1152 3 1271 19968 133 26104 2 256 κ1σ σ σ σ σ σ + κ1 κ1σ − κ1 κ1σ σ σ + κ2σ σ σ σ σ − κ1σ σ σ σ σ σ σ − κ2σ σ − −
− − + −
60 6393 350 1112 15 6272
5
κ1σ σ σ σ +
38
κ1σ σ σ κ2 + κ2 κ2σ +
15 34144 525
κ
3 1σ
−
κ1σ κ3σ +
7 24
7 1792
15 228 5
525
896
κ1 κ3σ σ −
15 179436
κ1 κ1σ κ2 −
κ3σ
κ κ
2 1 2σ
75
−
66904 525
κ1 κ1σ σ +
175 198336
κ1σ κ2σ σ −
2533
525
κ1σ κ1σ σ σ σ +
175 109824
κ12σ + 5 120496 −1 − κ1 κ1σ κ1σ σ ∂σ , 175
4200
23504
25
κ1 κ2σ σ σ − 2216 525
704 5
κ1σ σ κ2σ
κ1 κ1σ σ σ σ σ +
2588 105
κ1σ σ κ1σ σ σ
6
κ1σ + κ3σ σ σ σ 5
525
where Ω = ∂σ − 4κ1 − is the recursion operator of the KdV equation. For a given (E , F , G, H , L), the motion of the curves in four-dimensional projective geometry is determined by Eqs. (5.5) and (5.6). 2
2κ1σ ∂σ−1
6. Concluding remarks In this paper, we have carried out a study of inextensible motions of curves in two-, three- and four-dimensional projective geometries. In fact, the Kaup–Kupershmidt hierarchy, a coupled Kaup–Kupershmidt–KdV system and their extension can be obtained from such motions. So we can set up a correspondence between the Kaup–Kupershmidt hierarchy and the inextensible motions of curves in projective geometry. Similarly, we can discuss motions of inextensible space curves in Pn (n ≥ 5) and motions of surfaces in Pn (n ≥ 2); we believe that many 1 + 1- and 2 + 1-dimensional integrable systems are associated with such motions. Acknowledgements This work was supported by the China NSF for Distinguished Young Scholars (Grant No. 10925104) and the Program of Xianyang Normal University (08XSYK334) of Shaan Xi Province. References [1] R.E. Goldstein, D.M. Petrich, The Korteweg–de Vries hierarchy as dynamics of closed curves in the plane, Phys. Rev. Lett. 67 (1991) 3203–3206. [2] K. Nakayama, H. Segur, M. Wadati, Integrability and the motion of curves, Phys. Rev. Lett. 69 (1992) 2603–2606.
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Journal of Geometry and Physics journal homepage: www.elsevier.com/locate/jgp
Integrable motions of curves in projective geometries Yanyan Li a,b,∗ , Changzheng Qu b,c , Shichang Shu a a
Institute of Mathematics and Information Science, Xianyang Normal University, Xianyang 712000, PR China
b
Center for Nonlinear Studies, Northwest University, Xi’an 710069, PR China
c
Department of Mathematics, Northwest University, Xi’an 710069, PR China
article
info
Article history: Received 23 June 2009 Accepted 3 March 2010 Available online 12 March 2010 Keywords: Motion of curve Integrable system Projective geometry Kaup–Kupershmidt equation KdV equation
abstract In this paper, integrable systems satisfied by the curvatures of curves under inextensible motions in projective geometries are identified. It is shown that motions of curves in two-, three- and four-dimensional projective geometries are respectively related to the Kaup–Kupershmidt hierarchy, a coupled Kaup–Kupershmidt–KdV system and their extension. Crown Copyright © 2010 Published by Elsevier B.V. All rights reserved.
1. Introduction It has been known that integrable systems are closely related to the motions of curves or surfaces. There have been many works devoted to this subject. About the relationship between integrable systems and motions of curves in the plane, Goldstein and Petrich [1] in an intriguing paper showed that the celebrated mKdV equation naturally arises from inextensible motion of curves in Euclidean geometry, where ‘‘naturally’’ means that the normal velocity in the curve flow is the derivative of curvature of the curve with respect to the arc-length. It was of great interest that Nakayama, Segur and Wadati [2] set up a correspondence between mKdV hierarchy and inextensible motions of plane curves in Euclidean geometry. Later, it was also shown that the KdV hierarchy [3,4], Burgers hierarchy [5,6] and Sawada–Kotera hierarchy [7] naturally arise from inextensible motions of plane curves in centro-affine geometry, similarity geometry and affine geometry, respectively. They can be regarded as a centro-affine version, similarity version and affine version of the mKdV hierarchy. As compared to the integrable motions of plane curves, it is of great interest to study motions of space curves in various geometric settings. The pioneering work concerning the relationship between integrable systems and invariant curve motions is due to Hasimoto [8]. He showed that the vortex filament flow is related to the non-linear Schrödinger equation via the so-called Hasimoto transformation. Indeed, the Schrödinger flow is an inextensible motion of space curves in Euclidean geometry. Lamb [9] used the Hasimoto transformation to connect other motions of curves to the mKdV and sine-Gordon equations. Lakshmanan [10] interpreted the dynamics of a nonlinear string of fixed length in R3 through consideration of the motion of an arbitrary rigid body along it, deriving the AKNS spectral problem without spectral parameter. Langer and Perline [11] showed that the dynamics of a non-stretching vortex filament in R3 gives the NLS hierarchy. Motions of curves on S 2 and S 3 were considered by Doliwa and Santini [12]. Langer et al. [13] obtained the system of mKdV equations from
∗
Corresponding author at: Institute of Mathematics and Information Science, Xianyang Normal University, Xianyang 712000, PR China. E-mail address: [email protected] (Y. Li).
0393-0440/$ – see front matter Crown Copyright © 2010 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.geomphys.2010.03.001
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inextensible motions of space curves in Euclidean geometries. Nakayama [14,15] showed that the defocusing nonlinear Schrödinger equation and a coupled system of KdV equations and their hyperbolic type arise from motions of curves in hyperboloids in the Minkowski space. In [6], integrable motions of space curves in similarity geometries were also studied, a coupled system of mKdV–Burgers hierarchy and their higher-dimensional extension were obtained. The well-known Boussinesq equation can be obtained from the inextensible motion of space curves in affine geometry and centro-affine geometries [7]. It is also of great interest to study motions of curves in various Manifolds and Homogeneous spaces. A number of integrable systems were obtained in [16–32]. More interestingly, the Hamiltonian structure can be identified clearly from the invariant geometric flows, which provides a nice geometric interpretation for Hamiltonian structure to integrable systems. To obtain higher-dimensional integrable systems from invariant curves or surface motions in various geometries, Lakshmanan, Myrzakulov et al. [33–38] studied motions of curves in Euclidean geometries by adding an extra space variable; many higher-dimensional integrable systems can be obtained in this way. This provides a geometric interpretation for higher-dimensional integrable systems, although it is not natural. It has been known that there are three integrable fifth-order evolution equations of the form ut = u5 + auu3 + bu1 u2 + cu2 u1 where a, b and c are constants, u(t , x) is a function of t and x, which are the fifth-order KdV equation, the Sawada–Kotera equation and the Kaup–Kupershmidt equation. The fifth-order KdV equation and the Sawada–Kotera equation have been shown to associate with inextensible motions of curves respectively in centro-affine geometry and affine geometry. So a question arises: Does the Kaup–Kupershmidt equation arise from a motion of curves in certain geometries? The purpose of this paper is to investigate the relationship between integrable systems and inextensible motions of curves in two-, three- and four-dimensional projective geometries. The paper is outlined as follows: In Section 2, we give a brief account of projective differential geometry of curves. In Section 3, we discuss the inextensible motion of plane curves in the two-dimensional projective geometry P2 . It will be shown that the Kaup–Kupershmidt hierarchy naturally arises from such motion flow. In Section 4, we consider motion of curves in the three-dimensional projective geometry P3 , a coupled Kaup–Kupershmidt and KdV system is obtained. Motion of curves in the four-dimensional projective geometry P4 is studied in Section 5. Finally, Section 6 gives concluding remarks on this work. 2. Projective differential geometry of curves Projective differential geometry is an old subject; earlier work on this topic can be traced back to [39]. From Klein’s Erlanger program, it is the study of the differential geometry which is invariant with respect to the projective group, i.e. homography. Let us first give an account of projective differential geometry P2 of plane curves [39–42]. Let A be a point of γ parametrized by s with projective coordinates x0 (s), x1 (s) and x2 (s). Denote it by A = [x0 (s), x1 (s), x2 (s)]T . The coordinates xi of A satisfy the third order differential equation
000 z 000 A
z 00 A00
z0 A0
z = 0. A
It can be written as z 000 + pz 00 + qz 0 + rz = 0
(2.1)
where p=−
|A000 A0 A| , |A00 A0 A|
q=−
|A000 A00 A| , |A00 A0 A|
r =−
|A000 A00 A0 | . |A00 A0 A|
It is necessary to assume that |A00 A0 A| 6= 0, which is equivalent to that the curve has no inflection points. Eq. (2.1) defines a class of curves projectively equal to each other and to the curve γ , but it does not necessarily give all curves projectively ¯ = λ(s)A, we obtain a new equation (2.1) which from the same curve γ , also defines equal to γ . In fact, if we change A to A curves that are projectively equal to it. This also applies to the change of parameter s → s¯ = f (s). This allows us to simplify Eq. (2.1). We look for two functions λ(s) and f (s) in order to eliminate the terms in z 0 and z 00 in the equation. We shall see that there are infinitely many possible ways, as s¯ is only defined up to homography. Now, we want to transform the equation A000 + pA00 + qA0 + rA = 0
(2.2)
to the equation of the form
¯ 000 + r¯ A¯ = 0 A
(2.3)
by the transformations
¯ = λ(s)A, A
s¯ = f (s),
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provided f (s) and λ(s) satisfy the system
λ0 f 00 3 − 0 λ f
3λ00
= p,
λ
−
6λ0 f 00
λ f0
−
f 000 f0
+
3f 00
2
f 02
= q,
(2.4)
where r¯ satisfies 2 λ000 3λ00 f 00 λ0 f 000 3λ0 f 00 3 − − + + r¯ f 0 = r . λ λ f0 λ f0 λ f 02
(2.5)
From the above expression, we also have 1 f 000 2
f0
3 f 00
−
4
2
f 02
=−
1 12
p2 −
1 0 1 p + q. 4 4
(2.6)
The left hand side of (2.6) is the Schwarzian of s¯ with respect to s. Once s¯ is known we deduce λ by the first one of (2.4). It is easily to see that s¯ is defined only up to homography since the Schwarzian is invariant under homography. If we take into account the expression of s¯ and of λ(s) in (2.5), this reduces to 3
r¯ f 0 = r −
1 3
pq +
2 27
p3 −
1 0 1 1 q + pp0 + p00 ≡ H (s), 3 6
2
(2.7)
which can be written as r¯ ds¯3 = Hds3 . Thus we have 1
1
dσ = r¯ 3 ds¯ = H 3 ds. Notice that H involves derivatives of the fourth order; it is easy to check that the arc-length is invariant under the projective transformation group. Using the projective arc-length σ to parametrize the curve γ , and noting the expression (2.4) for p, Eq. (2.2) has no terms in d2 A/dσ 2 , so Eq. (2.2) can be written in the form d3 A dσ 3
+ 2κ
dA dσ
+ hA = 0.
(2.8)
The quantities h and κ are invariant with respect to a projective transformation, but they are not independent, since for any parameter s, we must have 1
dσ = H 3 ds. In particular, if we have dσ = ds, that is to say H = 1 which, taking into account Eq. (2.6), enables us to write h = κ 0 + 1. Hence if the curve is parametrized by arc-length, Eq. (2.8) becomes d3 A dσ 3
+ 2κ
dA dσ
+ (κ 0 + 1)A = 0.
(2.9)
This leaves only one invariant κ in the equation, and it is the projective curvature introduced previously. The Schwarzian {¯s}σ of s¯ with respect to σ is given by
{¯s}σ =
1 2
κ
and we can deduce
κ = 2{¯s}σ . To determine κ we need the following result. Proposition 2.1. Given two functions x(s) and y(s) of the same variable s, we have
{y}x dx2 = [{y}s − {x}s ]ds2 .
Y. Li et al. / Journal of Geometry and Physics 60 (2010) 972–985
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Applying this proposition to x(s) = σ (s) and y(s) = s¯(s), we obtain
{¯s}σ dσ 2 = [{¯s}s − {σ }s ]ds2 . Hence, taking into account the expression of κ , we have
{¯s}s − {σ }s . ( ddsσ )2
κ=2
Since {¯s}s is known, we need to compute {σ }s . In fact, we have dσ ds
1
= H3.
Deriving logarithmically
σ 00 1 H0 = σ0 3 H and a further derivative yields 2 σ 000 σ 00 2 1 H0 1 H 00 − − . = σ0 3 H 3 H2 σ 02
Thus we have
{σ }s =
1 σ 000 2 σ0
−
3 σ 00
2
4 σ 02
=
1 H 00 6 H
−
7 H0
2
36 H 2
.
Plugging them into the expression for κ , we get
" κ=H
2
1 1 1 1 H 00 7 H0 − p0 − p2 + q − + 2 6 2 3 H 18 H 2
− 32
# .
(2.10)
It is readily checked that the projective curvature is related to the affine curvature of the curve γ via
00 2 # − 23 " φ 1 φ 000 7 φ φ − − + κ= , 2 2 3 φ0 18 φ 0 where the 0 denotes the derivative with respect to the affine arc-length. The Frenet frame of the curve is the moving frame of P2 denoted by A, A(1) , A(2) , defined by the following formulas A(1) = A(2) =
dA
,
dσ d2 A dσ 2
+ κ A.
It gives a moving frame of P2 , where the point A(1) is the tangent to the curve in A and the line hA, A(2) i is the projective normal. Notice that det(A, A(1) , A(2) ) = 1. The following Frenet formulas are an immediate consequence of the definition of the Frenet frame and the third-order differential equation (2.8):
A A(1) = A(2) σ
0
κ −1
!
A 0 1 A(1) . 0 A(2)
1 0
κ
(2.11)
Now the Frenet frame is given by A(1) (s) = A(2) (s) =
dA dσ d2 A dσ 2
1
= H− 3
dA ds
, 2
+ κA = H− 3
d2 A ds2
1
5
− H− 3 H0 3
dA ds
+ κ A.
It is easy to verify the fundamental theorem for curves in P2 . Theorem 2.1. Given a continuous function κ(σ ) on a closed interval a ≤ σ ≤ b. Then there exists a unique plane curve up to the homography, its arc-length is σ and curvature is κ(σ ). It follows from the fundamental Theorem 2.1 that the curves with constant projective curvature fall in three categories:
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1. The general parabola: y = xm , m 6∈ {2, 21 , −1}; 2. The logarithmic spiral: ρ = emθ , m 6= 0; 3. The exponential: y = ex . Similarly, we can establish the theory of the curves in n-dimensional projective geometry Pn [42]. In the threedimensional case, we obtain the Serrent–Frenet formulas for curves in P3 [42] 0 A 3κ A(1) (2) = − 16 5 A (3) A σ −τ
1 0
0 1
4κ
0
4
3κ
−
5
0 A 0 A(1) 1 A(2) , A(3) 0
(2.12)
where κ and τ are the projective curvatures of the curves in P3 . In the four-dimensional case, the Serrent–Frenet formulas read as [42]
A 0 (1) A 4κ1 (2) A = 264 A(3) 104κ 2 (4) 24κ3 A
1 0 6κ1 108 8κ2
σ
0 1 0 6κ1 48
0 0 1 0 4κ1
A 0 (1) 0 A (2) 0 A , 1 A(3) 0 A(4)
(2.13)
where κ1 , κ2 and κ3 are the projective curvatures of the curves in P4 . The curves are determined uniquely by the curvatures κ1 , κ2 and κ3 up to the projective transformation group. 3. Motion of curves in two-dimensional projective geometry P2 In this section, we consider inextensible motion of curves in two-dimensional projective geometry P2 by using the previous frame structure to obtain the Kaup–Kupershmidt hierarchy. For convenience, instead of A by γ in (2.11), we have the Serret–Frenet formulas in P2
γ 0 ( 1 ) γ = κ −1 γ (2) σ
γ 0 1 γ (1) , 0 γ (2)
!
1 0
κ
(3.1)
where σ (p, t ) is the projective arc-length and κ is the projective curvature of the curve γ (t , σ ), p is a free parameter, independent of time t. One may parametrize the curve γ by
[γ , γ (1) , γ (2) ] = 1.
(3.2) 2
The projective curve flow in P is governed by
γt = F γ + Gγ (1) + H γ (2) ,
(3.3)
where F , G and H depend on the projective curvature κ and its derivatives. Let L be the perimeter for a closed curve, then
I L=
gdp,
where g = dσ /dp. Normally, we require that the curve is not inextensible and the arc-length does not depend on time t. In view of the formula for the projective arc-length, this is the case if d/dt [γ , γ (1) , γ (2) ] = 0. At the same time, using the projective Serret–Frenet formula (3.1), one can readily find gt g
1
1
3
3
= F + Gσ + Hσ σ + κ H .
Using it, one obtains the time variation of the perimeter dL dt
I =
F + Gσ +
1 3
Hσ σ +
1 3
κ H dσ .
So the inextensibility condition means
I F+
1 3
κ H dσ = 0,
F = −Gσ −
1 3
Hσ σ −
1 3
(3.4)
κH.
Y. Li et al. / Journal of Geometry and Physics 60 (2010) 972–985
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The first equation in (3.4) means that the perimeter for a closed curve is invariant under the motion (3.3), and the second one means that the arc-length σ commutes with time t. In view of Eqs. (3.1) and (3.3), one obtains the time evolution of the frame in two-dimensional projective geometry
γ F γ (1) = F1 F2 γ (2) t
G G1 G2
H H1 H2
! γ γ (1) , γ (2)
(3.5)
where F , G, H, Fi , Gi and Hi , i = 1, 2 are functions of curvature κ to be determined. Eqs. (3.1) and (3.5) are compatible, that is to say
γ γ ( 1 ) ( 1 ) γ = γ , γ (2) t σ γ (2) σ t
one has F1 = Fσ + κ G − H , G1 = F + Gσ + κ H , H1 = G + Hσ , F2 = −κt − κ F + F1σ + κ G1 − H1 , G2 = F1 − κ G + G1σ + κ H1 , H2 = G1 − κ H + H1σ , and
−F + κ F1 − F2σ − κ G2 + H2 = 0, κt = F2 + G − κ G1 + G2σ + κ H2 , G2 + H − κ H1 + H2σ = 0.
(3.6)
Clearing up and simplifying the above equations, one arrives at the following equations F = −Gσ − G=
1 18
1 3
Hσ σ −
Hσ σ σ σ −
5 9
1 3
κH,
κ Hσ σ −
5 18
κσ +
1
2
Hσ −
1 9
1
(κσ σ − 4κ 2 )H − ∂σ−1 [(κσ σ − 4κ 2 )Hσ ], 9
F1 = Fσ + κ G − H , G1 = F + Gσ + κ H , H1 = G + Hσ ,
(3.7)
1 3 1 1 F2 = − Fσ σ − Gσ σ σ − G − κ Hσ σ − (3κσ + 1)Hσ − 2 2 2 2
1 2
κσ σ − κ 2 H ,
G2 = 2Fσ + Gσ σ + κ G + 2κ Hσ + (κσ − 1)H , H2 = F + 2Gσ + Hσ σ ,
κt =
3 2
Fσ σ +
1 2
Gσ σ σ + 2κ Gσ + κσ G +
3 2
3
1
2
2
κ Hσ σ + (κσ − 1)Hσ + κσ σ H .
Substituting F and G into the last equation of system (3.7), we obtain the evolution equation satisfied by the curvature κ
κt = −18Ω Hσ ,
(3.8)
where
Ω = ∂σ6 − 12κ∂σ4 − 36κσ ∂σ3 − 49κσ σ ∂σ2 + 36κ 2 ∂σ2 − 35κσ σ σ ∂σ + 120κκσ ∂σ − 13κσ σ σ σ
+ 82κκσ σ + 69κσ2 − 32κ 3 + 27 − (2κσ σ σ σ σ − 20κκσ σ σ − 50κσ κσ σ + 40κ 2 κσ )∂σ−1 + 2κσ ∂σ−1 (κσ σ − 4κ 2 ). Indeed, taking κ = − 12 κ˜ , we arrive at
Ω = ∂σ6 + 6κ∂ ˜ σ4 + 18κ˜ σ ∂σ3 +
+
41 2
κ˜ κ˜ σ σ
49
35
13
κ˜ σ σ ∂σ2 + 9κ˜ 2 ∂σ2 + κ˜ σ σ σ ∂σ + 30κ˜ κ˜ σ ∂σ + κ˜ σ σ σ σ 2 2 2 69 2 25 1 3 + κ˜ σ + 4κ˜ + 27 + κ˜ σ σ σ σ σ + 5κ˜ κ˜ σ σ σ + κ˜ σ κ˜ σ σ + 5κ˜ 2 κ˜ σ ∂σ−1 + κ˜ σ ∂σ−1 (κ˜ σ σ + 2κ˜ 2 ). 4
2
2
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Y. Li et al. / Journal of Geometry and Physics 60 (2010) 972–985
It is the recursion operator of the Kaup–Kupershmidt equation [43]
κt = κσ σ σ σ σ − 10κκσ σ σ − 25κσ κσ σ + 20κ 2 κσ .
(3.9)
Its integrability has been explored extensively. Letting Hσ = −(1/18)Ω n−1 κσ , we get the Kaup–Kupershmidt hierarchy
κt = Ω n κσ ,
n ≥ 1.
Taking H = −18κ , we arrive at the seventh-order Kaup–Kupershmidt equation
κt = κσ σ σ σ σ σ σ − 14κκσ σ σ σ σ − 49κσ κσ σ σ σ − 84κσ σ κσ σ σ + 56κ 2 κσ σ σ + 252κκσ κσ σ + 70κσ3 −
224 3
κ 3 κσ + 27κσ .
4. Motion of curves in three-dimensional projective geometry P3 Now we study the motion of space curves in three-dimensional projective geometry P3 . In three-dimensional projective geometry, the Serret–Frenet formulas for curves read as [42]
0 γ 3κ ( 1 ) γ (2) = − 16 5 γ γ (3) σ −τ
1 0
0 1
4κ
0
−
4 5
0 γ 0
γ (1)
1 γ (2) .
3κ
0
γ
(4.1)
(3)
The three-dimensional projective curve flow is governed by
γt = E γ + F γ (1) + Gγ (2) + H γ (3) ,
(4.2)
where E, F , G and H are velocities of the curve motion, depending on the projective curvatures κ , τ and their derivatives with respect to the projective arc-length σ . In terms of (4.1) and (4.2), the time evolution of the frame in three-dimensional projective geometry is given by
γ E ( 1 ) γ E1 (2) = E γ 2 (3) E3 γ t
F F1 F2 F3
G G1 G2 G3
γ H γ (1) H1 , H2 γ (2) H3 γ (3)
where E, F , G, H, Ei , Fi , Gi and Hi , i = 1, 2, 3 are functions of the curvatures, to be determined. The commutativity condition ∂σ ∂t = ∂t ∂σ leads to the following equations E1 = Eσ + 3 κ F −
16 5
G − τ H,
F1 = E + Fσ + 4κ G − G1 = F + Gσ + 3κ H ,
4 5
H,
H1 = G + Hσ , E2 = −3κt − 3κ E + E1σ + 3κ F1 − F2 = E1 − 3κ F + F1σ + 4κ G1 − G2 = F1 − 3κ G + G1σ + 3κ H1 ,
4 5
16 5
G1 − τ H1 ,
H1 ,
H2 = G1 − 3κ H + H1σ , 16 16 E − 4κ E1 + E2σ + 3κ F2 − G2 − τ H2 , E3 = 5 5 16 4 F3 = E2 − 4κt + F − 4κ F1 + F2σ + 4κ G2 − H2 , 5 5 16 G3 = F2 + G − 4κ G1 + G2σ + 3κ H2 , 5 16 H3 = G2 + H − 4κ H1 + H2σ , 5
(4.3)
Y. Li et al. / Journal of Geometry and Physics 60 (2010) 972–985
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and 4 4 F1 − 3κ F2 + F3σ + 4κ G3 − H3 = 0, 5 5 4 G3 + τ H + H1 − 3κ H2 + H3σ = 0, 5 4 1 κt = (F3 + τ G + G1 − 3κ G2 + G3σ + 3κ H3 ), 3 5 4 16 τt = −τ E − E1 + 3κ E2 − E3σ − 3κ F3 + G3 + τ H3 . 5 5 Simplifying the above equations, one has the following equations E3 + τ F +
1 3 1 3 E = − Fσ − Gσ σ − 2κ G − Hσ σ σ − 2κ Hσ − κσ H − H , 2 4 2 5 5 5 1 8 2 1 1 8 2 1 1 −1 Gσ σ σ σ − κ Gσ σ − κσ + 1 Gσ − κσ σ − κ + τ G − ∂σ κσ σ − κ + τ Gσ F = 12 3 6 6 3 6 3 3 3 1 5 5 1 1 4 2 1 + Hσ σ σ σ σ − κ Hσ σ σ + − κσ + Hσ σ − κσ σ − κ + τ Hσ − 3κ H 24 6 6 6 3 12 12 4 1 5 1 − ∂σ−1 κσ σ − κ 2 + τ Hσ σ + κ Hσ , 6 3 12 3 16 E1 = Eσ + 3κ F − G − τ H, 5 4 F1 = E + Fσ + 4κ G − H , 5 G1 = F + Gσ + 3κ H ,
H1 = G + Hσ , 7 21 3 33 24 62 3 3 3 7 6 9 E2 = E1σ − κ E − F2σ + κ F1 − F − G1 − κ G2 − G3σ − τ G − τ H1 + H2 − κ H3 , 10 10 10 10 25 25 10 10 10 10 25 10 4 F2 = E1 − 3κ F + F1σ + 4κ G1 − H1 , 5 G2 = F1 − 3κ G + G1σ + 3κ H1 , H2 = G1 − 3κ H + H1σ , 16 16 E3 = E − 4κ E1 + E2σ + 3κ F2 − G2 − τ H2 , 5 5 3 9 3 3 24 38 33 7 7 3 6 21 F3 = E1σ − κ E + F2σ − κ F1 + F − G1 + κ G2 − τ G − G3σ − τ H1 − H2 − κ H3 , 10 10 10 10 25 25 10 10 10 10 25 10 16 G − 4κ G1 + G2σ + 3κ H2 , G3 = F2 + 5 16 H3 = G2 + H − 4κ H1 + H2σ , 5 1 1 6 3 3 3 κt = − Fσ σ σ + 2κ Fσ + κσ F − Gσ σ σ σ + 2κ Gσ σ + κσ − Gσ − Hσ σ σ σ σ − κσ + Hσ σ 2 5 2 5 2 20 9 12 2 3 18 3 − κσ σ − κ 2 + τ Hσ − κσ σ σ − κκσ + τσ H , 5 5 5 5 5 10
1 2 τt = 6Fσ σ + 4τ Fσ + τσ F − Gσ σ σ σ σ σ + 10κ Gσ σ σ σ + 25κσ + Gσ σ σ + (24κσ σ − 32κ 2 + 6τ )Gσ σ 2 5 208 1 + 11κσ σ σ − 96κκσ + κ + 4τσ Gσ + (2κσ σ σ σ − 32κκσ σ − 32κσ2 + 20κσ + τσ σ )G − Hσ σ σ σ σ σ σ 5 5 11 11 19 18 116 2 14 + κ Hσ σ σ σ σ + κσ − Hσ σ σ σ + κσ σ + κ + τ Hσ σ σ 2 5 5 5 5 5 1 348 244 27 272 4 576 3 + − κσ σ σ + κκσ + κ + τσ Hσ σ + −κσ σ σ σ + κκσ σ + 56κσ + 43κσ2 − κτ − κ 10 5 5 10 5 5 5 78 177 3 − 12 + τσ σ Hσ + − κσ σ σ σ σ + κκσ σ σ + κσ κσ σ + 18κσ σ + 2κσ τ 10 5 5 864 2 8 1 − κ κσ − κτσ + τσ σ σ H . 5
5
10
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Y. Li et al. / Journal of Geometry and Physics 60 (2010) 972–985
Substituting the expression for F into the equations for the projective curvatures κ and τ , one obtains the following equations
κ A = 1 A2 τ t
B1 B2
Gσ Hσ
,
(4.4)
where 1 3
τσ σ +
5
1
2
1
1
τσ ∂σ + τ ∂σ2 − κτ − κσ ∂σ−1 τ + Ω2 (τσ ∂σ−1 ), 6 3 6 12 1 7 4 1 8 2 29 53 104 2 9 27 B1 = Ω1 + τ Ω2 ∂σ − ∂σ + κ∂σ + κσ ∂σ + κσ σ − κ + κσ σ σ − κκσ ∂σ−1 2 12 30 3 15 10 5 6 15 5 5 2 1 1 1 3 −1 1 2 1 −1 − κσ ∂σ κ + τσ σ σ + τσ σ ∂σ + τσ ∂σ − κ − ∂σ + κσ − τ − κσ ∂σ−1 τ ∂σ , 3 24 6 24 6 10 12 5 12 5 1 20 28 5 64 208 1 τσ ∂σ3 − κτ + ∂σ2 − 10κσ τ + κτσ ∂σ − 6κσ σ τ + κσ τσ − κ 2 τ − κ A2 = τ ∂σ4 + 3 12 3 5 3 6 3 5 4 4 1 64 8 5 1 + τ2 − κσ σ σ τ + κσ σ τσ − κκσ τ − 20κσ − κ 2 τσ + τ τσ ∂σ−1 − τσ ∂σ−1 (2κσ σ − 16κ 2 + τ ), 3 3 3 3 3 6 6 1 14 196 2 5 1 6 1 14 10 42 ∂σ + τ ∂σ5 − κ − τσ ∂σ4 − 7κσ + κτ + ∂σ3 − κσ σ + 5κσ τ − κ + κτσ B2 = 20 24 3 5 5 5 6 6 5 37 588 5 32 2 104 41 2 2 28 2 − τ ∂σ − κσ σ σ + 3κσ σ τ − κκσ + κσ τσ − κ τ − κ − τσ + τ ∂σ − 2κσ σ σ σ A1 = Ω1 +
15 2
12
5
352
5 32
1
12
3
5
576
30
4
3 292
1
κκσ σ − κσ σ τσ + κκσ τ + 59κσ + 10κσ − κ + κ τσ − κτ − 12 + τσ σ − κσ σ σ τ + 3 6 3 5 3 15 2 5 5 3 78 177 864 2 23 1 −1 − τ τσ − κσ σ σ σ σ − κκσ σ σ − κσ κσ σ + κ κσ + 10κσ τ + κτσ − τσ σ σ ∂σ −
12 1 12
10
5
τσ ∂σ−1 ((2κσ σ
2
5
3
5
2
5
10
− 16κ + τ )∂σ + 20κ), 2
where
Ω1 = −
+
1
13
35
49
41
23
∂σ6 + κ∂σ4 + 3κσ ∂σ3 + κσ σ ∂σ2 − 6κ 2 ∂σ2 + κσ σ σ ∂σ − 20κκσ ∂σ + κσ σ σ σ − κκσ σ − κσ2 12 12 12 3 2 6 1 10 25 40 1 κ3 − + κσ σ σ σ σ − κκσ σ σ − κσ κσ σ + κ 2 κσ ∂σ−1 − κσ ∂σ−1 (κσ σ − 8κ 2 )
24 32 3
5
6
3
3
3
3
is the recursion operator of the Kaup–Kupershmidt equation, and
Ω2 = ∂σ2 − 4κ − 2κσ ∂σ−1 is the recursion operator of the KdV equation. Letting G = 6, H = 0, one gets the following system 1
κt = κσ σ σ σ σ − 20κκσ σ σ − 50κσ κσ σ + 80κ 2 κσ + τσ σ σ − 2κτσ − κσ τ , 2 τt = −8κσ σ σ τ − 2κσ σ τσ + 120κσ + 128κκσ τ − 5τ τσ + 16κ 2 τσ ,
which can be written as 1
κt = κσ σ σ σ σ − 20κκσ σ σ − 50κσ κσ σ + 80κ 2 κσ + (∂σ2 − 4κ − 2κσ ∂σ−1 )τσ , 2
τt = −(2κσ σ + 5τ − 16κ 2 )τσ − 8τ (∂σ2 − 4κ − 2κσ ∂σ−1 )κσ + (80κτ + 120)κσ . Note that
κt = κσ σ σ σ σ − 20κκσ σ σ − 50κσ κσ σ + 80κ 2 κσ is the fifth-order Kaup–Kupershmidt equation and ∂σ2 − 4κ − 2κσ ∂σ−1 is the recursion operator of the KdV equation
κt = κσ σ σ − 6κκσ . So the system (4.5) can be regarded as a coupled Kaup–Kupershmidt–KdV system.
(4.5)
Y. Li et al. / Journal of Geometry and Physics 60 (2010) 972–985
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5. Motion of curves in four-dimensional projective geometry P4
In four-dimensional projective geometry, the Serret–Frenet formulas for curves read as [42]
γ 0 (1) 4κ1 γ (2) γ = 264 γ (3) 104κ2 (4) γ 24κ3 σ
1
0
0
0
1
0
6κ1
0
1
108
6κ1
0
8κ2
48
4κ1
γ 0 γ (1) (2) 0 γ . (3) 1 γ γ (4) 0 0
(5.1)
The four-dimensional projective curve flow is governed by
γt = E γ + F γ (1) + Gγ (2) + H γ (3) + Lγ (4) ,
(5.2)
where E, F ,G, H and L are velocities, depending on the projective curvatures κi , i = 1, 2, 3. In terms of (5.1) and (5.2), the time evolution in four-dimensional projective geometry is given by
γ E (1) E1 γ (2) γ = E2 E γ (3) 3 (4) E4 γ t
F F1 F2 F3 F4
G G1 G2 G3 G4
H H1 H2 H3 H4
γ L (1) γ L1 (2) L2 γ L3 γ (3) L4 γ (4)
where E, F , G, H, L, Ei , Fi , Gi , Hi and Li , i = 1, 2, 3, 4 are functions of κi , i = 1, 2, 3, to be determined. The commutativity condition ∂σ ∂t = ∂t ∂σ leads to the following equations E1 = Eσ + 4κ1 F + 264G + 104κ2 H + 24κ3 L, F1 = E + Fσ + 6κ1 G + 108H + 8κ2 L, G1 = F + Gσ + 6κ1 H + 48L, H1 = G + Hσ + 4κ1 L, L1 = H + Lσ , E2 = −4κ1t − 4κ1 E + E1σ + 4κ1 F1 + 264G1 + 104κ2 H1 + 24κ3 L1 , F2 = E1 − 4κ1 F + F1σ + 6κ1 G1 + 108H1 + 8κ2 L1 , G2 = F1 − 4κ1 G + G1σ + 6κ1 H1 + 48L1 , H2 = G1 − 4κ1 H + H1σ + 4κ1 L1 , L2 = H1 − 4κ1 L + L1σ , E3 = −264E − 6κ1 E1 + E2σ + 4κ1 F2 + 264G2 + 104κ2 H2 + 24κ3 L2 , F3 = −6κ1t + E2 − 264F − 6κ1 F1 + F2σ + 6κ1 G2 + 108H2 + 8κ2 L2 , G3 = F2 − 264G − 6κ1 G1 + G2σ + 6κ1 H2 + 48L2 , H3 = G2 − 264H − 6κ1 H1 + H2σ + 4κ1 L2 , L3 = H2 − 264L − 6κ1 L1 + L2σ , E4 = −104κ2t − 104κ2 E − 108E1 − 6κ1 E2 + E3σ + 4κ1 F3 + 264G3 + 104κ2 H3 + 24κ3 L3 , F4 = E3 − 104κ2 F − 108F1 − 6κ1 F2 + F3σ + 6κ1 G3 + 108H3 + 8κ2 L3 , G4 = −6κ1t + F3 − 104κ2 G − 108G1 − 6κ1 G2 + G3σ + 6κ1 H3 + 48L3 , H4 = G3 − 104κ2 H − 108H1 − 6κ1 H2 + H3σ + 4κ1 L3 , L4 = H3 − 104κ2 L − 108L1 − 6κ1 L2 + L3σ ,
(5.3)
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Y. Li et al. / Journal of Geometry and Physics 60 (2010) 972–985
and F4 − 24κ3 G − 8κ2 G1 − 48G2 − 4κ1 G3 + G4σ + 6κ1 H4 + 48L4 = 0, H4 − 24κ3 L − 8κ2 L1 − 48L2 − 4κ1 L3 + L4σ = 0, 1
1 66 1 1 26 39 1 E1σ − F− κ1 F1 + F2σ − κ2 G + G1 + G3σ 20 5 10 20 5 5 20 24 1 1 6 2 12 1 − κ3 H + κ2 H1 + 3H2 + κ1 H3 + H4σ + κ3 L1 + κ2 L2 + L3 + κ1 L4 , 5 5 10 20 5 5 5 5 27 3 1 3 1 3 13 E3σ − κ3 F − κ2 F1 − F2 = − κ2 E − E1 − κ1 E2 + 14 28 56 112 14 14 7 1 33 3 13 27 3 1 + F4σ + G3 + κ1 G4 + κ2 H3 + H4 + κ3 L3 + κ2 L4 , 112 14 56 14 28 14 14 1 1 1 1 13 = −κ3 E − κ2 E1 − 2E2 − κ1 E3 + E4σ + κ1 F4 + 11G4 + κ2 H4 + κ3 L4 . 3 6 24 6 3
κ1t = − κ1 E + 5 6
κ2t
κ3t
(5.4)
Simplifying the above equations, one has the following equations 16 5
32
64
32
67
318
429
7338
κ1σ − 1260 Fσ − κ1σ σ F + Gσ σ σ σ σ − κ1 Gσ σ σ − κ1σ − Gσ σ 5 10 5 5 5 219 38 51 286 κ1σ σ − 224κ12 − 448κ2 Gσ − κ1σ σ σ − 224κ1 κ1σ − 224κ2σ G + Hσ σ σ σ σ σ − κ1 Hσ σ σ σ 5 5 10 5 21 1146 1408 Hσ σ σ + κ1σ σ + 224κ12 + κ2 Hσ σ 57κ1σ + 5 5 5 656 48 138 κ1σ σ σ + 224κ1 κ1σ + 13080κ1 + κ2σ + 120κ3 Hσ + κ1σ σ σ σ + 8856κ1σ 5 5 5 112 33 38 144 2004 κ2σ σ + 72κ3σ H + Lσ σ σ σ σ σ σ − κ1 Lσ σ σ σ σ + κ1σ − Lσ σ σ σ 5 25 5 5 5 2346 1208 2 2568 10748 10628 3376 κ1σ σ − κ1 − κ2 Lσ σ σ + κ1σ σ σ − κ1 κ1σ − κ2σ + 7800κ1 25 25 25 25 25 25 108 1278 7848 2 10548 204 κ 3 Lσ σ + κ1σ σ σ σ − κ1σ − κ3σ − κ1 κ1σ σ + 8208κ1σ + 60480 + 2688κ1 κ2 5 25 25 25 25 10428 246 3216 6048 κ2σ σ + 384κ13 Lσ + κ1σ σ σ σ σ − κ1 κ1σ σ σ − κ1σ κ1σ σ + 2520κ1σ σ + 768κ12 κ1σ
Fσ σ σ σ −
5
κ1 Fσ σ −
5
− − + − + + − −
25
3376 25
25
κ2σ σ σ + 1792κ1σ κ2 + 1792κ1 κ2σ −
25
192 5
25
κ3σ σ L = 0,
(5.5)
and
κ1 κ2 κ3
! = t
A1 A2 A3
B1 B2 B3
C1 C2 C3
D1 D2 D3
!
Fσ Gσ H , σ Lσ
(5.6)
where 1 A1 = − Ω , 2 3 B1 = − Ω ∂σ + 63, 4 9 4 9 1 1 C1 = − ∂ σ − κ1σ − 63 ∂σ − (27κ1σ σ − 36κ12 − 112κ2 ) − (9κ1σ σ σ − 54κ1 κ1σ − 8κ2σ )∂σ−1 , 20 2 5 5
1 22 62 ∂σ5 − κ1 ∂σ3 − (5κ1σ − 18)∂σ2 − (36κ1σ σ − 28κ12 − 56κ2 )∂σ − κ1σ σ σ − κ1 κ1σ 10 5 5 5 102 252 48 24 2 − 36κ1 − κ2σ − 6κ3 − κ1σ σ σ σ − 4κ1 κ1σ σ − 2κ1σ − κ1σ − κ2σ σ − κ3σ ∂σ−1 , 5 5 5 5 2 4 2 12 45 12 12 4 12 A2 = ∂σ − κ1 ∂σ2 − κ1σ − ∂σ − κ1σ σ − κ12 − 4κ2 − κ1σ σ σ − κ1 κ1σ − κ2σ ∂σ−1 , D1 = −
35
1
7
35
4
35
35
35
35
Y. Li et al. / Journal of Geometry and Physics 60 (2010) 972–985
99
191
619
747
57
983
58
κ1σ σ − κ12 − 10κ2 ∂σ 1120 280 560 280 70 35 129 15 135 186 3462 − κ1σ σ σ − κ3 − κ2σ − κ1 κ1σ − κ1 560 14 14 35 35 3 58 264 20 3 − κ1σ σ σ σ − κ1 κ1σ σ − 2κ12σ − κ1σ − κ2σ σ − κ3σ ∂σ−1 , 280 35 7 7 7 9 6 79 13 3981 81 52 2 24 4 3 C2 = ∂σ − κ1 ∂ σ + κ1σ − ∂σ + κ1σ σ − κ1 + κ2 ∂σ2 160 280 560 280 70 35 7 927 30 1941 33 15 + κ1σ σ σ − κ1 κ1σ + κ1 + κ2σ + κ3 ∂σ 560 7 10 70 7 166 31 2 1635 288 3 1208 127 15 261 κ1σ σ σ σ − κ1 κ1σ σ − κ1σ + κ1σ + κ1 + κ1 κ2 − κ2σ σ + κ3σ + 1890 + 280 35 14 7 35 35 35 7 27 57 177 1107 432 2 + κ1σ σ σ σ σ − κ1 κ1σ σ σ − κ1σ κ1σ σ + κ1σ σ + κ1 κ1σ 140 35 70 14 35 634 66 9 + 20κ1σ κ2 + κ1 κ2σ − κ2σ σ σ + κ3σ σ ∂σ−1 , 35 35 14 9 7 687 737 2 101 211 423 212 D2 = ∂σ + κ1 ∂σ5 + κ1σ − ∂σ4 + κ1σ σ − κ1 − κ2 ∂σ3 700 1400 280 140 200 350 175 567 6687 363 3667 69 2871 1903 2 + κ1σ σ σ − κ1 κ1σ + κ1 − κ2σ + κ3 ∂σ + κ1σ σ σ σ − κ1 κ1σ σ 200 700 70 700 140 1400 175 1387 2 1248 3 2729 33 1089 627 7356 − κ1σ − κ1σ + κ1 + κ1 κ2 − κ2σ σ − κ3σ + 1485 ∂σ + κ1σ σ σ σ σ B2 =
− +
∂σ5 −
κ1 ∂σ3 −
175 117
70 3903
20 3312
350
κ1 κ1σ σ σ −
7
175
κ3σ σ −
3
20 31
5
25
+ 300κ2σ − κ3σ σ σ −
∂σ2 −
175 3792 2
κ1σ κ1σ σ +
27
κ12 + 444κ2 −
κ1σ +
175 3557
κ1 κ1σ −
κ2σ σ σ
700 213
70
1400
12
1116
κ1σ σ + κ1 κ3 + κ1 κ2σ 25 35 171 837 306 2 + 62κ1σ κ2 + κ1σ σ σ σ σ σ − κ1σ σ σ − κ1σ σ 35
1400
− +
70 848
70 41
35 249
κ1 κ1σ σ −
280
κ1σ σ σ σ σ −
κ12σ −
35
175
35 8612 35
κ1 κ1σ σ σ σ +
2256
70 24
14
175
κ κ
175
+ 24κ1σ σ κ2 + κ1 κ3σ 35 19296 69 2536 1056 2 + 34κ1σ κ2σ + κ1 κ1σ − κ1σ κ1σ σ σ + κ1 κ2σ σ + κ1 κ1σ σ ∂σ−1 , 35 25 175 175 4 5 923 4 2 6 6 48 3 2 A3 = − ∂σ + κ1 ∂σ + κ1σ − ∂σ + κ1σ σ + κ1 − 8κ2 ∂σ 35 35 7 20 35 35 32 12 334 8 4 4 64 + κ1σ σ σ + κ1 κ1σ + κ1 + 5κ3 + κ1σ σ σ σ + κ1 κ1σ σ + κ12σ + κ1σ + κ3σ ∂σ−1 , 35 35 5 35 35 35 5 233 6 103 7543 389 429 773 B3 = − ∂σ + κ1 ∂σ4 + κ1σ − ∂σ3 + κ1σ σ − κ12 − 34κ2 ∂σ2 1120 56 80 280 112 35 2657 2689 585 75 145 479 + κ1σ σ σ − κ1 κ1σ − κ1 − κ2σ + κ3 ∂σ + κ1σ σ σ σ 112 2019
κ2σ σ σ σ −
350 972
14
κ1σ + 16κ13 + 112κ1 κ2 −
κ1 κ1σ σ σ −
3088 35
κ1σ σ −
2 1 1σ
634 35
112
65 2
3
20604
2
5
κ2σ σ − κ3σ +
κ1σ κ1σ σ + 16κ κ
2 1 1σ
169 6 κ1σ κ2 − κ2σ σ σ − κ3σ σ ∂σ−1 , 3 3 21 7 23 7 347 1403 8599 393 197 2 146 5 C3 = − ∂σ + κ1 ∂ σ + κ1σ + ∂σ4 + κ1σ σ − κ1 − κ2 ∂σ3 160 280 560 280 560 35 7 55 741 8989 153 5 − κ1σ σ σ + κ1 κ1σ + κ1 + κ2σ − κ3 ∂σ2 16 70 10 10 7 349 199 3 2 121439 16 2232 293 60 2328 − κ1σ σ σ σ + κ1 κ1σ σ − κ1σ + κ1σ + κ13 − κ1 κ2 + κ2σ σ + κ3σ + ∂σ +
112
κ1 κ2σ +
80
112
70
5
70
35
35
70
7
5
984
Y. Li et al. / Journal of Geometry and Physics 60 (2010) 972–985
17
159
κ1σ σ σ σ σ −
κ1 κ1σ σ σ −
67
89641
5
249
κ1 κ1σ σ +
2032
5344
664
κ1σ σ + κ12 κ1σ + κ1σ κ2 + κ1 κ2σ 8 35 5 70 35 105 105 12012 2 598 13368 99 27 248 − 10κ1 κ3 − κ1 − κ2σ σ σ − κ2 + κ3σ σ − κ1σ σ σ σ σ σ + κ1σ κ2σ −
105
5
80
− −
35 72
175 35356 105 46896
κ κ
2 1 1σ
+
κ1 κ1σ −
κ12 κ1σ σ +
14
70
35 23271
700 1792 2
15 7988
25 106382
7 6032
243
κ1 κ2 +
κ1 κ2σ σ − 10κ1σ κ3 +
1728
7
κ
5 334
κ1σ κ2σ +
25 50563
525 487
2 1σ σ
35
+
28
κ1 κ3σ −
κ2σ σ σ σ −
87302
35 25
175 53691 350
κ1 κ2σ σ −
525 576
κ1 κ12σ −
10188
κ1 κ1σ + 40κ2 κ1σ σ − 2κ1 κ3σ 177 2 97 − κ1 κ1σ σ σ σ + κ1σ σ σ − κ2σ − κ1σ σ + κ3σ σ σ − κ2σ σ σ σ ∂σ−1 , 35 70 5 35 14 35 143 8 83 296 11357 2253 1929 2 66 D3 = − ∂σ − κ1 ∂σ6 − κ1σ − ∂σ5 − κ1σ σ − κ1 + κ2 ∂σ4 4200 1400 175 700 350 350 175 343 4201 5933 37 233 κ1σ σ σ − κ2σ + κ1 − κ1 κ1σ − κ3 ∂σ3 − 20 20 14 140 140 8107 2 2981 412 18608 37724 46637 26637 41 + κ1σ + κ2σ σ − κ1σ σ σ σ − κ1σ − κ1 κ2 + κ1 κ1σ σ − − κ3σ 140 84 35 35 525 700 5 140 28292 3 8212 51372 2 102922 494 152713 14843 − κ1 ∂σ2 + κ2σ σ σ + κ2 + κ1 − κ1σ σ − κ1 κ2σ + κ1 κ3 525 420 25 175 350 525 35 23113 39101 134674 51598 2 967 81 + κ1σ κ1σ σ + κ1 κ1σ σ σ − κ1σ κ2 − κ1 κ1σ + κ3σ σ − κ1σ σ σ σ σ ∂σ −
κ1σ κ1σ σ σ +
864
5
28 73
κ1σ σ σ +
1238
5 6272
20
140 844
κ3σ σ σ +
κ1σ σ κ2 +
35 147587
1056
κ1σ κ3
κ1σ κ1σ σ σ
2100 38376
κ12 κ1σ σ + κ1 κ1σ σ σ σ + 43152κ1 + κ14 + κ22 − κ3 − κ2σ 525 5 15 5 25 2100 1152 3 1271 19968 133 26104 2 256 κ1σ σ σ σ σ σ + κ1 κ1σ − κ1 κ1σ σ σ + κ2σ σ σ σ σ − κ1σ σ σ σ σ σ σ − κ2σ σ − −
− − + −
60 6393 350 1112 15 6272
5
κ1σ σ σ σ +
38
κ1σ σ σ κ2 + κ2 κ2σ +
15 34144 525
κ
3 1σ
−
κ1σ κ3σ +
7 24
7 1792
15 228 5
525
896
κ1 κ3σ σ −
15 179436
κ1 κ1σ κ2 −
κ3σ
κ κ
2 1 2σ
75
−
66904 525
κ1 κ1σ σ +
175 198336
κ1σ κ2σ σ −
2533
525
κ1σ κ1σ σ σ σ +
175 109824
κ12σ + 5 120496 −1 − κ1 κ1σ κ1σ σ ∂σ , 175
4200
23504
25
κ1 κ2σ σ σ − 2216 525
704 5
κ1σ σ κ2σ
κ1 κ1σ σ σ σ σ +
2588 105
κ1σ σ κ1σ σ σ
6
κ1σ + κ3σ σ σ σ 5
525
where Ω = ∂σ − 4κ1 − is the recursion operator of the KdV equation. For a given (E , F , G, H , L), the motion of the curves in four-dimensional projective geometry is determined by Eqs. (5.5) and (5.6). 2
2κ1σ ∂σ−1
6. Concluding remarks In this paper, we have carried out a study of inextensible motions of curves in two-, three- and four-dimensional projective geometries. In fact, the Kaup–Kupershmidt hierarchy, a coupled Kaup–Kupershmidt–KdV system and their extension can be obtained from such motions. So we can set up a correspondence between the Kaup–Kupershmidt hierarchy and the inextensible motions of curves in projective geometry. Similarly, we can discuss motions of inextensible space curves in Pn (n ≥ 5) and motions of surfaces in Pn (n ≥ 2); we believe that many 1 + 1- and 2 + 1-dimensional integrable systems are associated with such motions. Acknowledgements This work was supported by the China NSF for Distinguished Young Scholars (Grant No. 10925104) and the Program of Xianyang Normal University (08XSYK334) of Shaan Xi Province. References [1] R.E. Goldstein, D.M. Petrich, The Korteweg–de Vries hierarchy as dynamics of closed curves in the plane, Phys. Rev. Lett. 67 (1991) 3203–3206. [2] K. Nakayama, H. Segur, M. Wadati, Integrability and the motion of curves, Phys. Rev. Lett. 69 (1992) 2603–2606.
Y. Li et al. / Journal of Geometry and Physics 60 (2010) 972–985 [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41]
[42] [43]
985
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