The Geometric Invariants of A Null Cartan Curve

The Geometric Invariants of A Null Cartan Curve Under Similarity Transformations Hakan Simsek

Mustafa Özdemir May 6, 2015

Abstract In this paper, we study the di¤erential geometry of null Cartan curves under the similarity transformations in the Minkowski space-time. Besides, we extend the fundamental theorem for a null Cartan curve according to a similarity mapping. We …nd the equations of all self-similar null curves which is given its shape Cartan curvatures. Keywords : Similarity Transformation, Pseudo-de Sitter parameter, Self-similar null curves, Cartan curves. MSC 2010 : 14H50, 14H81, 53A35, 53A55, 53B30.

1

Introduction

A similarity transformation (or similitude) of Euclidean space, which consists of a rotation, a translation and an isotropic scaling, is an automorphism preserving the angles and ratios between lengths. The structure, which forms geometric properties unchanged by similarity transformations, is called the similarity geometry. The whole Euclidean geometry can be considered as a glass of similarity geometry. The similarity transformations are being studied in most area of the pure and applied mathematics. Curve matching is an important research area in the computer vision and pattern recognition, which can help us determine what category the given curve belongs to. S. Li [22, 23] presented a system for matching and pose estimation of 3D space curves under the similarity transformation. Brook et al. [2] discussed various problems of image processing and analysis by using the similarity transformation. Sahbi [9] investigated a method for shape description based on kernel principal component analysis (KPCA) in the similarity invariance of KPCA. The idea of self-similarity is one of the most basic and fruitful ideas in mathematics. A selfsimilar object is exactly similar to a part of itself, which in turn remains similar to a smaller part of itself, and so on. In the last few decades, the self-similarity notion led to the areas such as fractal geometry, dynamical systems, computer networks and statistical physics. Mandelbrot presented the …rst description of self-similar sets, namely sets that may be expressed as unions of rescaled copies of themselves. He called these sets fractals, which are systems that present such self-similar behavior and the examples in nature are many. The Cantor set, the von Koch snow‡ake curve and the Sierpinski gasket are some of the most famous examples of such sets (see [7, 11, 13, 18]). Bonnor [24] introduced the Cartan frame to study the behaviors of a null curve and proved the fundamental existence and congruence theorems in Minkowski space-time. Bejancu [1] represented a method for the general study of the geometry of null curves in Lorentz manifolds and, more generally, 1

in semi-Riemannian manifolds (see also the book [14]). Ferrandez, Gimenez and Lucas [4] gave a reference along a null curve in an n-dimensional Lorentzian space. They showed the fundamental existence and uniqueness theorems and described the null helices in higher dimensions. Cöken and Ciftci [3] studied null curves in the Minkowski space-time and characterized pseudo-spherical null curves and Bertrand null curves. The study of the geometry of null curves has a growing importance in the mathematical physics. The null curves are useful to …nd the solution of some equations in the classical relativistic string theory (see [15, 16, 19]) Moreover, there exists a geometric particle model associated with the geometry of null curves in the Minkowski space-time (see [5, 6]). Berger [17] represented the broad content of similarity transformations of Euclidean space. Encheva and Georgiev [20, 21] studied the di¤erential geometric invariants of curves according to a similarity in the Euclidean n-space. Chou and Qu [12] showed that the motions of curves in two, three and n-dimensional (n > 3) similarity geometries correspond to the Burgers hierarchy, Burgers-mKdV hierarchy and a multi-component generalization of these hierarchies. Kamishima [26] examined the properties of compact Lorentzian similarity manifolds using developing maps and holonomy representations. The main idea of this paper is to introduce the di¤erential geometry of a null curve under the similarity mapping and determine the self-similar null curves in the Minkowski space-time. The scope of paper is as follows. First, we give basic informations about null Cartan curves. Then, we introduce a new parameter, which is called pseudo-de Sitter parameter that is invariant under the similarity transformation. We represent the di¤erential geometric invariants of a null Cartan curve, which are called shape Cartan curvatures, according to the group of similarity transformations in the Minkowski space-time. We prove the uniqueness theorem which states that two null Cartan curves having same the shape Cartan curvatures are equivalent according to a similarity mapping. Furthermore, we show the existence theorem that is a process for constructing a null Cartan curve by the shape Cartan curvatures under some initial conditions. Lastly, we obtain the equations of all self-similar null Cartan curves, whose shape Cartan curvatures are real constant.

2

Preliminaries

Let u = (u1 ; u2 ; u3 ; u4 ) ; v = (v1 ; v2 ; v3 ; v4 ) be two arbitrary vectors in Minkowski space-time M4 . The Lorentzian inner product of u and v can be stated as u v = uI vT where I = diag( 1; 1; 1; 1): We say that a vector u in M4 is called spacelike, null (lightlike) or timelike p if u u > 0; u u = 0 or u u < 0; respectively. The norm of the vector u is represented by kuk = ju uj. We can describe the pseudo-spheres in M4 as follows: The hyperbolic 3-space is de…ned by H 3 ( 1) = u 2 M4 : u u =

1

and de Sitter 3-space is de…ned by S13 = u 2 M4 : u u = 1 ; [8, 25]. A basis B = fL; N; W1 ; W2 g is said to pseudo-orthonormal if it satis…es the following conditions: L L = N N = 0;

L N=1

L Wi = N Wi = W1 W2 = 0 Wi Wi = 1 where i 2 f1; 2g ([14]) : 2

Now, we consider the mapping A : L; N; W1 ; W2 ! (L; N; W1 ; W2 ) of one pseudo-orthonormal basis onto another at any point P in M4 , de…ned by 2 3 2 3 32 L 0 0 0 L 1 1 6N7 6 76 N 7 "2 + 2 2 6 7=6 7 76 (1) 4W1 5 4 " cos + sin 0 cos sin 5 4W1 5 W2 " sin cos 0 sin cos W2

where ; "; and are real constant and 6= 0: The image of pseudo-orthonormal basis under the mapping A is a pseudo-orthonormal basis. Moreover, we have AT J A = J and det A = 1 where 3 2 0 1 0 0 61 0 0 07 7 J =6 40 0 1 05 0 0 0 1

and the orientation is preserved by (1). Bonnor [24] de…ned the mapping A as a null rotation. A null rotation at P is equivalent to a Lorentzian transformation between two sets of natural coordinate functions whose values coincide at P . A curve locally parameterized by : J R ! M4 is called a null curve if 0 (t) 6= 0 is a null vector for all t. We know that a null curve (t) satis…es 00 (t) 00 (t) 0 (see [14]). If 00 (t) 00 (t) = 1; then it is said that a null curve (t) in M4 is parameterized by pseudo-arc. If we assume that the acceleration vector of the null curve is not null, the pseudo-arc parametrization becomes as the following Z t 1=4 00 s= (u) 00 (u) du ([3, 24]). (2) t0

A null curve (t) in M4 with 00 (t) 00 (t) 6= 0 is a Cartan curve if F := 0 (t); 00 (t); (3) (t); (4) (t) is linearly independent for any t. ; There exists a unique Cartan frame C := fL; N; W1 ; W2 g of the Cartan curve has the same orientation with F according to pseudo arc-parameter t; such that the following equations are satis…ed; 0

= L;

0

L = W1 ; N0 = W1 + W2 W10 W20

=

L

=

L

(3)

N

where N is a null vector, which is called null transversal vector …eld, and C is pseudo-orthonormal and positively oriented. The functions and are called the Cartan curvatures of (t) and their values are given as (t) = (t) =

1 (3) (t) 2q

(4) (t)

(3)

(t)

(4) (t)

(4) (3) (t)

(3) (t) 2

for the pseudo-arc parameter t: It can be seen the materials [1], [4], [14] and [24] for more information about the geometry of null curves. 3

3

Geometric Invariants of Null Curves According to a Similarity

Now, we de…ne pseudo-similarity transformation for null curves in M4 : A pseudo-similarity (psimilarity) f : M4 ! M4 of Minkowski space-time is determined by f (x) = A (x) + C;

(5)

where > 0 is a real constant, A is a null rotation and C is a translation vector. The p-similarity transformations are a group under the composition of maps and denoted by Simp M4 . The psimilarity transformations in M4 preserve the orientation. Let (t) : J R ! M4 be a null curve in M4 . We denote image of under f 2 Simp M4 by . Then, the null curve can be stated as (t) = A ( (t)) + b; The pseudo-arc length function

t 2 J:

(6)

starting at t0 2 J is

s (t) =

Zt

00

00

(u)

(u)

1=4

du =

p

s (t)

(7)

t0

where s 2 I R is pseudo-arc parameter of : I ! M4 : From now on, we will denote by a prime ”0 ” p p s and s the di¤erentiation with respect to s. We can compute the Cartan curvatures of by using (4) as 1 1 ; = : (8) = We de…ne W2 indicatrix W2 of the null curve parameterized by W2 (s) = W2 (s). The W2 indicatrix is a pseudo-spherical null curve lies on the de Sitter 3-space S13 . The pseudo-arc p parameter of W2 can be given as d = ds by using the equation (2) : The parameter is called pseudo-de Sitter parameter of : It is invariant under the p-similarity transformation because it can be easily found d = d ; where is the pseudo-de Sitter parameter of : Therefore, we can reparametrize a null curve with the pseudo-de Sitter parameter in order to study di¤erential geometry of a null curve under the p-similarity transformation. The derivative formulas of and C with respect to are given by d d

d d2 = 2 d 2 d

1 = p L;

d d

+

1

W1

(9)

and dL d dN d dW1 d dW2 d

1 = p W1 =p

W1 +

=

p

L

=

p

L:

4

p

W2

1 p N

(10)

Similarly, we can …nd the same formulas (9) and (10) for the null curve . Now, we construct a new frame corresponding to p-similarity transformation for a null curve. Let’s denote the functions d and ~ := ; ~ := 2 d respectively. The functions ~ and ~ are invariant under the p-similarity because of ~ = ~ and ~ = ~ . We let Lsim =

p

1 Nsim = p N

L;

W1sim = W1 ;

W2sim = W2

such that it is satis…ed Lsim Nsim = 1 and C sim := Lsim ; Nsim ; W1sim ; W2sim orthonormal frame of . Then, the derivative formulas for C sim are d

C sim

d where

T

= P C sim

2

0 6 0 P =6 4 ~ 1

T

(11)

3 0 17 7: 05 0

0 1 0 ~ 1 0 0 0

Also, we consider the pseudo-orthogonal frame C H := fH1 ; H2 ; H3 ; H4 g for H1 =

1

Lsim ; H2 =

1

Nsim ; H3 =

1

is a pseudo-

W1sim and H4 =

1

where

W2sim :

Since, from (5) ; we can obtain f (Hi ) = Hi , i = 1; 4, the pseudo-orthogonal frame C H is invariant according to p-similarity map. Then, using (9) and (11) ; we get the derivative formulas of C H as the following d T T C H = P~ C H (12) d where 3 2 2~ 0 1 0 6 0 2~ ~ 1 7 7: P~ = 6 4 ~ 1 2~ 0 5 1 0 0 2~ We can think of the equation (12) as the Frenet equation of a null curve according to the pseudo-orthogonal moving frame C H under the group Simp M4 . As a result, the following lemma is obtained. Lemma 1 Let : I ! M4 be a null Cartan curve with pseudo-de Sitter parameter be Cartan curvatures of with the Cartan frame C : Then, the functions ~ =

d 2 d

;

~ =

and f

;

g

(13)

and the pseudo-orthogonal frame C H are invariant under the p-similarity transformation in the Minkowski space-time and the derivative formulas of C H with respect to are given by the equation (12) : 5

d

and the pseudo-orthonormal frame C H are called De…nition 2 The functions ~ = 2 d ; ~ = shape Cartan curvatures and shape Cartan frame of a null Cartan curve ; respectively. Remark 3 We consider the W1 indicatrix W1 of null Cartan curve parameterized by W1 (s) = W1 (s) ; where s is a pseudo-arc parameter of : The curve W1 is a pseudo-spherical spacelike curve if > 0 or pseudo-spherical timelike curve if < 0 on S13 : If u is an arc-length parameter of W1 ; p then we can …nd du = j2 jds: The arc length element du is invariant according to the p-similarity transformation. Remark 4 Instead of W1sim = W1 ; we take W1sim = new frame are d d

C sim

T

2

0 6 0 =6 4 ~

0 0 1 0

1

dLsim in C sim : The derivative formulas for a d 1 ~ 0 0

3 0 17 7 C sim 05 0

T

where ~ = ~2 + ~ 2 : It may be considered an alternative frame for a null Cartan curve under the p-similarity transformation. However, the problem is that the new frame is not the pseudo-orthogonal due to W1sim ; Nsim 6= 0 although W1sim is a unit spacelike vector.

4

The Fundamental Theorem for a Null Curve

The existence and uniqueness theorems were shown by [1], [4] and [24] for a null Cartan curve under the Lorentz transformations. This notion can be extended with respect to Simp M4 for the null Cartan curves parameterized by the pseudo-de Sitter parameter. Theorem 5 Let ; : I ! M4 be two null Cartan curves parameterized by the same pseudo-de Sitter parameter , where I R is an open interval. Suppose that and have the same shape Cartan curvatures ~ = ~ and ~ = ~ for any 2 I: Then, there exists a f 2Simp M4 such that = f : Proof. Let ; and ; be the Cartan curvatures and also s and s be the pseudo-arc length parameters of and ; respectively. Using the equality ~ = ~ ; we get = for some real constant > 0: Then, the equality ~ = ~ imply = : On the other hand, we can write ds = p1 ds from the de…nition of pseudo-de Sitter parameter : Let’s consider the map : M4 ! M4 de…ned by (x) = 1 ' (x) where ' is a null rotation. Using the equation (8) ; the null Cartan curves = ( ) and have the same Cartan curvatures. Then, there exists a Lorentz transformation : M4 ! M4 for the null Cartan curves (see [1, 4]) such 1 ' : M4 ! M4 such that that ( ) = : Therefore, we have a p-similarity transformation f = f( )= : The following theorem shows that every two functions determine a null Cartan curve according to a p-similarity under some initial conditions. Theorem 6 Let zi : I ! R; i = 1; 2 be two functions and L0sim ; N0sim ; W10sim ; W20sim be a pseudoorthonormal frame at a point x0 in the Minkowski space-time. According to a p-similarity, there exists a unique non-null Cartan curve : I ! M4 parameterized by a pseudo-de Sitter parameter such that satis…es the following conditions:

6

(i) There exists 0 2 I such that ( 0 ) = x0 and the shape Cartan frame of W10sim ; W20sim : (ii) ~ ( ) = z1 ( ) and ~ ( ) = z2 ( ) ; for any 2 I:

at x0 is L0sim ; N0sim ;

Proof. Let us consider the following system of di¤erential equations with respect to a matrix-valued T function K ( ) = Lsim ; Nsim ; W1sim ; W2sim dK ( ) = M( )K( ) d with a given matrix

2

0 6 0 M( ) = 6 4 z1 1

0 1 0 z1 1 0 0 0

(14)

3 0 17 7: 05 0

The system (14) has a unique solution K ( ) which satis…es the initial conditions K( for

0

0)

= L0sim ; N0sim ; W10sim ; W20sim

T

2 I: Then, we can write d d

d T d K J K + J KT J K d d = J KT MT J + J M K = 0

J KT J K = J

since we have the equation MT J + J M = 0 4 4 : Also, we have J KT ( 0 ) J K ( 0 ) = I where I is the unit matrix since L0sim ; N0sim ; W10sim ; W20sim is the pseudo-orthonormal 4-frame. As a result, we …nd J XT ( ) J X ( ) = I for any 2 I: It means that the vector …elds Lsim ; Nsim ; W1sim ; W2sim form pseudo-orthonormal frame …eld in the Minkowski space-time. Let : I ! M4 be the null curve given by Z R ( ) = x0 + e2 z2 ( )d Lsim ( ) d ; 2 I: (15) 0

By the equality (14), we get that ( ) is a null Cartan curve in Minkowski Rspace-time with shape Cartan curvatures ~ ( ) = z1 ( ) and ~ ( ) = z2 ( ) : Also, we …nd d = e z2 ( )d ds by using (2) and (14) ; where s is a pseudo-arc parameter; thus, is the pseudo-de Sitter parameter of the null Cartan curve . Besides, the pseudo-orthonormal 4-frame Lsim ; Nsim ; W1sim ; W2sim is a Cartan frame of the null Cartan curve under the p-similarity transformation. Remark 7 In case of ~ ( ) = 0; the Cartan curvature = c is a non-zero real constant. Then, the parametrization of a null curve : I ! M4 with ~ ( ) = 0 with respect to pseudo-de Sitter parameter is given by Z 1 ( ) = x0 + Lsim ( ) d ; 2I (16) c 0 from the equation (9) and (15) :

7

Example 8 We let ~ = 0 and ~ = with the initial conditions L0sim = W10sim =

1

for the shape Cartan curvatures of a null curve

1 1 1 1 p ; 0; p ; 0 ; N0sim = p ; 0; p ; 0 ; 2 2 2 2 1 1 1 1 0; p ; 0; p ; W20sim = 0; p ; 0; p : 2 2 2 2

: I ! M4 (17)

Then, the system (14) determines a null vector Lsim given by 1 Lsim ( ) = p (cosh ; sinh ; cos ; sin ) 2

(18)

with Lsim (0) = L0sim ; in M4 . Solving the equation (15) ; we obtain the null Cartan curve terized by 1 ( ) = p ( 2 + 2 sinh 2 cosh ; 2 + 2 cosh 2 2 2 2 sin + 2 cos ; 2 cos + 2 sin ) for any

5

parame-

2 sinh ;

2 I:

Self-similar Null Cartan Curves

A null Cartan curve : I ! M4 is called self-similar if any p-similarity f 2 G conserve globally and G acts transitively on where G is a one-parameter subgroup of Simp M4 : This means that shape Cartan curvatures ~ and ~ are constant. In fact, let p = (s1 ) and q = (s2 ) be two di¤erent points lying on for any s1 ; s2 2 I: Since G acts transitively on ; there is a p-similarity f 2 G such that f (p) = q. Then, we …nd ~ (s1 ) = ~ (s2 ) and ~ (s1 ) = ~ (s2 ) ; which it implies the invariability of shape Cartan curvatures. Now, we determine the parametrizations of all self-similar null curves by means of the constant shape Cartan curvatures in the Minkowski space-time. It can be separated into the four di¤erent cases as follows. We can take the initial conditions (17) in the example 8 for all cases. Case 1: Let’s take ~ 1 = 0 and ~ 1 = 0: Then, using the equation (14) ; we …nd the null vector Lsim given by (18) : From the equation (16) ; we obtain the self similar null curve parameterized by 1(

Case 2: Let’s take ~

1

1 ) = p (sinh ; cosh ; sin ; c 2

cos ) :

= 0 and ~ 1 = b 6= 0: Then, using the equation (14) ; it is obtained e2b Lsim ( ) = p (cosh ; sinh ; cos ; sin ) : 2 2

From the equation (15), we get the self-similar null curve given by 2(

1 cosh (w1 ) + sinh (w1 ) cosh (w2 ) + sinh (w2 ) )= p ( + ; w1 w2 2 2 cosh (w1 ) + sinh (w1 ) cosh (w2 ) + sinh (w2 ) ; w1 w2 4be2b cos + 2e2b sin 2e2b cos + 4be2b sin ; ) 4b2 + 1 4b2 + 1 8

where w1 = 2b + 1 and w2 = 2b 1: Case 3: Let’s take ~ 3 = a 6= 0 and ~ 3 = 0. Then, using the equation (14) and (16) ; we get the self-similar null curve given by 1 )= p c 2

sinh (q1 ) cosh (q1 ) sin (q2 ) cos (q2 ) ; ; ; q1 q1 q2 q2 p p p p where q1 = a + a2 + 1 and q2 = a + a2 + 1: Case 4: Let’s take ~ 4 = a 6= 0 and ~ 4 = b 6= 0: Using the equation (14) and (15) ; we obtain the self-similar null curve given by 3(

4(

1 cosh (m1 ) + sinh (m1 ) cosh (m2 ) + sinh (m2 ) )= p ( + ; m1 m2 2 2 cosh (m1 ) + sinh (m1 ) cosh (m2 ) + sinh (m2 ) ; m1 m2 4be2b cos (q2 ) + 2q2 e2b sin (q2 ) 2q2 e2b cos (q2 ) + 4be2b sin (q2 ) ; ) 4b2 + q22 4b2 + q22

where m1 = 2b + q1 and m2 = 2b q1 : A null curve is called a null helix if it has the constant Cartan curvatures which are not both zero in M4 : The equations of null helices satisfying 6= 0 are expressed by r 1 1 1 1 1 (s) = sinh vs; cosh vs; sin rs; cos rs (19) v2 + r2 v v r r pp pp 2+ 2 2 + 2 + ([24]). In case of and r = = 0; the equation (19) reduces where v = to p p p p 1 sinh s ; cosh s ; sin s ; cos s : 0 (s) = p 2 The Cartan curvatures of the self-similar null curves 1 and 3 are given by 1 = 0; 1 = c 6= 0 and 3 = ac; 3 = c 6= 0, respectively. Then, 1 and 3 are null helices which correspond to the null curves 0 and , respectively. Hence, we say that null helices satisfying 6= 0 are a class of self-similar null curves in M4 . Also, we can characterize the null helices by means of the shape Cartan curvatures in M4 .

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[25] W. Greub, Linear Algebra. 3rd ed., Springer Verlag, Heidelberg, 1967. [26] Y. Kamishima, Lorentzian similarity manifolds, Cent. Eur. J. Math., 10(5), 1771-1788, 2012.

Hakan Simsek and Mustafa Özdemir Department of Mathematics Akdeniz University Antalya, TURKEY; e-mail: [email protected]. [email protected]

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