on Indicatrices of Null Cartan Curves in R⁴₁

on Indicatrices of Null Cartan Curves in R⁴₁

1

1

Zafer anl 1, Yusuf Yayl 2

1Department of Mathematics,Faculty of Arts and Science, Mehmet Akif Ersoy University, Burdur,Turkey

2Department of Mathematics, Faculty of Science, Ankara University, Ankara, Turkey

Abstract

In this paper, we investigate indicatrices of null Cartan curves in Minkowski 4-space which lie on lightcone and pseudosphere, and give some characterizations for these curves to be a generalized helix in terms of Cartan curvatures.

1. Introduction

   There are three types curve in a (M,g) Semi- Riemannian manifolds according to its velocity vectors: spacelike, timelike and (null) lightlike. As non-null curves (spacelike and timelike) have some similarities with Riemannian case, but some diffuculties arise for null curves.

   1

   1

   Contrary to Riemannian case, general Frenet frame and its general Frenet equations are not unique as they depend on the parameter of the curve and the screen vector bundle. To deal this non-uniqueness problem, Bonnor [1] was the first who introduced a unique Frenet frame along null curves in R with the minimum number of curvature functions. Bonnor's work was recently generalized by Ferrandez-Gimenez- Lucas [3] for null Cartan curves in Mm+2 and they studied some classification theorem on null helices.

   Recently physical importance of null Cartan curves has shown by Ferrandez-Gimenez-Lucas [4]. After that Çöken-Çiftçi [2] have followed the 3- dimensional notion of Bertrand curves and proved a characterization theorem for null helices in R. Also Sakaki [7] has shown a correspondence between the evolute of a null curve and the involute of a certain spacelike curve in the 4-dimensional Minkowski spacetime.

   In this paper, we study relationships between a null Cartan curve and its indicatrices and give some characterizations.

2. Null Cartan Curves

   The Minkowski space-time R is the Euclidean 4-space equipped with the the indefinite flat metric given by <,>=-dx²+dx²+dx²+dx², where (x,x,x,x) is rectangular coordinate system of R. Recall that an arbitrary vector vR\{0} can be spacelike, timelike or null(lightlike), if <v,v>>0,

   <v,v><0 or <v,v>=0 and v0 respectively. In particular, the vector v=0 is spacelike. The norm of a vector v is given by v=J| < v, v > |, and two

   vectors v and w are said to be orthogonal, if <v,w>=0.

   An arbitrary curve (t) in R, can be locally spacelike, timelike or null(lightlike), if all its velocity vectors (t) are spacelike, timelike or null, respectively. A spacelike or timelike curve (s) is said to be parametrized by a pseudo-arclength parameter s, i.e. <(s),(s)>=±1.

   Let (s) be a spacelike curve in R, parametrized by arclenght function s. Similar to the Euclidean case, a spacelike helix can be defined according to the tangent indicatrix: A spacelike curve in R is said to be a general helix if its unit tangent indicatrix is contained in a hyperplane.

   Recall that the pseudo-sphere, and the lightcone are hyperquadratics in R, respectively defined by

   S³(m,r)={xR:<x-m,x-m>=r²}

   and

   ³(m)={xR:<x-m,x-m>=0}

   where r>0 is the radius and mR is the center of hyperquadratics. In particular, for r=1 and m=0 we denote this hyperquadratics S³ and ³ respectively.

   Now let (t) be a null curve in R. Since

   <(t),(t)>=0, classical methods for non-null curves does not work in this case. So we need a new construction for this curve. We say that a null curve (t) in R is parametrized by pseudo-arc if <(t),(t)>=1. If a null curve (t) in R satisfies <(t),(t)>0, then

   <(t),(t)>>0 and

   t

   t

   u(t)=ft < y(t), y(t) > dt

   o

   becomes the pseudo-arc parameter.

   Let us say that a null curve (t) in R with

   <(t),(t)>0 is a Cartan curve if

   {(t),(t),(t),(t)} is linearly indipendent for any t.

   where a and b are non-zero constants and

   {L,N,W,W} is the Cartan frame of .

   For a null Cartan curve (t) in R with pseudo- arc parameter t, there exist a unique Frenet Frame

   {L,N,W,W} such that

   = L, L=W, N=W+W 2.1

   W = -L-N, W=-L

   where

3. Conic Indicatrices of Null Cartan Curves

   Let be a null Cartan curve in R. From (2.1) and (2.2), its indicatrix curves (L) and (N) are spacelike curves which lie on null cone. Then the unit tangential vector fields of these curves are

   <L,L>=<N,N>=0 2.2

   <W,W>=<W,W>=1 2.3

   and

   T(L)=W 3.1

   T(N) = 1 (W+W). 3.2

   <L,W >=<N, W >=0 , <L,N>=1,i=1,2 2.4

   .jK²+T²

   respectively. Therefore we can give following results:

   and {L,N,W,W} and {,,,} have the same orientation and {L,N,W,W} is positively oriented. Also the functions and are called the Cartan curvatures of .

   Definition 2.1 A null Cartan curve :IR is said to be a generalized helix if there exists a constant vector

   Proposition 3.1 Let be a null Cartan curve with non- zero constant curvature . Then its conic indicatrix (L) is a spacelike generalized helix if and only if Cartan curvatures of satisfy following differential equation

   2

   1 = 1 (4² A²)

   v0 such that the product =<L(t),v>0 is constant. Theorem 2.2 Let be a null Cartan curve. Then is a generalized helix if and only if its Cartan curvatures

   T

   where A=(/µ)-1.

   K³

   3.3

   satisfy the following differential equation ()²=²(2+c) ,0 2.5

   where c is a constant.

   Proof Suppose that (L) is a spacelike generalized helix in R, i.e. there exists a constant vector u such that the product <T(L),u> is a non-zero constant. Then we have from (3.1)

   The axis of generalized helix , in above theorem,

   is

   v=-(L+N-KrW) 2.6

   T

   Let

   <W,u>=µ. 3.4

   u=fL+gN+µW+hW

   where =c is a non-zero constant [See 4].

   Teorem 2.3 Let be a Cartan curve in R. Then is a pseudo-spherical curve if and only if is a non-zero constant.

   Teorem 2.4 Let be a Cartan curve in R. Then is a

   where µ is a non-zero constant and f,g and h are differentiable functions of the parameter t. Then

   du = (f ht)L + (gr )N

   three-dimensional null helix if and only if there exists a dt r

   fixed direction u such that

   <L,u>=a and <N,u>=b

   +(f + g)W1 + (h + g)W2 . 3.5

   As u is a constant vector we get

   f=-2µ²((1/))((1/)),g=2µ((1/))((1/)),h=-2µ((1/)).

   Thus the axis of the (L) is

   T(W )= -1 (L+N)

   u=-2µ²((1/))((1/))L+2µ((1/))((1/))N+µW- 2µ((1/))W. 3.6

   On the other side, as <u,u>=±1= we obtain that equation (3.3).

   and

   respectively.

   1 2K

   T(W2)=-L

   Conversely, suppose the curvatures of satisfy the equation (3.3) and consider vector field along defined by (3.6) where µ=/|(/(A + 1))|.Then u is a constant vector and <T(L),u>=µ.

   Proposition 3.2 Let be a generalized helix in R.

   Then its conic indicatrix (N) is a generalized spacelike helix which has same axis of the if and only if Cartan curvatures of satisfy the following differential equations

   ()²=A(²+²) where A is ((µ²)/(²)).

   Proof Let assume that (N) be a generalized spacelike helix and (26) is its axis. As

   <T(N),v>=µ0,

   Thus we can give following results.

   Proposition 4.1 Let be a Cartan curve in R with is non-zero constant. Then is a three-dimensional null helix if and only if its pseudo-spherical indicatrix (W) is a generalized helix.

   Proof Since is a three-dimensional null helix, there exists a fixed direction u such that <L,u>=a and

   <N,u>=b, where a and b are non zero constants. Then direct computation shows that

   (W )

   (W )

   <T ,u>= 1 (<L,u>+<N,u>)

   1 2K

   is non-zero constant.

   Conversely, assume that (W) is a generalized helix,

   i.e. there exits a constant vector u such that

   <T(W1),u>=µ0.

   we obtain that

   µ=<((-1)/((²+²)))(W+W),-(L+N- (()/)W)>

   =((-)/((²+²))).

   Then we have

   ()²=A(²+²) 3.7

   Then we get

   If we set

   1 (<L,u>+<N,u>)=µ.

   2K

   <L,u>=f 4.1

   where A is ((µ²)/(²)).

   Conversely, using the equation (3.7)

   where f is a differentiable function of t, we have

   <T(N)

   then we obtain that

   ,v>²=((²()²)/(²+²)),

   <T(N),v>²=²A

   <N,u>=µ2-f. 4.2

   Then taking derivatives of (4.1) and (4.2) with respect to t, we have

   <W,u>=f 4.3

   is non-zero constant.

4. Pseudospherical Indicatrices of null Cartan Curves

   Let be a null Cartan curve in R. Again from (2.1) and (2.3), its indicatrix curves (W) and (W) is spacelike curve with >0 and lightlike curve respectively, which lie on pseudo-sphere. Then the unit tangential vector fields of these curves are

   and

   Now let asume that,

   <W,u>=0. 4.4

   <W,u>=0. 4.5

   If we take derivative of (4.5) and account that (4.1), we have

   -f=0

   so f=0.Then from (4.2) the product <N,u> is a non-zero constant. But there is no constant vector such that

   <N,u>=cons.[5]. Thus =0. Then is a 3-dimensional helix.

   Proposition 4.2 Let be a generalized helix in R. Then its pseudo-spherical indicatrix (W) is a generalized helix which has same axis of the if and only if is pseudo-spherical curve.

   Proof Let (W) be a generalized helix and consider its axis in above. Since

   <T(W2),v>=µ0,

   we obtain that

   µ=<-L,-(L+N-(()/)W)>=.

   Thus is a non-zero constant. This means that from Theorem 2.3 is a pseudo-spherical curve.

   Conversely, as is a non-zero constant the product

   <T(W2),v>=

   is also non-zero constant.

5. References

[1]. Bonnor, W.B., Null Curves in Minkowski Space- time, Tensor N.S. 20 1969, 229-242.

[2]. Cöken, A.C., and Ciftci, Ü., On the Cartan Curvatures of a Null Curve in Minkowski Space-time, Geometriae Dedicata, 114, 2005, 71-78.

[3]. Ferrandez, A., Gimenez, A. and Lucas, P., Null Helices in Lorentzian Space Forms, Int., J. Mod. Phys. A16, 2001, 4845-4863.

[4]. Ferrandez, A., Gimenez, A. and Lucas, P., Null Generalized Helices and the Betchov-Da Rios Equation in Lorentz-Minkowski Spaces, Proceedings of the XI Fall Workshop on Geometry and Physics, Madrid, 2004.

[5]. Karada, H.B., Karada, M., Null Generalized Slant Halices in Lorentzian Space, Differ. Geom. Dyn. Syst., 10,2008, 178-185.

[6]. Önder, M., Kocayiit, Kazaz, M., Spacelike Helices in Minkowski 4-Space E, Ann. Univ. Ferrara, 56(2), 2010, 335-343.

[7]. Sakaki, M., Notes on Null Curves in Minkowski Spaces, Turk., J. Math, 34, 2010, 417-424..