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The tangent spherical image and ccr-curve of a time-like curve in L 3

Journal of Inequalities and Applications20132013:55

https://doi.org/10.1186/1029-242X-2013-55

  • Received: 11 December 2012
  • Accepted: 28 January 2013
  • Published:

Abstract

In this work, we define the tangent spherical image of a unit speed time-like curve lying on the pseudohyperbolic space H 0 2 ( r ) in L 3 . In addition, we calculate a ccr-curve of this curve in L 3 . Besides, we determine a relation between harmonic curvature and a ccr-curve in L 3 , and so we obtain some new results.

Keywords

  • Lorentzian Space
  • Null Curve
  • Arbitrary Curve
  • Unit Vector Field
  • Harmonic Curvature

Introduction

Let X = ( x 1 , x 2 , x 3 ) and Y = ( y 1 , y 2 , y 3 ) be two non-zero vectors in the three-dimensional Lorentz-Minkowski space R 1 3 . We denoted R 1 3 shortly by L 3 . For X , Y L 3 ,
X , Y = x 1 y 1 + x 2 y 2 + x 3 y 3
is called a Lorentzian inner product. The couple { R 1 3 , , } is called a Lorentzian space and denoted by L 3 . Then the vector X of L 3 is called
  1. (i)

    time-like if X , X < 0 ,

     
  2. (ii)

    space-like if X , X > 0 or X = 0 ,

     
  3. (iii)

    a null (or light-like) vector if X , X = 0 , X 0 .

     
The norm of a vector X is given by X = | X , X | . Therefore, X is a unit vector if X , X = ± 1 . Next, vectors X, Y in L 3 are said to be orthogonal if X , Y = 0 . The velocity of a curve α ( s ) is given by α ( s ) . Space-like or time-like α ( s ) is said to be parametrized by an arclength function s if α ( s ) , α ( s ) = ± 1 [1]. For any X = ( x 1 , x 2 , x 3 ) , Y = ( y 1 , y 2 , y 3 ) R 1 3 , the pseudo-vector product of a X and Y is defined as follows:
X Λ Y = ( ( x 2 y 3 x 3 y 2 ) , x 3 y 1 x 1 y 3 , x 1 y 2 x 2 y 1 )

[2].

1 Basic concepts

Definition 1.1 An arbitrary curve α : I L 3 in the space L 3 can locally be space-like, time-like or a null curve if, respectively, all of its velocity vectors α ( s ) are space-like, time-like or null [3].

Definition 1.2 Let α L 3 be a given time-like curve. If the Frenet vector { V 1 ( s ) , V 2 ( s ) , V 3 ( s ) } which corresponds to s I is defined as
k i : I R , k i ( s ) = V i ( s ) , V i + 1 ( s ) ,

then the function k i is called an i th curvature function of the time-like curve α, and the real k i ( s ) is also called an i th curvature at the point α ( s ) [4].

Definition 1.3 Let α : I L 3 be a unit speed non-null curve in L 3 . The curve α is called a Frenet curve of osculating order d ( d 3 ) if its 3rd order derivatives α ( s ) , α ( s ) , α ( s ) are linearly independent and α ( s ) , α ( s ) , α ( s ) , α ı v ( s ) are no longer linearly independent for all s I . For each Frenet curve of order 3, one can associate an orthonormal 3-frame { V 1 ( s ) , V 2 ( s ) , V 3 ( s ) } along α (such that α ( s ) = V 1 ) called the Frenet frame and k 1 , k 2 : I R called the Frenet curvatures, such that the Frenet formulas are defined in the usual way:

where is the Levi-Civita connection of L 3 .

Definition 1.4 A non-null curve α : I L 3 is called a W-curve (or helix) of rank3, if α is a Frenet curve of osculating order 3 and the Frenet curvatures k i , 1 i 2 , are non-zero constants.

2 Harmonic curvatures and constant curvature ratios in L 3

Definition 2.1 Let α be a non-null curve of osculating order 3. The harmonic functions
H j : I R , 0 j 1 ,
defined by
{ H 0 = 0 , H 1 = k 1 k 2

are called harmonic curvatures of α, where k 1 , k 2 are Frenet curvatures of α which are not necessarily constant.

Definition 2.2 Let α be a time-like curve in L 3 with α ( s ) = V 1 . X ϵ χ ( L 3 ) being a constant unit vector field, if
V 1 , X = cosh φ ( constant ) ,

then α is called a general helix (inclined curves) in L 3 , φ is called a slope angle and the space S p { X } is called a slope axis [5].

Definition 2.3 Let α be a non-null of osculating order 3. Then α is called a general helix of rank1 if
H 1 2 = c ,

holds, where c 0 is a real constant.

We have the following results.

Corollary 2.1
  1. (i)

    If H 1 = 0 , then α is a straight line.

     
  2. (ii)

    If H 1 is constant, then α is a general helix of rank1.

     

Proof By the use of the above definition, we obtain the proof. □

Proposition 2.1 Let α be a curve in L 3 of osculating order 3. Then

where H 1 is harmonic curvature of α.

Proof By using the Frenet formulas and the definitions of harmonic curvatures, we get the result. □

Now, we will give the relation between harmonic curvature and a ccr-curve in L 3 .

Definition 2.4 A curve α : I L 3 is said to have constant curvature ratios (that is to say, it is a ccr-curve) if all the quotients ε i ( k i + 1 k i ) are constant, where ε i = V i , V i = ± 1 .

Corollary 2.2 For i = 1 , the ccr-curve is ε 1 H 1 .

Proof The proof can be easily seen by using the definitions of harmonic curvature and ccr-curve. □

Corollary 2.3 Let α : I L 3 be a ccr-curve. If ε 1 H 1 = c , c is a constant, then ( ε 1 H 1 ) = 0 .

Proof The proof is obvious. □

3 Tangent spherical image

Definition 3.1 [1]

Let n 2 and 0 v n . Then the pseudohyperbolic space of radius r > 0 in R 1 3 is the hyperquadric
H 0 2 ( r ) = { p R 1 3 : p , p = r 2 }

with dimension 2 and index 0.

Definition 3.2 Let α = α ( s ) be a unit speed time-like curve in L 3 . If we translate the tangent vector to the center 0 of the pseudohyperbolic space H 0 2 ( r ) , we obtain a curve δ = δ ( s δ ) . This curve is called the tangent spherical image of a curve α in L 3 .

Theorem 3.1 [6]

  1. (i)

    Let α = α ( s ) be a unit speed time-like curve and δ = δ ( s δ ) be its tangent spherical image. Then δ = δ ( s δ ) is a space-like curve.

     
  2. (ii)

    Let α = α ( s ) be a unit speed time-like curve and δ = δ ( s δ ) be its tangent spherical image. If α is a ccr-curve or a helix (i.e. W-curve), then δ is also a helix.

     

Proof From [6] it is easy to see the proof of the theorem. □

4 An example

Example 4.1 Let us consider the following curve in the space L 3 :
where α ( s ) , α ( s ) = 1 , which shows α ( s ) is a unit speed time-like curve. Thus α ( s ) = 1 . We express the following differentiations:
and
V 2 ( s ) = α ( s ) α ( s ) = α ( s ) .
So, we have the first curvature as
k 1 ( s ) = V 1 ( s ) , V 2 ( s ) = 1 = constant .
Moreover, we can write the third Frenet vector of the curve as follows:
V 3 ( s ) = V 1 ( s ) Λ V 2 ( s ) = ( 1 , 2 sin s , 2 cos s ) .
Finally, we have the second curvature of α ( s ) as
k 2 ( s ) = V 2 ( s ) , V 3 ( s ) = 2 = constant .
Now, we will calculate a ccr-curve of α ( s ) in L 3 . If the vector V 1 is time-like, then ε 1 = 1 ,
ε 1 k 2 k 1 = 2 = constant .

Thus α ( s ) is a ccr-curve in L 3 .

Declarations

Acknowledgements

Dedicated to Prof. Hari M. Srivastava.

Authors’ Affiliations

(1)
Department of Mathematics, Art and Science Faculty, Uludağ University, Bursa, 16059, Turkey

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