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

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.

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, X0.

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:

XΛY= ( ( 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 sI is defined as

k i :IR, 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 (d3) 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 sI. 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 :IR 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 , 1i2, 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 :IR,0j1,

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 . Xϵχ( L 3 ) being a constant unit vector field, if

V 1 ,X=coshφ(constant),

then α is called a general helix (inclined curves) in L 3 , φ is called a slope angle and the space Sp{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 c0 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 n2 and 0vn. 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 sins, 2 coss).

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 .

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Acknowledgements

Dedicated to Prof. Hari M. Srivastava.

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Correspondence to Esen İyigün.

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İyigün, E. The tangent spherical image and ccr-curve of a time-like curve in L 3 . J Inequal Appl 2013, 55 (2013). https://doi.org/10.1186/1029-242X-2013-55

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Keywords

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