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

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Research
Part of the following topical collections:
  1. Proceedings of the International Congress in Honour of Professor Hari M. Srivastava

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 ) Open image in new window in L 3 Open image in new window. In addition, we calculate a ccr-curve of this curve in L 3 Open image in new window. Besides, we determine a relation between harmonic curvature and a ccr-curve in L 3 Open image in new window, and so we obtain some new results.

Keywords

Lorentzian Space Null Curve Arbitrary Curve Unit Vector Field Harmonic Curvature 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Introduction

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

    time-like if X , X < 0 Open image in new window,

     
  2. (ii)

    space-like if X , X > 0 Open image in new window or X = 0 Open image in new window,

     
  3. (iii)

    a null (or light-like) vector if X , X = 0 Open image in new window, X 0 Open image in new window.

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

[2].

1 Basic concepts

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

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

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

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

where ∇ is the Levi-Civita connection of L 3 Open image in new window.

Definition 1.4 A non-null curve α : I L 3 Open image in new window 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 Open image in new window, 1 i 2 Open image in new window, are non-zero constants.

2 Harmonic curvatures and constant curvature ratios in L 3 Open image in new window

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

are called harmonic curvatures of α, where k 1 Open image in new window, k 2 Open image in new window are Frenet curvatures of α which are not necessarily constant.

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

then α is called a general helix (inclined curves) in L 3 Open image in new window, φ is called a slope angle and the space S p { X } Open image in new window 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

holds, where c 0 Open image in new window is a real constant.

We have the following results.

Corollary 2.1
  1. (i)

    If H 1 = 0 Open image in new window, then α is a straight line.

     
  2. (ii)

    If H 1 Open image in new windowis 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 Open image in new window of osculating order 3. Then

where H 1 Open image in new window 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 Open image in new window.

Definition 2.4 A curve α : I L 3 Open image in new window 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 ) Open image in new window are constant, where ε i = V i , V i = ± 1 Open image in new window.

Corollary 2.2 For i = 1 Open image in new window, the ccr-curve is ε 1 H 1 Open image in new window.

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 Open image in new window be a ccr-curve. If ε 1 H 1 = c Open image in new window, c is a constant, then ( ε 1 H 1 ) = 0 Open image in new window.

Proof The proof is obvious. □

3 Tangent spherical image

Definition 3.1 [1]

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

with dimension 2 and index 0.

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

Theorem 3.1 [6]

  1. (i)

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

     
  2. (ii)

    Let α = α ( s ) Open image in new window be a unit speed time-like curve and δ = δ ( s δ ) Open image in new window 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 Open image in new window:
where α ( s ) , α ( s ) = 1 Open image in new window, which shows α ( s ) Open image in new window is a unit speed time-like curve. Thus α ( s ) = 1 Open image in new window. We express the following differentiations:
and
V 2 ( s ) = α ( s ) α ( s ) = α ( s ) . Open image in new window
So, we have the first curvature as
k 1 ( s ) = V 1 ( s ) , V 2 ( s ) = 1 = constant . Open image in new window
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 ) . Open image in new window
Finally, we have the second curvature of α ( s ) Open image in new window as
k 2 ( s ) = V 2 ( s ) , V 3 ( s ) = 2 = constant . Open image in new window
Now, we will calculate a ccr-curve of α ( s ) Open image in new window in L 3 Open image in new window. If the vector V 1 Open image in new window is time-like, then ε 1 = 1 Open image in new window,
ε 1 k 2 k 1 = 2 = constant . Open image in new window

Thus α ( s ) Open image in new window is a ccr-curve in L 3 Open image in new window.

Notes

Acknowledgements

Dedicated to Prof. Hari M. Srivastava.

References

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Copyright information

© İyigün; licensee Springer 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Department of Mathematics, Art and Science FacultyUludağ UniversityBursaTurkey

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