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

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

### Keywords

Lorentzian Space Null Curve Arbitrary Curve Unit Vector Field Harmonic Curvature

## Introduction

Let and be two non-zero vectors in the three-dimensional Lorentz-Minkowski space . We denoted shortly by . For ,
is called a Lorentzian inner product. The couple is called a Lorentzian space and denoted by . Then the vector X of is called
1. (i)

time-like if ,

2. (ii)

space-like if or ,

3. (iii)

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

The norm of a vector X is given by . Therefore, X is a unit vector if . Next, vectors X, Y in are said to be orthogonal if . The velocity of a curve is given by . Space-like or time-like is said to be parametrized by an arclength function s if [1]. For any , , the pseudo-vector product of a X and Y is defined as follows:

[2].

## 1 Basic concepts

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

Definition 1.2 Let be a given time-like curve. If the Frenet vector which corresponds to is defined as

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

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

where ∇ is the Levi-Civita connection of .

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

## 2 Harmonic curvatures and constant curvature ratios in

Definition 2.1 Let α be a non-null curve of osculating order 3. The harmonic functions
defined by

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

Definition 2.2 Let α be a time-like curve in with . being a constant unit vector field, if

then α is called a general helix (inclined curves) in , φ is called a slope angle and the space 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 is a real constant.

We have the following results.

Corollary 2.1
1. (i)

If , then α is a straight line.

2. (ii)

If 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 of osculating order 3. Then

where 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 .

Definition 2.4 A curve is said to have constant curvature ratios (that is to say, it is a ccr-curve) if all the quotients are constant, where .

Corollary 2.2 For , the ccr-curve is .

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

Corollary 2.3 Let be a ccr-curve. If , c is a constant, then .

Proof The proof is obvious. □

## 3 Tangent spherical image

Definition 3.1 [1]

Let and . Then the pseudohyperbolic space of radius in is the hyperquadric

with dimension 2 and index 0.

Definition 3.2 Let be a unit speed time-like curve in . If we translate the tangent vector to the center 0 of the pseudohyperbolic space , we obtain a curve . This curve is called the tangent spherical image of a curve α in .

Theorem 3.1 [6]

1. (i)

Let be a unit speed time-like curve and be its tangent spherical image. Then is a space-like curve.

2. (ii)

Let be a unit speed time-like curve and 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 :
where , which shows is a unit speed time-like curve. Thus . We express the following differentiations:
and
So, we have the first curvature as
Moreover, we can write the third Frenet vector of the curve as follows:
Finally, we have the second curvature of as
Now, we will calculate a ccr-curve of in . If the vector is time-like, then ,

Thus is a ccr-curve in .

## Notes

### Acknowledgements

Dedicated to Prof. Hari M. Srivastava.

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## Authors and Affiliations

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