# 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
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 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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