# The tangent spherical image and ccr-curve of a time-like curve in ${\mathbb{L}}^{3}$

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## 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 ${\mathbb{L}}^{3}$. In addition, we calculate a ccr-curve of this curve in ${\mathbb{L}}^{3}$. Besides, we determine a relation between harmonic curvature and a ccr-curve in ${\mathbb{L}}^{3}$, and so we obtain some new results.

### Keywords

Lorentzian Space Null Curve Arbitrary Curve Unit Vector Field Harmonic Curvature## Introduction

*Lorentzian inner product*. The couple $\{{\mathbb{R}}_{1}^{3},\u3008\phantom{\rule{0.25em}{0ex}},\phantom{\rule{0.25em}{0ex}}\u3009\}$ is called a

*Lorentzian space*and denoted by ${\mathbb{L}}^{3}$. Then the vector

*X*of ${\mathbb{L}}^{3}$ is called

- (i)
time-like if $\u3008X,X\u3009<0$,

- (ii)
space-like if $\u3008X,X\u3009>0$ or $X=0$,

- (iii)
a null (or light-like) vector if $\u3008X,X\u3009=0$, $\phantom{\rule{0.25em}{0ex}}X\ne 0$.

*X*is given by $\parallel X\parallel =\sqrt{|\u3008X,X\u3009|}$. Therefore,

*X*is a unit vector if $\u3008X,X\u3009=\pm 1$. Next, vectors

*X*,

*Y*in ${\mathbb{L}}^{3}$ are said to be orthogonal if $\u3008X,Y\u3009=0$. The velocity of a curve $\alpha (s)$ is given by $\parallel {\alpha}^{\mathrm{\prime}}(s)\parallel $. Space-like or time-like $\alpha (s)$ is said to be parametrized by an arclength function

*s*if $\u3008{\alpha}^{\mathrm{\prime}}(s),{\alpha}^{\mathrm{\prime}}(s)\u3009=\pm 1$ [1]. For any $X=({x}_{1},{x}_{2},{x}_{3})$, $Y=({y}_{1},{y}_{2},{y}_{3})\in {\mathbb{R}}_{1}^{3}$, the pseudo-vector product of a

*X*and

*Y*is defined as follows:

[2].

## 1 Basic concepts

**Definition 1.1** An arbitrary curve $\alpha :I\u27f6{\mathbb{L}}^{3}$ in the space ${\mathbb{L}}^{3}$ can locally be space-like, time-like or a null curve if, respectively, all of its velocity vectors ${\alpha}^{\mathrm{\prime}}(s)$ are space-like, time-like or null [3].

**Definition 1.2**Let $\alpha \subset {\mathbb{L}}^{3}$ be a given time-like curve. If the Frenet vector $\{{V}_{1}(s),{V}_{2}(s),{V}_{3}(s)\}$ which corresponds to $s\in I$ is defined as

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 $\alpha (s)$ [4].

**Definition 1.3**Let $\alpha :I\u27f6{\mathbb{L}}^{3}$ be a unit speed non-null curve in ${\mathbb{L}}^{3}$. The curve

*α*is called a Frenet curve of osculating order

*d*($d\le 3$) if its 3rd order derivatives ${\alpha}^{\mathrm{\prime}}(s)$, ${\alpha}^{\mathrm{\prime}\mathrm{\prime}}(s)$, ${\alpha}^{\mathrm{\prime}\mathrm{\prime}\mathrm{\prime}}(s)$ are linearly independent and ${\alpha}^{\mathrm{\prime}}(s)$, ${\alpha}^{\mathrm{\prime}\mathrm{\prime}}(s)$, ${\alpha}^{\mathrm{\prime}\mathrm{\prime}\mathrm{\prime}}(s)$, ${\alpha}^{\u0131v}(s)$ are no longer linearly independent for all $s\in I$. 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 ${\alpha}^{\mathrm{\prime}}(s)={V}_{1}$) called the Frenet frame and ${k}_{1},{k}_{2}:I\u27f6\mathbb{R}$ called the Frenet curvatures, such that the Frenet formulas are defined in the usual way:

where ∇ is the Levi-Civita connection of ${\mathbb{L}}^{3}$.

**Definition 1.4** A non-null curve $\alpha :I\u27f6{\mathbb{L}}^{3}$ is called a *W*-curve (or helix) of rank3, if *α* is a Frenet curve of osculating order 3 and the Frenet curvatures ${k}_{i}$, $1\le i\le 2$, are non-zero constants.

## 2 Harmonic curvatures and constant curvature ratios in ${\mathbb{L}}^{3}$

**Definition 2.1**Let

*α*be a non-null curve of osculating order 3. The harmonic functions

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 ${\mathbb{L}}^{3}$ with ${\alpha}^{\mathrm{\prime}}(s)={V}_{1}$. $X\u03f5\chi ({\mathbb{L}}^{3})$ being a constant unit vector field, if

then *α* is called a general helix (inclined curves) in ${\mathbb{L}}^{3}$, *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

holds, where $c\ne 0$ is a real constant.

We have the following results.

**Corollary 2.1**

- (i)
*If*${H}_{1}=0$,*then**α**is a straight line*. - (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*${\mathbb{L}}^{3}$

*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 ${\mathbb{L}}^{3}$.

**Definition 2.4** A curve $\alpha :I\u27f6{\mathbb{L}}^{3}$ is said to have constant curvature ratios (that is to say, it is a ccr-curve) if all the quotients ${\epsilon}_{i}(\frac{{k}_{i+1}}{{k}_{i}})$ are constant, where ${\epsilon}_{i}=\u3008{V}_{i},{V}_{i}\u3009=\pm 1$.

**Corollary 2.2** *For* $i=1$, *the ccr*-*curve is* $\frac{{\epsilon}_{1}}{{H}_{1}}$.

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

**Corollary 2.3** *Let* $\alpha :I\u27f6{\mathbb{L}}^{3}$ *be a ccr*-*curve*. *If* $\frac{{\epsilon}_{1}}{{H}_{1}}=c$, *c* *is a constant*, *then* ${(\frac{{\epsilon}_{1}}{{H}_{1}})}^{\mathrm{\prime}}=0$.

*Proof* The proof is obvious. □

## 3 Tangent spherical image

**Definition 3.1** [1]

with dimension 2 and index 0.

**Definition 3.2** Let $\alpha =\alpha (s)$ be a unit speed time-like curve in ${\mathbb{L}}^{3}$. If we translate the tangent vector to the center 0 of the pseudohyperbolic space ${H}_{0}^{2}(r)$, we obtain a curve $\delta =\delta ({s}_{\delta})$. This curve is called the tangent spherical image of a curve *α* in ${\mathbb{L}}^{3}$.

**Theorem 3.1** [6]

- (i)
*Let*$\alpha =\alpha (s)$*be a unit speed time*-*like curve and*$\delta =\delta ({s}_{\delta})$*be its tangent spherical image*.*Then*$\delta =\delta ({s}_{\delta})$*is a space*-*like curve*. - (ii)
*Let*$\alpha =\alpha (s)$*be a unit speed time*-*like curve and*$\delta =\delta ({s}_{\delta})$*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 ${\mathbb{L}}^{3}$:

Thus $\alpha (s)$ is a ccr-curve in ${\mathbb{L}}^{3}$.

## Notes

### Acknowledgements

Dedicated to Prof. Hari M. Srivastava.

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