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

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

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 ${\mathbb{R}}_1^3$. We denoted ${\mathbb{R}}_1^3$ shortly by ${\mathbb{L}}^3$. For $X,Y\in {\mathbb{L}}^3$,

$$〈X,Y〉=-x_1{y}_1+x_2{y}_2+x_3{y}_3$$

is called a Lorentzian inner product. The couple $\{{\mathbb{R}}_1^3,〈,〉\}$ 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 $〈X,X〉<0$,

• (ii) space-like if $〈X,X〉>0$ or $X=0$,

• (iii) a null (or light-like) vector if $〈X,X〉=0$, $X\ne 0$.

The norm of a vector X is given by $\parallel X\parallel =\sqrt{|〈X,X〉|}$. Therefore, X is a unit vector if $〈X,X〉=\pm 1$. Next, vectors X, Y in ${\mathbb{L}}^3$ are said to be orthogonal if $〈X,Y〉=0$. 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 $〈{\alpha }^{\mathrm{\prime }}(s),{\alpha }^{\mathrm{\prime }}(s)〉=\pm 1$ [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:

$$X\mathrm{\Lambda }Y=(-(x_2{y}_3-x_3{y}_2),x_3{y}_1-x_1{y}_3,x_1{y}_2-x_2{y}_1)$$

[2].

1 Basic concepts

Definition 1.1 An arbitrary curve $\alpha :I⟶{\mathbb{L}}^3$ 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

$$k_i:I⟶\mathbb{R},k_i(s)=〈V_i^{\mathrm{\prime }}(s),V_{i+1}(s)〉,$$

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 $\alpha (s)$ [4].

Definition 1.3 Let $\alpha :I⟶{\mathbb{L}}^3$ 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 ${\alpha }^{\mathrm{\prime }}(s)$, ${\alpha }^{\mathrm{\prime }\mathrm{\prime }}(s)$, ${\alpha }^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}(s)$, ${\alpha }^{ıv}(s)$ 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 $k_1,k_2:I⟶\mathbb{R}$ 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 $\alpha :I⟶{\mathbb{L}}^3$ 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

$$H_j:I⟶\mathbb{R},0\le j\le 1,$$

defined by

$$\{\begin{array}{c}{H}_0=0,\hfill \\ H_1=\frac{k_1}{k_2}\hfill \end{array}$$

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 ${\alpha }^{\mathrm{\prime }}(s)=V_1$. $Xϵ\chi ({\mathbb{L}}^3)$ being a constant unit vector field, if

$$〈V_1,X〉=cosh\phi (\text{constant}),$$

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

$$H_1^2=c,$$

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 $\alpha :I⟶{\mathbb{L}}^3$ 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 ${\epsilon }_i=〈V_i,V_i〉=\pm 1$.

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 $\alpha :I⟶{\mathbb{L}}^3$ 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]

Let $n⩾2$ and $0\le v\le n$. Then the pseudohyperbolic space of radius $r>0$ in ${\mathbb{R}}_1^3$ is the hyperquadric

$$H_0^2(r)=\{p\in {\mathbb{R}}_1^3:〈p,p〉=-r^2\}$$

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 ${\mathbb{L}}^3$:

where $〈{\alpha }^{\mathrm{\prime }}(s),{\alpha }^{\mathrm{\prime }}(s)〉=-1$, which shows $\alpha (s)$ is a unit speed time-like curve. Thus $\parallel {\alpha }^{\mathrm{\prime }}(s)\parallel =1$. We express the following differentiations:

and

$$V_2(s)=\frac{{\alpha }^{\mathrm{\prime }\mathrm{\prime }}(s)}{\parallel {\alpha }^{\mathrm{\prime }\mathrm{\prime }}(s)\parallel }={\alpha }^{\mathrm{\prime }\mathrm{\prime }}(s).$$

So, we have the first curvature as

$$k_1(s)=〈V_1^{\mathrm{\prime }}(s),V_2(s)〉=1=\text{constant}.$$

Moreover, we can write the third Frenet vector of the curve as follows:

$$V_3(s)=V_1(s)\mathrm{\Lambda }{V}_2(s)=(-1,\sqrt{2}sins,-\sqrt{2}coss).$$

Finally, we have the second curvature of $\alpha (s)$ as

$$k_2(s)=〈V_2^{\mathrm{\prime }}(s),V_3(s)〉=\sqrt{2}=\text{constant}.$$

Now, we will calculate a ccr-curve of $\alpha (s)$ in ${\mathbb{L}}^3$. If the vector $V_1$ is time-like, then ${\epsilon }_1=-1$,

$${\epsilon }_1\frac{k_2}{k_1}=-\sqrt{2}=\text{constant}.$$

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