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

[2].

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].

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.

are called harmonic curvatures of α, where $k_1$, $k_2$ are Frenet curvatures of α which are not necessarily 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].

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

We have the following results.

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. □

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]

Proof From [6] it is easy to see the proof of the theorem. □

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:

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