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The tangent spherical image and ccr-curve of a time-like curve in
Journal of Inequalities and Applications volume 2013, Article number: 55 (2013)
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.
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
time-like if ,
space-like if or ,
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 . For any , , the pseudo-vector product of a X and Y is defined as follows:
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 .
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 .
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
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 .
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.
If , then α is a straight line.
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 
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 
Let be a unit speed time-like curve and be its tangent spherical image. Then is a space-like curve.
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  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:
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 .
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Dedicated to Prof. Hari M. Srivastava.
The author declares that they have no competing interests.
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Cite this article
İyigün, E. The tangent spherical image and ccr-curve of a time-like curve in . J Inequal Appl 2013, 55 (2013). https://doi.org/10.1186/1029-242X-2013-55
- Lorentzian Space
- Null Curve
- Arbitrary Curve
- Unit Vector Field
- Harmonic Curvature