 Research
 Open Access
 Published:
The tangent spherical image and ccrcurve of a timelike curve in {\mathbb{L}}^{3}
Journal of Inequalities and Applications volume 2013, Article number: 55 (2013)
Abstract
In this work, we define the tangent spherical image of a unit speed timelike curve lying on the pseudohyperbolic space {H}_{0}^{2}(r) in {\mathbb{L}}^{3}. In addition, we calculate a ccrcurve of this curve in {\mathbb{L}}^{3}. Besides, we determine a relation between harmonic curvature and a ccrcurve 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 nonzero vectors in the threedimensional LorentzMinkowski space {\mathbb{R}}_{1}^{3}. We denoted {\mathbb{R}}_{1}^{3} shortly by {\mathbb{L}}^{3}. For X,Y\in {\mathbb{L}}^{3},
is called a 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)
timelike if \u3008X,X\u3009<0,

(ii)
spacelike if \u3008X,X\u3009>0 or X=0,

(iii)
a null (or lightlike) vector if \u3008X,X\u3009=0, \phantom{\rule{0.25em}{0ex}}X\ne 0.
The norm of a vector 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. Spacelike or timelike \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 pseudovector 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 spacelike, timelike or a null curve if, respectively, all of its velocity vectors {\alpha}^{\mathrm{\prime}}(s) are spacelike, timelike or null [3].
Definition 1.2 Let \alpha \subset {\mathbb{L}}^{3} be a given timelike 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 timelike 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 nonnull 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 3frame \{{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 LeviCivita connection of {\mathbb{L}}^{3}.
Definition 1.4 A nonnull curve \alpha :I\u27f6{\mathbb{L}}^{3} is called a Wcurve (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 nonzero constants.
2 Harmonic curvatures and constant curvature ratios in {\mathbb{L}}^{3}
Definition 2.1 Let α be a nonnull curve of osculating order 3. The harmonic functions
defined by
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 timelike 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 nonnull 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 ccrcurve 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 ccrcurve) 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 ccrcurve is \frac{{\epsilon}_{1}}{{H}_{1}}.
Proof The proof can be easily seen by using the definitions of harmonic curvature and ccrcurve. □
Corollary 2.3 Let \alpha :I\u27f6{\mathbb{L}}^{3} be a ccrcurve. 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\u2a7e2 and 0\le v\le n. Then the pseudohyperbolic space of radius r>0 in {\mathbb{R}}_{1}^{3} is the hyperquadric
with dimension 2 and index 0.
Definition 3.2 Let \alpha =\alpha (s) be a unit speed timelike 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 timelike curve and \delta =\delta ({s}_{\delta}) be its tangent spherical image. Then \delta =\delta ({s}_{\delta}) is a spacelike curve.

(ii)
Let \alpha =\alpha (s) be a unit speed timelike curve and \delta =\delta ({s}_{\delta}) be its tangent spherical image. If α is a ccrcurve or a helix (i.e. Wcurve), 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 \u3008{\alpha}^{\mathrm{\prime}}(s),{\alpha}^{\mathrm{\prime}}(s)\u3009=1, which shows \alpha (s) is a unit speed timelike curve. Thus \parallel {\alpha}^{\mathrm{\prime}}(s)\parallel =1. 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 \alpha (s) as
Now, we will calculate a ccrcurve of \alpha (s) in {\mathbb{L}}^{3}. If the vector {V}_{1} is timelike, then {\epsilon}_{1}=1,
Thus \alpha (s) is a ccrcurve in {\mathbb{L}}^{3}.
References
O’Neill B: SemiRiemannian Geometry with Applications to Relativity. Academic Press, New York; 1983.
Öğrenmiş AO, Balgetir H, Ergüt M:On the ruled surfaces in Minkowski 3space {\mathbb{R}}_{1}^{3}. J. Zhejiang Univ. Sci. A 2006, 7(3):326–329. 10.1631/jzus.2006.A0326
PetrovicTorgasev M, Sucurovic E: Wcurves in Minkowski spacetime. Novi Sad J. Math. 2002, 32(2):55–65.
Bektaş M, Ergüt M, Soylu D: The characterization of the spherical timelike curves in 3dimensional Lorentzian space. Bull. Malays. Math. Soc. 1998, 21: 117–125.
Ekmekçi N, Hacisalihoglu HH, İlarslan K: Harmonic curvatures in Lorentzian space. Bull. Malays. Math. Soc. 2000, 23: 173–179.
Yılmaz S, Özyılmaz E, Yaylı Y, Turgut M:Tangent and trinormal spherical images of a timelike curve on the pseudohyperbolic space {H}_{0}^{3}. Proc. Est. Acad. Sci. 2010, 59(3):216–224. 10.3176/proc.2010.3.04
Acknowledgements
Dedicated to Prof. Hari M. Srivastava.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The author declares that they have no competing interests.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
İyigün, E. The tangent spherical image and ccrcurve of a timelike curve in {\mathbb{L}}^{3}. J Inequal Appl 2013, 55 (2013). https://doi.org/10.1186/1029242X201355
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029242X201355
Keywords
 Lorentzian Space
 Null Curve
 Arbitrary Curve
 Unit Vector Field
 Harmonic Curvature