On The Characterizations of Timelike Curves in 𝑹𝟐𝟒 M. Aykut AKGÜN
Adıyaman University, Technical Sciences Vocational High School, Adıyaman, Turkey muslumakgun@adiyaman.edu.tr, ORCID: 0000-0002-8414-5228
Abstract
In this paper by establishing the Frenet frame for a timelike curve we study the different position vectors of timelike curves in Semi-Euclidean space . We gave the position vectors of timelike curves in terms of curvature functions which lie on the three dimensional subspaces of .
Keywords: Timelike curve, Frenet frame, Semi-Euclidean space
𝑹𝟐𝟒deki Timelike Eğrilerin Karakterizasyonları Üzerine
Özet
Bu çalışmada semi-Öklidyen uzayda bir timelike eğrinin farklı yer vektörleri Frenet çatısı kullanılarak çalışılmıştır. uzayının 3-boyutlu alt uzaylarında yatan timelike eğrilerin yer vektörleri araştırılmış ve bu yer vektörleri eğrinin eğrilik fonksiyonları türünden ifade edilmiştir.
Anahtar Kelimeler: Timelike eğri, Frenet çatısı, Semi-Öklidyen uzay
1. Introduction
The classical differential geometry of curves have been studied by several authors. Adıyaman University Journal of Science
https://dergipark.org.tr/en/pub/adyujsci DOI: 10.37094/adyujsci.493589
ADYUJSCI
9 (2) (2019) 342-351
dimensional Lorentzian space with the minimum number of curvatures [1]. Çöken and Çiftçi reconstructed the Cartan frame of a null curve in Minkowski spacetime for an arbitrary parameter, and they characterized pseudo-spherical null curves and Bertrand null curves in [2]
İlarslan and Boyacıoğlu studied position vectors of a timelike and a null helice in [3]. İlarslan and Nesovic gave the necessary and sufficient conditions for null curves in to be osculating curves in terms of their curvature functions [4].
İlarslan studied spacelike curves with different normal vectors in Minkowski space [5]. İlarslan, Nesovic and Petrovic-Torgasev characterized rectifying curves in the [6].
Ali and Önder characterized rectifying spacelike curves with curvature functions in Minkowski spacetime [7].
Keleş, Perktaş and Kılıç studied Biharmonic Curves in LP-Sasakian Manifolds [8]. Akgün and Sivridağ gave some theorems for null Cartan curves in 4-dimensional Minkowski space [9]. Also, in [10], Akgün and Sivridağ studied the spacelike curves of Minkowski 4-space.
In that work, some basic knowledge about curves in is given in the second section. The original part of this paper is the third section. In the third section we gave the conditions for timelike curves to lie on subspaces of and gave theorems for such curves.
2. Preliminaries
Let denotes semi-Euclidean 4-space together with two index metric of signature . A vector is called timelike if , spacelike if
and null (lightlike) if and , respectively. The norm of a vector
is denoted by and defined by .
curve if and spacelike curve if , for
Let be a timelike curve in with the Frenet frame and let N be a spacelike vector, and be null vectors. In this case there exists a unique Frenet frame
for the timelike curve with Frenet equations [11]
(1)
where , , and are mutually orthogonal vectors satisfying
. (2)
3. On The Timelike Curves in
In this section we will give the conditions under which the timelike curves lie subspaces of .
Let be a timelike curve in with the Frenet frame . Then, the
3-dimensional subspaces of are spanned by , , and
.
Case 1. Let the timelike curve lies on the space Span . We can write (3) for differentiable functions , and of the parameter s. If we Differentiate we have
From (4) we have
(5)
If we see that . From this result if we find . So we find
(6) If , we find the equations
(7)
From (7) we find the differential equation
(8) By using exchange variable in (3.6) we have
(9) The general solution of (9) is
, (10)
where . Replacing variable in (10) we obtain
(11)
(12) So we have (13) If , then we have (14)
From (14) we find the differential equation
(15) Here and are nonzero constants. From (15) we find
(16)
If we use the equation we have
(17)
From the equation we find
Theorem 1. Let be a timelike curve in . If the curve lies on the subspace
spanned by , then it is one of the following forms
or
where or
,
where and are nonzero constant.
Case 2. Let the timelike curve lies on the space Span . In that case we can write
. (19) Differentiating (19) we have
(20)
From (20) we have the following equations:
(21)
From (21) we see that and . If we use these equations in , we obtain
(22) So we have
(23)
So we can give the following theorem.
Theorem 2. Let be a timelike curve in . If it lies on the subspace spanned by
, then it is in the form
,
where .
Case 3. If we suppose that the timelike curve lies on the space Span we can write
, (24) Differentiating (24) we have
(25)
(26)
From (26) we can write and . If we take account the
equation , we obtain
(27)
So we have
Theorem 3. Let be a timelike curvein . If it lies on the subspace spanned by
, then it is in the form
Here c is a constant.
Case 4. Let the timelike curve lies on the space Span . In that case we have
. (28) Differentiating (28) we find that
(29)
From (29) we have the following equations:
From (30) we can write and . If we take account the
equation , we obtain
So we have
(31)
So we can give the theorem.
Theorem 4. Let be a timelike curve in . If it lies on the subspace spanned by , then it is in the form
References
[1] Fernandez, A., Gimenez, A., Lucas, P., Null helices in Lorentzian space forms, International Journal of Modern Physics A, 16, 4845-4863, 2001.
[2] Çöken, A.C., Çiftçi, Ü., On the Cartan curvatures of a null curve in Minkowski
spacetime, Geometriae Dedicate, 114 71-78, 2005.
[3] İlarslan, K., Boyacıoğlu, Ö., Position vectors of a timelike and a null helix in
Minkowski 3-space, Chaos, Solitions and Fractals, 38, 1383-1389, 2008.
[4] İlarslan, K., Nesovic, E., Some characterizations of null osculating curves in the
Minkowski space-time, Proceedings of the Estonian Academy of Sciences, 61(I), 1-8,
2012.
[5] İlarslan, K., Spacelike normal curves in Minkowski , Turkish Journal of Mathematics, 29, 53-63, 2005.
rectifying curves in Minkowski 3-space, Novi Sad Journal of Mathematics, 33(2), 23-32,
2003.
[7] Ali, A.T., Önder, M., Some characterizations of rectifying spacelike curves in
the Minkowski space-time, Global Journal of Science Frontier Research Mathematics,
12(1), 2249-4626, 2012.
[8] Keleş, S., Yüksel Perktaş S., Kılıç, E., Biharmonic curves in lp-sasakian
manifolds, Bulletin of the Malasyian Mathematical Society, 33(2), 325-344, 2010.
[9] Akgün, M.A., Sivridağ, A.İ., On the null Cartan curves of , Global Journal of Mathematics, 1(1), 41-50, 2015.
[10] Akgün, M.A., Sivridağ, A.İ., On the spacelike curves of , British Journal of Mathematics and Computer Science, 15(6), 1-10, 2016.
[11] İlarslan, K., Nesovic, E., Spacelike and timelike normal curves, Publications de l’Institut Mathematique, 85(99), 111-118, 2009.
