Characterization of proper curves and proper helix lying on $S_{1}^2(r)$

In this paper, we analyse the proper curve $\gamma(s)$ lying on the pseudo-sphere. We develop an orthogonal frame $\lbrace V_{1}, V_{2}, V_{3} \rbrace$ along the proper curve, lying on pseudosphere. Next, we find the condition for $\gamma(s)$ to become $V_{k} -$ slant helix in Minkowski space. Moreover, we find another curve $\beta(\bar{s})$ lying on pseudosphere or hyperbolic plane heaving $V_{2} = \bar{V_{2}}$ for which $\lbrace \bar{V_{1}},\bar{V_{2}},\bar{V_{3}} \rbrace$, an orthogonal frame along $\beta(\bar{s})$. Finally, we find the condition for curve $\gamma(s)$ to lie in a plane.

Keywords:

Minkowski space, pseudo sphere, proper helix of order 2 proper curve of order 2, $V_{k}$-slant helix,

PDF

___

[1] N. Abe, Y. Nakanishi and S. Yamaguchi, Circles and spheres in pseudo-Riemannian geometry, Aequationes Mathematicae, 39, 134-145, 1990.
[2] T. Adachi, Kähler magnetic flows for a manifold of constant holomorphic sectional curvature, Tokyo J. Math. 18(2), 473-483, 1995.
[3] A.T. Ali, Position vectors of slant helices in Euclidean 3-space, J. Egypt. Math. Soc. 20, 1-6, 2012.
[4] A.T. Ali and R. López, On slant helices in Minkowski space $E_{1}^3$, J. Korean Math. Soc. 48, 159-167, 2011.
[5] A.T. Ali and M. Turgut, Position vector of a time-like slant helix in Minkowski 3- space, J. Math. Anal. Appl. 365, 559-569, 2010.
[6] M. Barros, General helices and a theorem of Lancret, Proc. Amer. Math. Soc. 125, 1503-1509, 1997.
[7] B. Bukcu, M.K. Karacan and N. Yuksel, New characterizations for Bertrand curves in Minkowski 3 - space, International J. Math. Combin. 2, 98-103, 2011.
[8] B.Y. Chen, When does the position vector of a space curve always lie in its rectifying plane?, Amer. Math. Monthly, 110, 147-152, 2003.
[9] İ. Gök, Ç. Camci and H. Hilmi Hacisalihoğlu, $V_{n}$ - slant helices in Euclidean n - space En, Math. Commun. 14, 317-329, 2009.
[10] S. Izumiya and N. Takeuchi, New special curves and developable surfaces, Turk. J. Math. 28, 153-163, 2004.
[11] S. Kiziltuğ and Y. Yayli, Timelike curves on Timelike parallel surfaces in Minkowski 3-space $E_{1}^3$ , Mathematica Aeterna 2, 689-700, 2012.
[12] S. Maeda and B. Hak Kim, A certain two-parameter family of helices of order 6 in Euclidean sphere, Differ. Geom. Appl. 35, 117-124, 2014.
[13] K. Sakamoto, Helical immersions into a unit sphere, Math. Ann. 261, 63-80, 1982.
[14] H.H. Song, T. Kimura and N. Koike, On proper helices and extrinsic spheres in pseudo-Riemannian geometry, Tsukuba J. Math. 20, 263-280, 1996.
[15] C. Zhang and D. Pei, Generalized bertrand curves in minkowski 3-space, Mathematics MDPI, 8(12), 2199, 2020.
[16] E. Zıplar, Y. Yaylı and İ. Gök, A New Approach on Helices in Pseudo-Riemannian Manifold, Abstr. Appl. Anal. 2014, ID 718726, 2014.

Hacettepe Journal of Mathematics and Statistics-Cover

Yayın Aralığı: Yılda 6 Sayı
Başlangıç: 2002
Yayıncı: Hacettepe Üniversitesi Fen Fakultesi

Arşiv