Mannheim B-curve couples in Minkowski 3-space

Article Sidebar

PDF
Published: Oct 1, 2020
DOI: https://doi.org/10.5556/j.tkjm.51.2020.3059
Keywords:
Bishop frame Mannheim curve null Cartan curve pseudo-null curve Minkowski space

Main Article Content

Kazim ILARSLAN
KIRIKKALE UNIVERSITY
Ali UCUM
KIRIKKALE UNIVERSITY
Emilija NESOVIC
KRAGUJEVAC UNIVERSITY
Nihal KILIC ASLAN
KIRIKKALE UNIVERSITY

Abstract

In this paper, we define null Cartan and pseudo null Mannheim curves in Minkowski 3-space according to their Bishop frames. We obtain the necessary and the sufficient conditions for pseudo null curves to be Mannheim B-curves in terms of their Bishop curvatures. We prove that there are no null Cartan curves in Minkowski 3-space which are Mannheim B-curves, by considering the cases when their Mannheim B-mate curves are spacelike, timelike, null Cartan and pseudo null curves. Finally, we give some examples of pseudo null Mannheim B-curve pairs.

Article Details

How to Cite
ILARSLAN, K., UCUM, A., NESOVIC, E., & KILIC ASLAN, N. (2020). Mannheim B-curve couples in Minkowski 3-space. Tamkang Journal of Mathematics, 51(3), 219–232. https://doi.org/10.5556/j.tkjm.51.2020.3059
Issue
Vol. 51 No. 3 (2020)
Section
Papers
Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

References

L.R. Bishop, There is more than one way to frame a curve, Amer. Math. Monthly 82(3)(1975), 246-251.

W.B. Bonnor, Null curves in a Minkowski space-time, Tensor 20, 229-242 (1969).

J.H. Choi, T.H. Kang, Y.H. Kim, Mannheim curves in 3-dimensional space forms, Bull.Korean Math. Soc. 50(4) (2013), 1099-1108.

K. L. Duggal, D. H. Jin, Null Curves and Hypersurfaces of Semi Riemann Manifolds, World Scientic, Singapore, 2007.

L.P. Eisenhart, A Treatise on the Differential Geometry of Curves and Surfaces. Dover Edition.Dover Publication, New York (1960)

M. Erdogdu, Parallel frame of non-lightlike curves in Minkowski space-time, Int. J. Geom. Methods Mod. Phys. 12 (2015) 16 pages.

S. Ersoy, M. Tosun, H. Matsuda, Generalized Mannheim curves in Minkowski space-time E^{4}_{1} ,Hokkaido Math. J. 41(3) (2012), 441-461.

F. Gokcelik, Z. Bozkurt, I. Gok, F. N. Ekmekci, Y. Yayl, Parallel transport frame in 4-dimensional Euclidean space E^{4}, Caspian J. Math. Sci. 3 (2014) 91-103.

M. Grbovic, K. Ilarslan, E. Nesovic, On null and pseudo null Mannheim curves in Minkowski 3-space, J. Geom. 105 (2014), 177-183.

M. Grbovic, K. Ilarslan, E. Nesovic, On generalized null Mannhem curves in Minkowski space-time, Publ. Inst. Math. (Belgrade) Vol. 99(113) (2016), 77-98.

M. Grbovic, E. Nesovic, On the Bishop frames of pseudo null and null Cartan curves in Minkowski 3-space, J. Math. Anal. Appl. 461 (2018) 219-233.

K. Ilarslan, E. Nesovic, On Bishop frame of a null Cartan curve in Minkowski space-time, Int. J. Geom. Meth. Mod. Phys. Vol. 15, No. 8 (2018) 1850142 (16 pages).

H. Liu, F. Wang, Mannheim partner curves in 3-space, J. Geom. 88, (2008) 120-126.

H. Matsuda, S. Yorozu, On generalized Mannheim curves in Euclidean 4-space, Nihonkai Math. J. 20, (2009), 33-56.

B. O'Neill, Semi Riemannian geometry with applications to relativity, Academic Press, New York, 1983.

M. Ozdemir, A. A. Ergin, Parallel frame of non-lightlike curves, Missouri J. Math. Sci. 20(2) (2008) 127-137.

J. Warave, Curves and surfaces in Minkowski space, Ph. D. Thesis, Leuven University, 1995.

S. Yilmaz, M. Turgut, A new version of Bishop frame and an application to spherical images, J. Math. Anal. Appl. 371 (2010), 764-776.

Most read articles by the same author(s)