Null (Lightlike) $f$-Rectifying Curves in the Three Dimensional Minkowski Space $\mathbb{E}^3_1$

Year 2020, Volume: 3 Issue: 1, 8 - 16, 10.06.2020

Zafar Iqbal Joydeep Sengupta

https://doi.org/10.33401/fujma.708816

Cited By: 3

Abstract

A rectifying curve $\gamma$ in the Euclidean $3$-space $\mathbb{E}^3$ is defined as a space curve whose position vector always lies in its rectifying plane (i.e., the plane spanned by the unit tangent vector field $T_\gamma$ and the unit binormal vector field $B_\gamma$ of the curve $\gamma$), and an $f$-rectifying curve $\gamma$ in the Euclidean $3$-space $\mathbb{E}^3$ is defined as a space curve whose $f$-position vector $\gamma_f$, defined by $\gamma_f(s) = \int f(s) d\gamma$, always lies in its rectifying plane, where $f$ is a nowhere vanishing real-valued integrable function in arc-length parameter $s$ of the curve $\gamma$. In this paper, we introduce the notion of $f$-rectifying curves which are null (lightlike) in the Minkowski $3$-space $\mathbb{E}^3_1$. Our main aim is to characterize and classify such null (lightlike) $f$-rectifying curves having spacelike or timelike rectifying plane in the Minkowski $3$-Space $\mathbb{E}^3_1$.

Keywords

Curvature, Minkowski 3-space, Null (lightlike) curve, Rectifying curve, Serret-Frenet formulae, Torsion

References

• [1] A. Pressley, Elementary Differential Geometry, 2nd ed., Springer, 2010.
• [2] M. P. do Carmo, Differential Geometry of Curves and Surfaces: Revised and Updated Second Edition, Courier Dover Publications, 2016.
• [3] M. Spivak, A Comprehensive Introduction to Differential Geometry, Vol. 2, 3rd ed., Publish or Perish, Houston, Texas, 1999.
• [4] R. S. Millman, G. D. Parker, Elements of Differential Geometry, Prentice-Hall, Inc., New Jersey, 1977.
• [5] B. Y. Chen, When does the position vector of a space curve always lie in its rectifying plane?, Amer. Math. Monthly, 110 (2003), 147–152.
• [6] B. Y. Chen, Rectifying curves and geodesics on a cone in the Euclidean 3-space, Tamkang J. Math., 48 (2017), 209-214.
• [7] B. Y. Chen, F. Dillen, Rectifying curves as centrodes and extremal curves, Bull. Inst. Math. Acad. Sinica, 33 (2005), 77-90.
• [8] S. Deshmukh, B. Y. Chen, S. Alshamari, On rectifying curves in Euclidean 3-space, Turk. J. Math., 42 (2018), 609-620.
• [9] K. Ilarslan, E. Nésovic, Timelike and null normal curves in Minkowski space E31, Indian J. Pure Appl. Math., 35(7) (2004), 881-888.
• [10] K. Ilarslan, E. Nésovic, On rectifying curves as centrodes and extremal curves in the Minkowski 3-Space, Novi Sad J. Math., 37 (2007), 53-64.
• [11] K. Ilarslan, E. Nésovic, T. M. Petrovic, Some characterization of rectifying curves in the Minkowski 3-Space, Novi Sad J. Math., 33 (2003), 23-32.
• [12] R. Lopez, Differential Geometry of curves and surfaces in Lorentz-Minkowski space, Int. Electron. J. Geom., 7(1) (2014), 44–107.
• [13] B. O’Neill, Semi–Riemannian Geometry with Applications to Relativity, Academic Press, New York, 1983.