A Study on $f$-Rectifying Curves in Euclidean $n$-Space

Yıl 2021, , 107 - 113, 30.09.2021

Zafar Iqbal Joydeep Sengupta

https://doi.org/10.32323/ujma.937479

Cited By: 1

Öz

A rectifying curve in the Euclidean $n$-space $\mathbb{E}^n$ is defined as an arc-length parametrized curve $\gamma$ in $\mathbb{E}^n$ such that its position vector always lies in its rectifying space (i.e., the orthogonal complement of its principal normal vector field) in $\mathbb{E}^n$. In this paper, in analogy to this, we introduce the notion of an $f$-rectifying curve in $\mathbb{E}^n$ as a curve $\gamma$ in $\mathbb{E}^n$ parametrized by its arc-length $s$ such that its $f$-position vector field $\gamma_f$, defined by $\gamma_f(s) = \int f(s) d\gamma$, always lies in its rectifying space in $\mathbb{E}^n$, where $f$ is a nowhere vanishing real-valued integrable function in parameter $s$. The main purpose is to characterize and classify such curves in $\mathbb{E}^n$.

Anahtar Kelimeler

Euclidean space, Frenet-Serret formulae, Higher curvatures, Rectifying curve, $f$-position vector field, $f$-rectifying curve

Kaynakça

• M. P. Do Carmo, Differential Geometry of Curves and Surfaces: Revised and Updated Second Edition, Courier Dover Publications, 2016.
• A. N. Pressley, Elementary Differential Geometry, Springer Science & Business Media, 2010. B.Y. Chen, When does the position vector of a space curve always lie in its rectifying plane? Amer. Math. Month., 110(2) (2003), 147–152.
• B.Y. Chen, F. Dillen, Rectifying curves as centrodes and extremal curves, Bull. Inst. Math. Acad. Sinica, 33(2) (2005), 77-90.
• B.Y. Chen, Rectifying curves and geodesics on a cone in the Euclidean 3-space, Tamkang J. Math., 48(2) (2017), 209-214.
• S. Deshmukh, B.Y. Chen, S. Alshamari, On rectifying curves in Euclidean 3-space, Turk. J. Math., 42(2) (2018), 609-620.
• K. {\.I}larslan, E. Nesovic, Some characterizations of rectifying curves in the Euclidean space $\mathbf{E}^4$, Turk. J. Math., 32(1) (2008), 21-30.
• S. Cambie, W. Goemans, I. Van den Bussche, Rectifying curves in the $n$-dimensional Euclidean space,Turk. J. Math., 40(1) (2016), 210-223.
• P. Lucas, J.A. Ortega-Yag{\"u}es, Rectifying curves in the three-dimensional sphere, J. Math. Anal. Appl., 421(2) (2015), 1855-1868.
• P. Lucas, J.A. Ortega-Yag{\"u}es, Rectifying curves in the three-dimensional hyperbolic space, Medit. J. Math., 13(4) (2016), 2199-2214.
• K. Ilarslan, E. Ne\v{s}ovi\'{c}, T. M. Petrovi\'{c}, Some characterization of rectifying curves in the Minkowski 3-Space, Novi Sad J. Math., 33(2) (2003), 23-32.
• K. Ilarslan, E. Ne\v{s}ovi\'{c}, On rectifying curves as centrodes and extremal curves in the Minkowski 3-Space, Novi Sad J. Math., 37(1) (2007), 53-64.
• K. Ilarslan, E. Ne\v{s}ovi\'{c}, Some characterizations of null, pseudo null and partially null rectifying curves in Minkowski space-time, Taiwanese J. Math., 12(5) (2008), 1035-1044.
• T.A. Ali, M. Onder, Some characterizations of space-like rectifying curves in the Minkowski space-time, Glob. J. Sci. Fron. Res. Math. Des. Sci., 12(1) (2012), 57-63.
• K. Ilarslan, E. Ne\v{s}ovi\'{c}, Some relations between normal and rectifying curves in Minkowski space-time, Inter. Elec. J. Geom., 7(1) (2014), 26-35.
• F. Hathout, A new class of curves generalizing helix and rectifying curves, arXiv: Diff. Geom., (2018).
• Z. Iqbal, J. Sengupta, Non-null (spacelike or timelike) f-rectifying curves in the Minkowski 3-space $\mathbb{E}_1^3$, Eurasian Bul. Math., 3(1) (2020), 38-55.
• Z. Iqbal, J. Sengupta, Null (lightlike) f-rectifying curves in the Minkowski 3-space $\mathbb{E}_1^3$, Fundam. J. Math. Appl., 3(1) (2020), 8-16.
• Z. Iqbal, J. Sengupta, Differential geometric aspects of lightlike $f$-rectifying curves in Minkowski space-time, Diff. Geom. - Dyn. Syst., 22 (2020), 113-129. Z. Iqbal, J. Sengupta, On $f$-Rectifying Curves in the Euclidean 4-Space, Acta Univ. Sapientiae Matem. (Accepted Manuscript).
• H. Gluck, Higher curvatures of curves in Euclidean space, Amer. Math. Mon., 73(7) (1966), 699--704.
• H. Gluck, Higher curvatures of curves in Euclidean space II, Amer. Math. Mon., 74(9) (1967), 1049--1056.