Conjugate mates for non-null Frenet curves

Abstract

For each non-null Frenet curve \gamma in Minkowski 3-space, there exists a unique unit speed non-null curve tangent

to the principal binormal vector field of \gamma. We briefly call this curve the conjugate mate of\gamma . The aim of this

paper is to prove some relationships between a non-null Frenet curve and its non-null conjugate mate.

Keywords

• slant helix,conjugate mate,general helix,adjoint curve,Salkowski curve,anti-Salkowski curve

References

1. [1] M. P. O'Neill, Semi-Riemannian Geometry with Applications to Relativity. World Scientific, New York, 1983.
2. [2] W. Kuhnel, Differential geometry: curves surfaces-manifolds. Weisbaden: Braunschweig 1999.
3. [3] S. Desmukh, BY. Chen and A. Alghanemi , Natural mates of Frenet curves in Euclidean 3-space, Turk. J. Math, (2018) 42: 2826-2840.
4. [4] J. Choi, Y. H. Kim, Associated curves of a Frenet curve and their applications, Appl. Math. Comput, 2012; 218: 9116-9124.
5. [5] J. Choi , Y. H. Kim, Ali A.T., Some associated curves of Frenet non-lightlike curves in E31, J. Math. Anal. Appl. 394 (2012) 712723.
6. [6] S. Deshmukh , I. Al-Dayel, K. Ilarslan, Frenet curves in Euclidean 4-space, Int. Electron. J Geom 2017; 10: 56-66.
7. [7] S. Deshmukh, BY Chen , NB Turki, A differential equations for Frenet curves in Euclidean 3-space and its applications, Rom. J. Math. Comput. Sci., 2018; 8: 1.
8. [8] H. Balgetir, M. Bektaş , M. Ergut, Bertrand curves for nonnull curves in 3-dimensional Lorentzian space, Hadronic Journal 27 (2004) 229-236.

9. [9] J. H. Choi, T. H. Kang and Y. H. Kim, Bertrand curves in 3-dimensional space forms, Applied Mathematics and Computation, 219 (2012) 1040-1046. [10] K. Ilarslan, E. Nesovic , M. Petrovic-Torgasev, Some characterizations of rectifying curves in the Minkowski 3-space, Novi Sad J. Math. 33(2) (2003), 23-32.
10. [11] A. T. Ali, R. Lopez, Slant helices in Minkowski space E31, J. Korean Math. Soc. 48 (2011) 159167.
11. [12] B. Bükçü, M. K. Karacan, On the Involute and Evolute Curves of the Spacelike Curve with a Spacelike Binormal in Minkowski 3−Space , Int. J. Contemp. Math. Sciences, Vol. 2, 2007, no. 5, 221 – 232.
12. [13] B. Bükçü, M. K. Karacan, On the Involute and Evolute Curves of the Timelike Curve in Minkowski 3−Space, Demonstratio Mathematica, Vol. XL No 3 2007.
13. [14] S. K. Nurkan, I. A. Güven, M. K. Karacan, Characterizations of adjoint curves in Euclidean 3-space. Proc Natl Acad Sci. India Sect A Phys Sci. https://doi.org/10.1007/s40010-017-0425-y.
14. [15] A.T. Ali, Spacelike Salkowski and anti-Salkowski curves with spacelike principal normal in Minkowski 3-space, Int. J. Open Problems Comp. Math. 2 (2009) 451–460.
15. [16] A.T. Ali, Timelike Salkowski and anti-Salkowski curves in Minkowski 3- space, J. Adv. Res. Dyn. Cont. Syst. 2 (2010) 17–26.
16. [17] A.T. Ali, Spacelike Salkowski and anti-Salkowski curves with timelike principal normal in Minkowski 3-space, Mathematica Aeterna, Vol. 1, 2011, no. 04, 201 - 210.