Conjugate Mates For Non-Null Frenet Curves

For each non-null Frenet curve ? in Minkowski 3-space, there exists a unique unit speed non-null curve ?̅ tangent to the principal binormal vector field of ?. We briefly call this curve ?̅ the conjugate mate of ?. The aim of this paper is to prove some relationships between a non-null Frenet curve and its non-null conjugate mate.

___

M. P. O'Neill, Semi-Riemannian Geometry with Applications to Relativity. World Scientific, New York, 1983.

W. Kuhnel, Differential geometry: curves surfaces-manifolds. Weisbaden: Braunschweig 1999.

S. Desmukh, BY. Chen and A. Alghanemi, Natural mates of Frenet curves in Euclidean 3- space, Turk. J. Math, (2018) 42: 2826-2840.

J. Choi, Y. H. Kim, Associated curves of a Frenet curve and their applications, Appl. Math. Comput, 2012; 218: 9116-9124.

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.

S. Deshmukh, I. Al-Dayel, K. Ilarslan, Frenet curves in Euclidean 4-space, Int. Electron. J Geom 2017; 10: 56-66.

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.

H. Balgetir, M. Bektaş, M. Ergut, Bertrand curves for nonnull curves in 3-dimensional Lorentzian space, Hadronic Journal 27 (2004) 229-236.

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.

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.

A. T. Ali, R. Lopez, Slant helices in Minkowski space E31, J. Korean Math. Soc. 48 (2011) 159167.

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.

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.

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.

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.

A.T. Ali, Timelike Salkowski and anti- Salkowski curves in Minkowski 3- space, J. Adv. Res. Dyn. Cont. Syst. 2 (2010) 17–26.

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.