Null Mannheim curves in the Minkowski 3-space $Bbb{E}^3 _1$

In this study, we give the deﬁnition of null Mannheim curve with timelike or spacelike Mannheim partner curve in the Minkowski 3-space $Bbb {E} ^3 _1$ . We get the necessary and suﬃcient conditions for the null Mannheim curves. Then we investigate the null and timelike or spacelike generalized helix as the null Mannheim curve and timelike or spacelike Mannheim partner curve, respectively.

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[1] Balgetir, H., Bekta¸ s, M. and Inoguchi, J.: Null Bertrand Curves in Minkowski 3-space and their characterizations, Note di Matematica, 23, no.1, 7-13 (2004/2005).
[2] Balgetir, H., Bekta¸ s, M. and Erg¨ ut, M.: On a Characterization of Null Helix, Bull. of the Inst. of Math. Acad. Sinica, 29, no.1, 71-78, March (2001).
[3] Blum, R.: A remarkable class of Mannheim curves, Canad. Math. Bull. 9, 223-228 (1966).
[4] Duggal, K. L. and Bejancu, A.: Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, Kluwer Academic Publishers, Dordrecht, (1996).
[5] Ekmek¸ ci,N.andHacısalihoğlu, H. H.: On Helices of a Lorentzian Manifold, Commun. Fac. sci. Univ. Ank. Series A1, 45, 45-50 (1996).
[6] Honda, K. and Inoguchi, J.: On a characterisation of null cubics in Minkowski 3-space, Diﬀerential Geo.-Dynamical Systems, 5, no.1, 1-6 (2003).
[7] İlarslan, K. and Nesovic, E.: Some Characterizations of Rectifying Curves in the Euclidean Space E4 Turk J.Math. 32, 21-30 (2008).
[8] Kula, L., Ekmekci, N., Yaylı, Y. and İlarslan, K.: Characterizations of Slant Helices in Euclidean 3-Space, Turk J. Math. 34, 261-273 (2010).
[9] Külahci, M. and Erg¨ ut, M.: Bertrand curves of AW(k)-type in Lorentzian space , Nonlinear Analysis, 1725-1731 (2009).
[10] Liu, H. and Wang, F.: Mannheim Partner curves in 3-space, Journal of Geometry, 88, 120-126 (2008).
[11] O’Neill, B.: Semi-Riemannian Geometry with Application to Relativity, Academic Press, New York, (1983).
[12] Tigano, O.: Sulla determinazione delle curve di Mannheim, Matematiche Catania 3, 25-29 (1948).