Generalized Bertrand Curves with Spacelike (1,3)(1,3)-Normal Plane in Minkowski Space-Time

Generalized Bertrand Curves with Spacelike (1,3)(1,3)-Normal Plane in Minkowski Space-Time

In this paper, we reconsider the (1, 3) -Bertrand curves with respect to the casual characters of a (1, 3) -normal plane that is a plane spanned by the principal normal and the second binormal vector fields of the given curve. Here, we restrict our investigation of (1, 3) -Bertrand curves to the spacelike (1, 3) -normal plane in Minkowski space-time. We obtain the necessary and sufficient conditions for the curves with spacelike (1, 3) -normal plane to be (1, 3) -Bertrand curves and we give the related examples for these curves.

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  • [1] Balgetir H, Bekta¸s M, Erg¨ut M. Bertrand curves for non-null curves in 3-dimensional Lorentzian space. Hadronic J 2004; 27: 229–236.
  • [2] Balgetir H, Bekta¸s M, Inoguchi J. Null Bertrand curves in Minkowski 3-space and their characterizations. Note Mat 2004; 23: 7–13.
  • [3] Bertrand JM. M´emoire sur la th´eorie des courbes ´a double courbure. Comptes Rendus 1850; 15: 332–350 (in French).
  • [4] Bioche C. Sur les courbes de M. Bertrand. Bull Soc Math France 1889; 17: 109–112 (in French).
  • [5] Bonnor WB. Null curves in a Minkowski space-time. Tensor 1969; 20: 229–242.
  • [6] Bonnor WB. Curves with null normals in Minkowski space-time. In: Dadhich NJ, editor. A Random Walk in Relativity and Cosmology. New York, NY, USA: Wiley, 1985, pp. 33–47.
  • [7] Burke JF. Bertrand curves associated with a pair of curves. Mathematics Magazine 1960; 34: 60–62.
  • [8] Ekmekci N, ˙Ilarslan K. On Bertrand curves and their characterization. Differ Geom Dyn Syst 2001; 3: 17–24.
  • [9] Ersoy S, ˙Inalcık A. Generalized spacelike Bertrand curves in Minkowski 5-space. Quaestiones Mathematicae 2014; 37: 19–29.
  • [10] G¨ok ˙I, Nurkan SK, ˙Ilarslan K. On pseudo null Bertrand curves in Minkowski space-time. Kyungpook Math J 2014; 54: 685–697.
  • [11] ˙Ilarslan K, Nesovic E. Spacelike and timelike normal curves in Minkowski space-time. Publications de L’Institut Math 2009; 85: 111–118.
  • [12] Jin DH. Null Bertrand curves in a Lorentz manifold. J Korea Soc Math Educ Ser B Pure Appl Math 2008; 15: 209–215.
  • [13] Kahraman F, G¨ok ˙I, ˙Ilarslan K. Generalized null Bertrand curves in Minkowski space-time. Scientic Annals of ”Al.I. Cuza” University of Iasi Mathematica 2014; 60: 489–502.
  • [14] Kim CY, Park JH, Yorozu S. Curves on the unit 3-sphere S 3 (1) in Euclidean 4-space R 4 . Bull Korean Math Soc 2013; 50: 1599–1622.
  • [15] Kuhnel W. Differential Geometry: Curves-Surfaces-Manifolds. Wiesbaden, Germany: Braunschweig, 1999.
  • [16] Lucas P, Ortega-Yag¨ues JA. Bertrand curves in the three-dimensional sphere. J Geom Phys 2012; 62: 1903–1914.
  • [17] Matsuda H, Yorozu S. Notes on Bertrand curves. Yokohama Math J 2003; 50: 41–58.
  • [18] O’Neill B. Semi-Riemannian Geometry with Applications to Relativity. New York, NY, USA: Academic Press, 1983.
  • [19] Oztekin HB. Weakened Bertrand curves in the Galilean space G 3 . Journal of Advanced Mathematical Studies 2009; 2: 69–76.
  • [20] Pears LR. Bertrand curves in Riemannian space. J London Math Soc 1935; 1–10: 180–183.
  • [21] Saint Venant B. M´emoire sur les lignes courbes non planes. Journal de l’Ecole Polytechnique 1845; 18: 1–76 (in French).
  • [22] U¸cum A, Ke¸cilio˘glu O, ˙Ilarslan K. Generalized Bertrand curves with timelike (1,3)-normal plane in Minkowski space-time. Kuwait J Sci 2015; 42: 10–27.
  • [23] Whittemore JK. Bertrand curves and helices. Duke Math J 1940; 6: 235–245.
  • [24] Yilmaz MY, Bekta¸s M. General properties of Bertrand curves in Riemann-Otsuki space. Nonlinear Anal-Theor 2008; 69: 3225–3231.