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Generalized bertrand curves with spacelike (1, 3) normal plane in minkowski space time

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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. | Turk J Math (2016) 40: 487 – 505 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ doi:10.3906/mat-1503-12 Research Article Generalized Bertrand Curves with Spacelike (1, 3)-Normal Plane in Minkowski Space-Time ∗ ˙ IO ˙ GLU, ˘ ˙ Ali UC ¸ UM, Osman KEC ¸ IL Kazım ILARSLAN Department of Mathematics, Faculty of Sciences and Arts, Kırıkkale University, Kırıkkale, Turkey Received: 03.03.2015 • Accepted/Published Online: 25.06.2015 • Final Version: 08.04.2016 Abstract: 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. Key words: Bertrand curve, Minkowski space-time, Frenet planes 1. Introduction Much work has been done about the general theory of curves in a Euclidean space (or more generally in a Riemannian manifold). Now we have extensive knowledge on their local geometry as well as their global geometry. Characterization of a regular curve is one of the important and interesting problems in the theory of curves in Euclidean space. There are two ways widely used to solve these problems: figuring out the relationship between the Frenet vectors of the curves [15], and determining the shape and size of a regular curve by using its curvatures k1 (or κ) and k2 (or τ ). In 1845, Saint Venant [21] proposed the question of whether the principal normal of a curve is the principal normal of another on the surface generated by the principal normal of the given one. Bertrand answered this question in [3], published in 1850. He proved that a necessary and .