On Pseudospherical Smarandache Curves in Minkowski 3-Space

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Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 404521, 14 pages http://dx.doi.org/10.1155/2014/404521

Research Article On Pseudospherical Smarandache Curves in Minkowski 3-Space Esra Betul Koc Ozturk,1,2 Ufuk Ozturk,1,2 Kazim Ilarslan,3 and Emilija NeĹĄoviT4 1

Department of Mathematics, Faculty of Sciences, University of C ¸ ankiri Karatekin, 18100 C ¸ ankiri, Turkey School of Mathematics & Statistical Sciences, Room PSA442, Arizona State University, Tempe, AZ 85287-1804, USA 3 Department of Mathematics, Faculty of Sciences and Arts, University of Kirikkale, 71450 Kirikkale, Turkey 4 Department of Mathematics and Informatics, Faculty of Science, University of Kragujevac, 34000 Kragujevac, Serbia 2

Correspondence should be addressed to Ufuk Ozturk; uuzturk@asu.edu Received 3 June 2013; Accepted 21 November 2013; Published 24 February 2014 Academic Editor: Laurent Gosse Copyright Š 2014 Esra Betul Koc Ozturk et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this paper we define nonnull and null pseudospherical Smarandache curves according to the Sabban frame of a spacelike curve lying on pseudosphere in Minkowski 3-space. We obtain the geodesic curvature and the expressions for the Sabban frame’s vectors of spacelike and timelike pseudospherical Smarandache curves. We also prove that if the pseudospherical null straight lines are the Smarandache curves of a spacelike pseudospherical curve đ?›ź, then đ?›ź has constant geodesic curvature. Finally, we give some examples of pseudospherical Smarandache curves.

1. Introduction It is known that a Smarandache geometry is a geometry which has at least one Smarandachely denied axiom [1]. An axiom is said to be Smarandachely denied, if it behaves in at least two different ways within the same space. Smarandache geometries are connected with the theory of relativity and the parallel universes. Smarandache curves are the objects of Smarandache geometry. By definition, if the position vector of a curve đ?›˝ is composed by the Frenet frame’s vectors of another curve đ?›ź, then the curve đ?›˝ is called a Smarandache curve [2]. Special Smarandache curves in the Euclidean and Minkowski spaces are studied by some authors [3–7]. The curves lying on a pseudosphere đ?‘†12 in Minkowski 3-space đ??¸13 are characterized in [8]. In this paper we define nonnull and null pseudospherical Smarandache curves according to the Sabban frame of a spacelike curve lying on pseudosphere in Minkowski 3-space. We obtain the geodesic curvature and the expressions for the Sabban frame’s vectors of spacelike and timelike pseudospherical Smarandache curves. We also prove that if the pseudospherical null straight lines are the Smarandache curves of a spacelike pseudospherical curve đ?›ź, then đ?›ź has nonzero constant geodesic curvature. Finally, we give some examples

of pseudospherical Smarandache curves in Minkowski 3space.

2. Basic Concepts The Minkowski 3-space R31 is the Euclidean 3-space R3 provided with the standard flat metric given by â&#x;¨â‹…, â‹…â&#x;Š = − đ?‘‘đ?‘Ľ12 + đ?‘‘đ?‘Ľ22 + đ?‘‘đ?‘Ľ32 ,

(1)

where (đ?‘Ľ1 , đ?‘Ľ2 , đ?‘Ľ3 ) is a rectangular coordinate system of R31 . Since đ?‘” is an indefinite metric, recall that a nonzero vector đ?‘Ľâƒ— ∈ R31 can have one of three Lorentzian causal characters: it can be spacelike if â&#x;¨đ?‘Ľ,⃗ đ?‘Ľâ&#x;Šâƒ— > 0, timelike if â&#x;¨đ?‘Ľ,⃗ đ?‘Ľâ&#x;Šâƒ— < 0, and null (lightlike) if â&#x;¨đ?‘Ľ,⃗ đ?‘Ľâ&#x;Šâƒ— = 0. In particular, the norm (length) of a ⃗ and two vectors đ?‘Ľâƒ— vector đ?‘Ľâƒ— ∈ R31 is given by ‖đ?‘Ľâ€–⃗ = √|â&#x;¨đ?‘Ľ,⃗ đ?‘Ľâ&#x;Š| and đ?‘Śâƒ— are said to be orthogonal if â&#x;¨đ?‘Ľ,⃗ đ?‘Śâ&#x;Šâƒ— = 0. Next, recall that an arbitrary curve đ?›ź = đ?›ź(đ?‘ ) in E31 can locally be spacelike, timelike, or null (lightlike) if all of its velocity vectors đ?›źó¸€ (đ?‘ ) are, respectively, spacelike, timelike, or null (lightlike) for all đ?‘ ∈ đ??ź [9]. A spacelike or a timelike curve đ?›ź is parameterized by arclength parameter đ?‘ if â&#x;¨đ?›źó¸€ (đ?‘ ), đ?›źó¸€ (đ?‘ )â&#x;Š = 1 or â&#x;¨đ?›źó¸€ (đ?‘ ), đ?›źó¸€ (đ?‘ )â&#x;Š = −1, respectively. For any two vectors đ?‘Ľâƒ— = (đ?‘Ľ1 , đ?‘Ľ2, đ?‘Ľ3 ) and

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