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Finslerian viewpoint to the rectifying, normal, and osculating curves. (English) Zbl 1485.53023

Summary: The theory of Finsler metric was introduced by Paul Finsler, in 1918. The author defines this metric using the Minkowski norm instead of the inner product. Therefore, this geometry is a more general metric and includes the Riemannian metric. In the present work, using the Finsler metric, we investigate the position vector of the rectifying, normal and osculating curves in Finslerian 3-space \(\mathbb{F}^3 \). We obtain the general characterizations of these curves in \(\mathbb{F}^3 \). Furthermore, we show that rectifying curves are extremal curves derived from the Finslerian spherical curve. We also plotted various examples by using the Randers metrics.

MSC:

53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)
53A04 Curves in Euclidean and related spaces
58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
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[1] [1] Akbar-Zadeh, H.: Initiation to global Finslerian geometry, North-Holland Math. Library (2006). · Zbl 1103.53044
[2] [2] Asanjarani, A., Bidabad, B.: Classification of complete Finsler manifolds through a second order differential equation, Differential Geom. Appl. 26, 434-444 (2008). · Zbl 1147.53060
[3] [3] Bejancu, A., Farran, H. R.: Geometry of pseudo-Finsler submanifolds, Kluwer Academic Publishers (2000). · Zbl 0999.53046
[4] [4] Bidabad, B., Shen, Z.: Circle-preserving transformations on Finsler spaces, Publication Mathematicae, in press. · Zbl 1299.53143
[5] [5] Bozkurt, Z., Gök, İ, Okuyucu, O.Z., Ekmekci, F.N.: Characterizations of rectifying, normal and osculating curves in three dimensional compact Lie groups, Life Science Journal. 10(3), 819-823 (2013).
[6] [6] Chen, B. Y., Dillen, F.: Rectifying curves as centrodes and extremal curves, Bull. of the Ins. of Maths. Academia Snica. 33(2), 77-90 (2005). · Zbl 1082.53012
[7] [7] Chen, B.Y.: When does the position vector of a space curve always lie in its rectifying plane?, Amer. Math. Monthly. 110, 147-152 (2003). · Zbl 1035.53003
[8] [8] Chern, S.S., Shen Z.: Riemann Finsler Geometry, World Scientific (2005). · Zbl 1085.53066
[9] [9] Deshmukh, S., Chen, B.Y., Alshammari, S.H.: On rectifying curves in Euclidean 3􀀀space, Turk. J Math. 42, 609-620 (2018). · Zbl 1424.53021
[10] [10] Ergüt, M., Külahcı, M.: Special curves in three dimensional Finsler manifold F3, TWMS J. Pure Appl. Math. (5)2, 147-151 (2014). · Zbl 1312.53030
[11] [11] Çetin, E.D, Gök, İ., Yayli, Y.: A New Aspect of Rectifying Curves and Ruled Surfaces in Galilean 3-Space, Filomat. 32(8), 2953-2962 (2018). · Zbl 1499.53047
[12] [12] İlarslan, K., Nešović, E., Petrovic-Torgasev, M.: Some characterizations of rectifying curves in the Euclidean space E4, Turk. J. Math. 32, 21-30 (2008). · Zbl 1148.53007
[13] [13] İlarslan, K., Nešović, E.: Some characterizations of osculating curves in the Euclidean space, Demonstratio Mathematica. 4, 931-939 (2008). · Zbl 1169.53003
[14] [14] İlarslan, K., Nešović, E. Spacelike and timelike normal curves in Minkowski spacetime, Publ. Inst. Math. Belgrade 85(99), 111-118 (2009). · Zbl 1289.53135
[15] [15] Öztekin, H., Ögrenmis, A.O.: Normal and rectifying curves in pseudo-Galilean space G31 and their characterizations, J. Math. Comput. Sci. 2(1), 91-100 (2012).
[16] [16] Öztekin, H.: Normal and rectifying curves in Galilean space G3, Proc. of IAM. 5(1), 98-109 (2016). · Zbl 1353.53013
[17] [17] Remizov, A.O.: Geodesics in generalized Finsler spaces: singularities in dimension two, Journal of Singularities. 14, 172-193 (2016). · Zbl 1358.53029
[18] [18] Shen, Z.: Lecture on Finsler geometry, World Scientific Publishing Co (2001). · Zbl 0974.53002
[19] [19] Yildirim, M.Y., Bektas, M.: Helices of the 3-dimensional Finsler manifold, J. Advanced. Math. Stud. 2(1), 107-113 (2009). · Zbl 1181.53025
[20] [20] Yildirim, M.Y.: Biharmonic general helices in 3􀀀dimensional Finsler manifold, Karaelmas Sci. Engineering J. 7(1), 1-4 (2017).
[21] [21] Yildiz, O.G., Özkaldi, Karakus S.: On the quaternionic normal curves in the semi-Euclidean space E^4_2 , Int. J. Math. Comb. 3, 68-76 (2016).
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