Research Article
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Year 2022, , 30 - 38, 30.04.2022
https://doi.org/10.36890/iejg.993188

Abstract

References

  • [1] Akbar-Zadeh, H.: Initiation to global Finslerian geometry, North-Holland Math. Library (2006).
  • [2] Asanjarani, A., Bidabad, B.: Classification of complete Finsler manifolds through a second order differential equation, Differential Geom. Appl. 26, 434-444 (2008).
  • [3] Bejancu, A., Farran, H. R.: Geometry of pseudo-Finsler submanifolds, Kluwer Academic Publishers (2000).
  • [4] Bidabad, B., Shen, Z.: Circle-preserving transformations on Finsler spaces, Publication Mathematicae, in press.
  • [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] 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).
  • [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).
  • [8] Chern, S.S., Shen Z.: Riemann Finsler Geometry, World Scientific (2005).
  • [9] Deshmukh, S., Chen, B.Y., Alshammari, S.H.: On rectifying curves in Euclidean 3􀀀space, Turk. J Math. 42, 609–620 (2018).
  • [10] Ergüt, M., Külahcı, M.: Special curves in three dimensional Finsler manifold F3, TWMS J. Pure Appl. Math. (5)2, 147-151 (2014).
  • [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).
  • [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).
  • [13] İlarslan, K., Nešović, E.: Some characterizations of osculating curves in the Euclidean space, Demonstratio Mathematica. 4, 931-939 (2008).
  • [14] İlarslan, K., Nešović, E. Spacelike and timelike normal curves in Minkowski spacetime, Publ. Inst. Math. Belgrade 85(99), 111-118 (2009).
  • [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] Öztekin, H.: Normal and rectifying curves in Galilean space G3, Proc. of IAM. 5(1), 98–109 (2016).
  • [17] Remizov, A.O.: Geodesics in generalized Finsler spaces: singularities in dimension two, Journal of Singularities. 14, 172-193 (2016).
  • [18] Shen, Z.: Lecture on Finsler geometry, World Scientific Publishing Co (2001).
  • [19] Yildirim, M.Y., Bektas, M.: Helices of the 3-dimensional Finsler manifold, J. Advanced. Math. Stud. 2(1), 107-113 (2009).
  • [20] Yildirim, M.Y.: Biharmonic general helices in 3􀀀dimensional Finsler manifold, Karaelmas Sci. Engineering J. 7(1), 1-4 (2017).
  • [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).

Finslerian Viewpoint to the Rectifying, Normal, and Osculating Curves

Year 2022, , 30 - 38, 30.04.2022
https://doi.org/10.36890/iejg.993188

Abstract

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.

References

  • [1] Akbar-Zadeh, H.: Initiation to global Finslerian geometry, North-Holland Math. Library (2006).
  • [2] Asanjarani, A., Bidabad, B.: Classification of complete Finsler manifolds through a second order differential equation, Differential Geom. Appl. 26, 434-444 (2008).
  • [3] Bejancu, A., Farran, H. R.: Geometry of pseudo-Finsler submanifolds, Kluwer Academic Publishers (2000).
  • [4] Bidabad, B., Shen, Z.: Circle-preserving transformations on Finsler spaces, Publication Mathematicae, in press.
  • [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] 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).
  • [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).
  • [8] Chern, S.S., Shen Z.: Riemann Finsler Geometry, World Scientific (2005).
  • [9] Deshmukh, S., Chen, B.Y., Alshammari, S.H.: On rectifying curves in Euclidean 3􀀀space, Turk. J Math. 42, 609–620 (2018).
  • [10] Ergüt, M., Külahcı, M.: Special curves in three dimensional Finsler manifold F3, TWMS J. Pure Appl. Math. (5)2, 147-151 (2014).
  • [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).
  • [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).
  • [13] İlarslan, K., Nešović, E.: Some characterizations of osculating curves in the Euclidean space, Demonstratio Mathematica. 4, 931-939 (2008).
  • [14] İlarslan, K., Nešović, E. Spacelike and timelike normal curves in Minkowski spacetime, Publ. Inst. Math. Belgrade 85(99), 111-118 (2009).
  • [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] Öztekin, H.: Normal and rectifying curves in Galilean space G3, Proc. of IAM. 5(1), 98–109 (2016).
  • [17] Remizov, A.O.: Geodesics in generalized Finsler spaces: singularities in dimension two, Journal of Singularities. 14, 172-193 (2016).
  • [18] Shen, Z.: Lecture on Finsler geometry, World Scientific Publishing Co (2001).
  • [19] Yildirim, M.Y., Bektas, M.: Helices of the 3-dimensional Finsler manifold, J. Advanced. Math. Stud. 2(1), 107-113 (2009).
  • [20] Yildirim, M.Y.: Biharmonic general helices in 3􀀀dimensional Finsler manifold, Karaelmas Sci. Engineering J. 7(1), 1-4 (2017).
  • [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).
There are 21 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Zehra Ozdemir 0000-0001-9750-507X

Fatma Ates 0000-0002-3529-1077

Nejat Ekmekçi 0000-0003-1246-2395

Publication Date April 30, 2022
Acceptance Date April 12, 2022
Published in Issue Year 2022

Cite

APA Ozdemir, Z., Ates, F., & Ekmekçi, N. (2022). Finslerian Viewpoint to the Rectifying, Normal, and Osculating Curves. International Electronic Journal of Geometry, 15(1), 30-38. https://doi.org/10.36890/iejg.993188
AMA Ozdemir Z, Ates F, Ekmekçi N. Finslerian Viewpoint to the Rectifying, Normal, and Osculating Curves. Int. Electron. J. Geom. April 2022;15(1):30-38. doi:10.36890/iejg.993188
Chicago Ozdemir, Zehra, Fatma Ates, and Nejat Ekmekçi. “Finslerian Viewpoint to the Rectifying, Normal, and Osculating Curves”. International Electronic Journal of Geometry 15, no. 1 (April 2022): 30-38. https://doi.org/10.36890/iejg.993188.
EndNote Ozdemir Z, Ates F, Ekmekçi N (April 1, 2022) Finslerian Viewpoint to the Rectifying, Normal, and Osculating Curves. International Electronic Journal of Geometry 15 1 30–38.
IEEE Z. Ozdemir, F. Ates, and N. Ekmekçi, “Finslerian Viewpoint to the Rectifying, Normal, and Osculating Curves”, Int. Electron. J. Geom., vol. 15, no. 1, pp. 30–38, 2022, doi: 10.36890/iejg.993188.
ISNAD Ozdemir, Zehra et al. “Finslerian Viewpoint to the Rectifying, Normal, and Osculating Curves”. International Electronic Journal of Geometry 15/1 (April 2022), 30-38. https://doi.org/10.36890/iejg.993188.
JAMA Ozdemir Z, Ates F, Ekmekçi N. Finslerian Viewpoint to the Rectifying, Normal, and Osculating Curves. Int. Electron. J. Geom. 2022;15:30–38.
MLA Ozdemir, Zehra et al. “Finslerian Viewpoint to the Rectifying, Normal, and Osculating Curves”. International Electronic Journal of Geometry, vol. 15, no. 1, 2022, pp. 30-38, doi:10.36890/iejg.993188.
Vancouver Ozdemir Z, Ates F, Ekmekçi N. Finslerian Viewpoint to the Rectifying, Normal, and Osculating Curves. Int. Electron. J. Geom. 2022;15(1):30-8.