Generalized osculating curves of type (n-3) in the n-dimensional Euclidean space

Abstract

In this paper, we give a generalization of the osculating curves to the $n$-dimensional Euclidean space. Based on the definition of an osculating curve in the 3 and 4 dimensional Euclidean spaces, a new type of osculating curve has been defined such that the curve is independent of the $(n-3)$th binormal vector in the n-dimensional Euclidean space, which has been called ”a generalized osculating curve of type $(n-3)$”. We find the relationship between the curvatures for any unit speed curve to be congruent to this osculating curve in $E^n$. In particular, we characterize the osculating curves in $E^n$ in terms of their curvature functions. Finally, we show that the ratio of the $(n-1)$th and $(n-2)$th curvatures of the osculating curve is the solution of an $(n-2)$th order linear nonhomogeneous differential equation.

Keywords

• Osculating curve
• curvatures
• unit speed curve
• higher order linear differential equation

References

1. Chen, B. Y., When does the position vector of a space curve always lie in its rectifying plane?, The American Mathematical Monthly, 110(2) (2003), 147-152. https://doi.org/10.1080/00029890.2003.11919949
2. Chen, B. Y., Dillen, F., Rectifying curves as centrodes and extremal curves, Bulletin of the Institute of Mathematics Academia Sinica, 33(2) (2005), 77-90.
3. Ilarslan, K., Nesovic, E., Petrovic-Torgasev, M., Some characterizations of rectifying curves in the Minkowski 3-space, Novi Sad Journal of Mathematics, 33(2) (2003), 23-32.
4. Ilarslan, K., Nesovic, E., On rectifying curves as centrodes and extremal curves in the Minkowski 3-space, Novi Sad Journal of Mathematics, 37(1) (2007), 53-64.
5. Ilarslan, K., Nesovic, E., Some characterizations of rectifying curves in the Euclidean space $E^{4}$, Turkish Journal of Mathematics, 32(1) (2008), 21-30.
6. Cambie, S., Goemans, W., Van den Bussche, I., Rectifying curves in the n-dimensional Euclidean space, Turkish Journal of Mathematics, 40(1) (2016), 210-223.
7. Ilarslan, K., Spacelike normal curves in Minkowski space, Turkish Journal of Mathematics $E^{3}_{1}$, 29(1) (2005), 53-63.
8. Ilarslan, K., Nesovic, E., Timelike and null normal curves in Minkowski space $E^{3}_{1}$, Indian Journal of Pure & Applied Mathematics, 35(7) (2004), 881-888.

9. Ilarslan, K., Nesovic, E., Some characterizations of osculating curves in the Euclidean spaces, Demonstratio Mathematica, 41(4) (2008), 931-940. https://doi.org/10.1515/dema-2008-0421
10. Yildiz, O. G., Ozkaldi Karakus, O., On the quaternionic normal curves in the Euclidean space, 2nd International Eurasian Conference on Mathematical Sciences and Applications, (2013), 26-29.
11. Yildiz, O. G., Ozkaldi Karakus, O., On the quaternionic normal curves in the semi-Euclidean space $E^{4}_{2}$ , International Journal of Mathematical Combinatorics, 3 (2016), 68-76. https://doi.org/10.5281/zenodo.825065
12. Ucum, A., Sakaki, M., Ilarslan, K., On osculating, normal and rectifying bi-null curves in $R^{6}_{3}$, Journal of Dynamical Systems and Geometric Theories, 15(1) (2017), 1-13. https://doi.org/10.1080/1726037X.2017.1323415
13. Ilarslan, K., Sakaki, M., Ucum, A., On osculating, normal and rectifying binull curves in $R^{5}_{2}$, Novi Sad Journal of Mathematics, 48(1) (2018), 9-20. https://doi.org/10.30755/NSJOM.05268
14. Ilarslan, K., Kilic, N., Erdem, H. A., Osculating curves in 4-dimensional semi-Euclidean space with index 2, Open Mathematics, 15(1) (2017), 562-567. https://doi.org/10.1515/math-2017- 0050
15. Kulahci, M., Almaz, F., Some characteristics of osculating curves in the lightlike cone, Boletim da Sociedade Paranaense de Matematica, 35(2) (2017), 39-48. https://doi.org/10.5269/bspm.v35i2.26227
16. Bektas, O., Gurses, N., Yuce, S., Quaternionic osculating curves in Euclidean and semi- Euclidean space, Journal of Dynamical Systems and Geometric Theories, 14(1) (2016), 65-84. https://doi.org/10.1080/1726037X.2016.1177935
17. Ilarslan, K., Nesovic, E., Some characterizations of null osculating curves in the Minkowski space-time, Proceedings of the Estonian Academy of Sciences, 61(1) (2012), 1-8. https://doi.org/10.3176/proc.2012.1.01
18. Ilarslan, K., Nesovic, E., The first kind and the second kind osculating curves in Minkowski space-time, Comptes rendus de l’Academie bulgare des Sciences, 62(6) (2009), 677-686.
19. Gluck, H., Higher curvatures of curves in Euclidean space, The American Mathematical Monthly, 73(7) (1966), 699-704. https://doi.org/10.2307/2313974
20. Kuhnel, W., Differential Geometry: Curves-Surfaces-Manifolds, Braunschweig, Wiesbaden, 1999.
21. O’Neill, B., Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York, 1983.
22. Struik, D. J., Lectures on Classical Differential Geometry, Addison-Wesley Press, Reading, Massachusetts, 1950.
23. Zill, D. G.,Wright, W. S., Differential Equations with Boundary-Value Problems, 8th Edition, Cengage Learning, 2012.
24. Boyce, W. E., DiPrima, R. C., Meade, D. B., Elementary Differential Equations and Boundary Value Problems, Wiley, New York, 1992.
25. Gungor, M. A., Tosun, M., Some characterizations of quaternionic rectifying curves, Differential Geometry - Dynamical Systems, 13 (2011), 89-100.
26. Ucum, A., Kecilioglu, O., Ilaslan, K., Generalized Bertrand curves with timelike (1,3)-normal plane in Minkowski space-time, Kuwait Journal of Science, 42(3) (2015), 10-27.
27. Ilarslan, K., Nesovic, E., Tensor product surfaces of a Lorentzian space curve and a Euclidean plane curve, Kuwait Journal of Science and Engineering, 34(2A) (2007), 41-55.