AW(k)-Type Curves in Modified Orthogonal Frame

Abstract

The goal of this article is to examine AW(k)-type curves in Euclidean 3- space according to a modified orthogonal frame with the non-zero curvature. Firstly, the relations between the curvatures for AW(k)-type curves in modified orthogonal frame are given. Also, harmonic curvatures of AW(k)-type curves according to this frame are obtained. The results are illustrated on examples. Finally, slant helices are analysed for the modified orthogonal frame and some relations are obtained for the curvatures of the curve to be of AW(1) and AW(2) type in case the curve be a slant helix.

Keywords

• AW(k)-type curves
• Modified orthogonal frame
• Harmonic curvatures
• Slant helices

References

1. Arslan K, West A. Product submanifolds with point-wise 3-planar normal sections. Glasgow Mathematical Journal. 1995; 37(1):73–81.
2. Arslan K, Özgür C. Curves and surfaces of AW (k) type. In Geometry and Topology of SubmanifoldsIX. 1999; 21–26.
3. Özgür C, Gezgin F. On some curves of AW(k)-type. Differential Geometry Dynamical Systems. 2005; 7:74.
4. Kılıç B, Arslan K. On curves and surfaces of AW(k)-type. Journal of the Institute of Science and Technology of Balıkesir University. 2004; 6(1):52-61.
5. Kişi İ, Öztürk G. Bishop Çatısına Göre AW (k)-Tipinden Eğriler. Afyon Kocatepe University Journal of Science and Engineering. 2019; 19(3):620-625.
6. Külahcı M, Öğrenmiş AO, Ergüt M. New characterizations of curves in the Galilean space G^3. International Journal of Physical and Mathematical Sciences. 2010; 1(1).
7. Sun J, Pei D. Null Cartan Bertrand curves of AW (k)-type in Minkowski 4-space. Physics Letters A. 2012; 376(33):2230-2233.
8. Gür S, Şenyurt S. Frenet vectors and geodesic curvatures of spheric indicators of Salkowski curve in E^3. Hadronic Journal. 2010; 33(5):485.

9. Hacisalihoğlu HH. Differential geometry. Ankara University Faculty of Science Press. Ankara; 2000.
10. Şenyurt S, Çalışkan A. Smarandache curves of Mannheim curve couple according to Frenet frame. Mathematical Sciences and Applications E-Notes. 2017; 5(1):122-136.
11. Do Carmo MP. Differential geometry of curves and surfaces: revised and updated second edition. Courier Dover Publications; 2016.
12. Bishop RL. There is more than one way to frame a curve. The American Mathematical Monthly. 1975; 82(3):246–251.
13. Yılmaz S, Turgut M. A new version of Bishop frame and an application to spherical images. Journal of Mathematical Analysis and Applications. 2010; 371(2):764-776.
14. Damar E, Yüksel N, Vanlı AT. The ruled surfaces according to type-2 Bishop frame in E^3. In International Mathematical Forum 2017; 12(3):133-143.
15. Damar E, Yüksel N, Karacan MK. Ruled surfaces according to parallel trasport frame in E^3. Mathematical Combinatorics.2020; 20.
16. Yüksel N. The ruled surfaces according to Bishop frame in Minkowski 3-space. In Abstract and Applied Analysis.2013; (2013): 810640
17. Hord RA. Torsion at an inflection point of a space curve. The American Mathematical Monthly.1972; 79(4):371-374.
18. Sasai T. The fundamental theorem of analytic space curves and apparent singularities of Fuchsian differential equations. Tohoku Mathematical Journal, Second Series.1984; 36(1):17-24.
19. Sasai T. Geometry of analytic space curves with singularities and regular singularities of differential equations. Funkcial. Ekvac.1987; 30, 283-303.
20. Bükçü B, Karacan MK. On the modified orthogonal frame with curvature and torsion in 3-Space. Mathematical Sciences and Applications E-Notes. 2016; 4(1):184-188.
21. Bükçü B, Karacan MK. Spherical curves with modified orthogonal frame. Journal of New Results in Science.2016; 5(10):60-68.
22. Gür S, Bektaş M. On the modified orthogonal frames of the non-unit speed curves in Euclidean Space E^3. Turkish Journal of Science.2022; 7(2):58-74.
23. Lone MS, Es H, Karacan MK, Bükçü B. On some curves with modified orthogonal frame in Euclidean 3-space. Iranian Journal of Science and Technology, Transactions A: Science.2019; 43:1905-1916.
24. Uddin S, Stankovi´c MS, Iqbal M, Yadav SK, Aslam M. Slant helices in Minkowski 3-space E_1^3 with Sasai’s modified frame fields. Filomat.2022; 36(1):151-164.
25. Yüksel N, Karacan MK, Demirkıran T. Spherical curves with modified orthogonal frame with torsion. Turkish Journal of Science.2022; 7(3):177-184.
26. Yüksel N, Ogras N. Canal surfaces with modified orthogonal frame in Minkowski 3-Space. Acta Universitatis Apulensis Mathematics-Informatics.2022;70.
27. Gür Mazlum S, Şenyurt S, Bektaş, M. Salkowski curves and their modified orthogonal frames in E3. Journal of New theory. 2022; 40, 12-26.
28. Arslan K, Çelik Y, Hacısalihoglu HH. On harmonic curvatures of Frenet curve. Commum. Fac. Sci. Univ. Ank. Series A1.2000; 49:15-23.
29. Izumiya S, Takeuchi N. New special curves and developable surfaces. Turkish Journal of Mathematics.2004; 28(2):153-164.