On Darboux Frames of Indicatrices of Spacelike Salkowski Curve with Spacelike Binormal in E13

Abstract

The aim of this study is to examine Darboux frames and some other geometric properties (geodesic curvatures, geodesic torsions, normal curvatures, Darboux derivative formulas, Darboux vectors, angles, etc.) of the spherical indicatrices on Lorentzian unit sphere S_1^2 and hyperbolic unit sphere H_0^2 of the spacelike Salkowski curve with spacelike binormal in Lorentzian 3-space E_1^3. In this context, new and interesting results have been obtained for this curve. Thus, relationships between the newly obtained curvatures and torsions and the curvature and torsion of the original curve are given. Moreover, the matrix relationship between the Darboux and Frenet frames of these indicatrices is shown. Finally, the Darboux vectors belong to the Darboux frames and the Darboux vectors belong to the Frenet frames of these curves are compared.

Keywords

• Lorentzian space
• Salkowski curve
• Darboux frame
• Spherical indicatrices
• Curvatures

References

1. Aksan B, Gür Mazlum S. 2023. On the Spherical Indicatrix Curves of the Spacelike Salkowski Curve with Timelike Principal Normal in Lorentzian 3-Space. Honam Math J, 45(3): 513-541.
2. Ali AT. 2011. Spacelike Salkowski and anti-Salkowski curves with timelike principal normal in Minkowski 3-space. Math Aeterna, 1(4): 201-210.
3. Babaarslan M, Yaylı Y. 2017. On space-like constant slope surfaces and bertrand curves in Minkowski 3-space. Analele Stiintifice ale Universitatii Al I Cuza din Iasi-Matematica, 63(F2): 323-339.
4. Bilici M, Çalışkan M. 2019. Some new results on the curvatures of the spherical indicatrix curves of the involutes of a spacelike curve with a spacelike binormal in Minkowski 3-space. MathLAB J, 2(1): 110-119.
5. Birman GS, Nomizu K. 1984. Trigonometry in Lorentzian geometry. Ann Math Mont, 91: 534-549.
6. Bükcü B, Karacan MK. 2007. On the involute and evolute curves of the spacelike curve with a spacelike binormal in Minkowski 3-space. Int J Contemp Math Sci, 2(5): 221-232.
7. Fenchel W. 1951. On the differential geometry of closed space curves. Bull Am Math Soc, 57: 44-54.
8. Gür Mazlum S, Şenyurt S, Bektaş M. 2022. Salkowski curves and their modified orthogonal frames in E3. J New Theory, 40: 12-26.

9. Gür Mazlum S. 2023. Geometric properties of timelike surfaces in Lorentz-Minkowski 3-space. Filomat, 37(17): 5735-5749.
10. Gür S, Şenyurt S. 2010. Frenet vectors and geodesic curvatures of spheric indicatrix curves of Salkowski curve in E3. Hadronic J, 33(5): 485-512.
11. Hacısalihoğlu HH. 1983. Differential geometry. İnönü University, Publication of Faculty of Sciences and Arts, Malatya, Türkiye.
12. Kahveci D, Yaylı Y. 2002. Geometric kinematics of persistent rigid motions in three-dimensional Minkowski space. Mechanism Machine Theory, 167: 104535.
13. Kula L, Yaylı Y. 2005. On slant helix and its spherical indicatrix. Appl Math Comput, 169(1): 600-607.
14. Li Y, Gür Mazlum S, Şenyurt S. 2023. The Darboux trihedrons of timelike surfaces in the Lorentzian 3-space. Internationa J Geomet Methods Modern Physics, 20(2): 2350030-82.
15. Lopez R. 2014. Differential geometry of curves and surfaces in Lorentz-Minkowski space. Int E-J Geomet, 7: 44-107.
16. Monterde J. 2009. Salkowski curves revisited: A family of curves with constant curvature and non-constant torsion. Comp Aided Geomet Design, 26(3): 271-278.
17. O’Neill B. 1983. Semi-Riemannian geometry with applications to relativity. Academic Press, London, UK, pp: 488.
18. Özdemir M. 2020. Diferansiyel geometri. Altın Nokta Yayınevi, İzmir, Türkiye, pp: 132.
19. Ratcliffe JG. 1994. Foundations of hyperbolic manifolds. Springer-Verlag, Tokyo, Japan, pp: 779.
20. Salkowski E. 1909. Zur transformation von raumkurven. Math Annalen, 66(4): 517-557.
21. Şentürk GY, Yüce S. 2015. Characteristic properties of the ruled surface with Darboux frame in E-3. Kuwait J Sci, 42(2), 14-33.
22. Uğurlu HH, Çalışkan A. 2012. Darboux ani dönme vektörleri ile spacelike ve timelike yüzeyler geometrisi. Celal Bayar University Press, Manisa, Türkiye, pp: 12.
23. Uğurlu HH, Kocayiğit H. 1996. The Frenet and Darboux instantaneous rotation vectors of curves on time-like surface. Math Comp Appl, 1(2): 133-141.
24. Uğurlu HH. 1997. On the geometry of time-like surfaces. Communications, Faculty of Sciences, University of Ankara, A1 Series, No: 46, pp: 211-223.
25. Yakıcı Topbaş ES, Gök İ, Ekmekci FN, Yaylı Y. 2016. Darboux frame of a curve lying on a lightlike surface. Math Sci Appl E-Notes, 4(2): 121-130.
26. Yüksel N, Saltık B, Damar E. 2014. Parallel curves in Minkowski 3-space. Gümüşhane Univ J Sci Tech, 12(2): 480-486.