Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
권호 표지
DOI QR코드

DOI QR Code

ON THE SPHERICAL INDICATRIX CURVES OF THE SPACELIKE SALKOWSKI CURVE WITH TIMELIKE PRINCIPAL NORMAL IN LORENTZIAN 3-SPACE

  • Birkan Aksan (Department of Mathematical Engineering, Gumushane University) ;
  • Sumeyye Gur Mazlum (Department of Computer Technology, Gumushane University)
  • Received : 2023.01.11
  • Accepted : 2023.03.28
  • Published : 2023.09.14
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we calculate Frenet frames, Frenet derivative formulas, curvatures, arc lengths, geodesic curvatures according to the Lorentzian 3-space ℝ31, Lorentzian sphere 𝕊21 and hyperbolic sphere ℍ20 of the spherical indicatrix curves of the spacelike Salkowski curve with the timelike principal normal in ℝ31 and draw the graphs of these indicatrix curves on the spheres.

Keywords

Acknowledgement

The authors sincerely thank the editors of the journal who devoted their attention to the paper, and the referees who contributed their valuable time to the study.

References

  1. R. A. Abdel-Baky, N. Alluhaibi, A. Ali, and F. Mofarreh, A study on timelike circular surfaces in Minkowski 3-space, International Journal of Geometric Methods in Modern Physics 17 (2020), no. 6, 2050074. 
  2. K. Akutagava and S. Nishikawa, The Gauss map and space-like surfaces with prescribed mean curvature in Minkowski 3-space, Tohoku Math. J. 42 (1990), 67-82. 
  3. A. T. Ali, Spacelike Salkowski and anti-Salkowski curves with timelike principal normal in Minkowski 3-space, Mathematica Aeterna 1 (2011), no. 4, 201-210. 
  4. A. T. Ali, Position vectors of slant helices in Euclidean 3-space, J. of the Egyptian Math. Soc. 20 (2012), 1-6.  https://doi.org/10.1016/j.joems.2011.12.005
  5. A. T. Ali and R. Lopez, Slant helices in Minkowski space E31, J. Korean Math. Soc. 48 (2011), no. 1, 159-167.  https://doi.org/10.4134/JKMS.2011.48.1.159
  6. N. Alluhaibi and R. A. Abdel-Baky, Kinematic Geometry of Timelike Ruled Surfaces in Minkowski 3-Space E31, Symmetry 14 (2022), no. 4, 749. 
  7. J. K. Beem, E. E. Paul and L. E. Kevin, Global Lorentzian Geometry, Routledge, 2017. 
  8. M. Bilici and M. Caliskan, Some new results on the curvatures of the spherical indicatrices of the involutes of a spacelike curve with a spacelike binormal in Minkowski 3-space, MathLAB J. 2 (2019), no. 1, 110-119. 
  9. M. Bilici and M. Caliskan, On the involutes of the space-like curve with a time-like binormal in Minkowski 3-space, Int. Math. Forum 4 (2009), no. 31, 1497-1509. 
  10. M. Bilici, E. Ergun, and M. C aliskan, A new approach to natural lift curves of the spherical indicatrices of timelike Bertrand mate of a spacelike curve in Minkowski 3-Space, International Journal of Mathematical Combinatorics 1 (2015), 35-48. 
  11. M. Bilici and M. C aliskan, Some geometrical calculations for the spherical indicatrices of involutes of a timelike curve in Minkowski 3-Space, Journal of Advances in Mathematics 5 (2014), no. 2, 668-677. 
  12. M. Bilici and M. C aliskan, New characterizations for spherical indicatrices of involutes of a spacelike curve with a timelike binormal in Minkowski 3-space, Journal of Science and Arts 22 (2022), no. 3, 629-638.  https://doi.org/10.46939/J.Sci.Arts-22.3-a09
  13. M. Bilici and S. Palavar, New-type tangent indicatrix of involute and ruled surface according to Blaschke frame in dual space, Maejo Int. J. Sci. Technol. 16 (2022), no. 3, 199-207. 
  14. M. Bilici and M. C aliskan, Some new results on the curvatures of the spherical indicatrices of the involutes of a spacelike curve with a spacelike binormal in Minkowski 3-space, MathLAB Journal 2 (2019), no. 1, 110-119. 
  15. G. S. Birman and K. Nomizu, Trigonometry in Lorentzian Geometry, Ann. Math. Mont. 91 (1984), 534-549. 
  16. B. Bukcu and M. K. Karacan, On the involute and evolute curves of the spacelike curve with a spacelike binormal in Minkowski 3-space, Int. J. Contemp. Math. Sciences 2 (2007), no. 5, 221-232.  https://doi.org/10.12988/ijcms.2007.07015
  17. F. Bulut and M. Bektas, Special helices on equiform differential geometry of spacelike curves in Minkowski space-time, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 69 (2020), 1045-1056. 
  18. S. Gur and S. Senyurt, Frenet vectors and geodesic curvatures of spheric indicators of Salkowski curve in E31, Hadronic J. 33 (2010), no. 5, 485-512. 
  19. S. Gur and S. Senyurt, Spacelike-timelike involute-evolute curve couple on dual Lorentzian space, J. Math. Comput. Sci. 3 (2013), no. 4, 1054-1075. 
  20. S. Gur Mazlum, Geometric properties of timelike surfaces in Lorentz-Minkowski 3-space, Filomat 37 (2023), no. 17, 5735-5749. 
  21. S. Gur Mazlum, S. Senyurt, and M. Bektas, Salkowski curves and their modified orthogonal frames in E3, Journal of New Theory 40 (2022), 12-26.  https://doi.org/10.53570/jnt.1140546
  22. H. H. Hacisalihoglu, Differential Geometry, ˙Inonu University, Publication of Faculty of Sciences and Arts: Malatya, Turkiye, 1983. 
  23. S. Izumiya and N. Tkeuchi, New special curves and developable surfaces, Turk J. Math. 28 (2004), 153-163. 
  24. S. Kilicoglu and H. Hacisalihoglu, On the b-scrolls with time-like generating vector in 3-dimensional Minkowski space, Beykent University Journal of Science and Technology 3 (2008), no. 2, 55-67. 
  25. M. Kulahci, M. Bektas, and M. Ergut, On harmonic curvatures of a Frenet curve in Lorentzian space, Chaos, Solitons and Fractals 41 (2009), 1668-1675.  https://doi.org/10.1016/j.chaos.2008.07.013
  26. Y. Li, M. T. Aldossary, and R. A. Abdel-Baky, Spacelike circular surfaces in Minkowski 3-Space, Symmetry 15 (2023), no. 1, 173. 
  27. Y. Li, S. Gur Mazlum, and S. Senyurt, The Darboux trihedrons of timelike surfaces in the Lorentzian 3-space, International Journal of Geometric Methods in Modern Physics 2022 (2022), 1-35. 
  28. Y. Li, Z. Chen, S. H. Nazra, and R. A. Abdel-Baky, Singularities for Timelike Developable Surfaces in Minkowski 3-Space, Symmetry 15 (2023), no. 2, 277. 
  29. Y. Li, F. Mofarreh, S. Dey, S. Roy, and A. Ali, General relativistic space-time with η1-Einstein metrics, Mathematics 10 (2022), no. 14, 2530. 
  30. R. Lopez, Differential geometry of curves and surfaces in Lorentz-Minkowski space, International Electronic Journal of Geometry 7 (2014), 44-107.  https://doi.org/10.36890/iejg.594497
  31. J. Monterde, Salkowski curves revisited: A family of curves with constant curvature and non-constant torsion, Computer Aided Geometric Design 26 (2009), no. 3, 271-278. 
  32. B. O'Neill, Semi-Riemannian Geometry with Applications to Rrelativity, Academic Press: London, England, 1983. 
  33. M. Ozdemir, Diferansiyel Geometri, Altin Nokta Printing and Distribution, Izmir, Turkiye, 2020. 
  34. J. G. Ratcliffe, Foundations of Hyperbolic Manifolds, Springer-Verlag, Tokyo, Japanese, 1994. 
  35. E. Salkowski, Zur transformation von raumkurven, Mathematische Annalen 66 (1909), no. 4, 517-557.  https://doi.org/10.1007/BF01450047
  36. M. Senatalar, Differential Geometry (Curves and Surfaces Theory), ˙Istanbul State Engineering and Architecture Academy Publications: Istanbul, Turkiye, 1978. 
  37. S. Senyurt and S. Gur, Spacelike surface geometry, International Journal of Geometric Methods in Modern Physics 14 (2017), 689-700. 
  38. S. Senyurt, S. Gur, and E. Ozyilmaz, The Frenet vectors and the geodesic curvatures of spherical indicatrix of the timelike Salkowski curve in Minkowski 3-space, Journal of Advanced Research in Dynamical and Control Systems 7 (2015), no. 4, 20-42. 
  39. S. Senyurt and E. Kemal, Smarandache curves of spacelike anti-Salkowski curve with a spacelike principal normal according to Frenet frame, Gumushane University, Journal of Science and Technology 10 (2020), 251-260. 
  40. S. Senyurt and B. Ozturk, Smarandache curves of Salkowski curve according to Frenet frame, Turkish Journal of Mathematics and Computer Science 10 (2018), 190-201. 
  41. D. J. Struik, Lectures on Classical Differential Geometry, Courier Corporation, 1961. 
  42. H. H. Ugurlu, On the geometry of time-like surfaces, Communications, Faculty of Sciences, University of Ankara, Al Series 46 (1997), 211-223. 
  43. H. H. Ugurlu and A. C aliskan, Darboux Ani Donme Vektorleri ile Spacelike ve Timelike Yuzeyler Geometrisi, Celal Bayar University Press, Manisa, Turkiye, 2012. 
  44. H. H. Ugurlu and A. Topal, Relation between Darboux instantaneous rotation vectors of curves on time-like surface, Mathematical and Computational Applications 1 (1996), 149-157.  https://doi.org/10.3390/mca1020149
  45. J. Walrave, Curves and surfaces in Minkowski space, Ph.D. Thesis, Katholieke Universiteit, Leuven, Belgium, 1995. 
  46. I. V. Woestijne, Minimal surfaces of the 3-dimensional Minkowski space, Geometry and Topology of Submanifolds: II, Word Scientific, Singapore, 1990. 
  47. N. Yuksel, B. Saltik, and E. Damar, Parallel curves in Minkowski 3-space, Gumushane University Journal of Science and Technology 12 (2014), no. 2, 480-486.