Geometry of Moving Spacelike Curves and their Evolution Equations in de Sitter 3-Space

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-23-2025-130
Published: May 26, 2025
Cite as:
M. Khalifa Saad, H. S. Abdel-Aziz, I. K. Youssef, Geometry of Moving Spacelike Curves and their Evolution Equations in de Sitter 3-Space, Int. J. Anal. Appl., 23 (2025), 130.

Main Article Content

M. Khalifa Saad, H. S. Abdel-Aziz, I. K. Youssef

Abstract

In this paper, we study the geometry of moving spacelike curves in the three-dimensional de Sitter space \(S_{1}^{3}\). Then, the evolution equations of the pseudo-orthonormal frame and the curvatures for these curves are derived. Moreover, some conditions for an inelastic curve flow in \(S_{1}^{3}\) are presented. Finally, interesting illustrative examples of the obtained results are given and plotted.

Article Details

References

  1. M.J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transformation, SIAM, Philadelphia, 1981.
  2. G.L. Lamb, Solitons on Moving Space Curves, J. Math. Phys. 18 (1977), 1654–1661. https://doi.org/10.1063/1.523453.
  3. H. Hasimoto, A Soliton on a Vortex Filament, J. Fluid Mech. 51 (1972), 477–485. https://doi.org/10.1017/S0022112072002307.
  4. P. Guha, Moving Space Curve Equations and a Family of Coupled KdV Type Systems, Chaos Solitons Fractals 15 (2003), 41–46. https://doi.org/10.1016/S0960-0779(02)00002-4.
  5. K.S. Chou, C. Qu, Motions of Curves in Similarity Geometries and Burgers-mKdV Hierarchies, Chaos Solitons Fractals 19 (2004), 47–53. https://doi.org/10.1016/S0960-0779(03)00060-2.
  6. K. Nakayama, H. Segur, M. Wadati, Integrability and the Motion of Curves, Phys. Rev. Lett. 69 (1992), 2603–2606. https://doi.org/10.1103/PhysRevLett.69.2603.
  7. M. Hisakado, M. Wadati, Moving Discrete Curve and Geometric Phase, Phys. Lett. A 214 (1996), 252–258. https://doi.org/10.1016/0375-9601(96)00207-1.
  8. K. Nakayama, Motion of Curves in Hyperboloid in the Minkowski Space, J. Phys. Soc. Japan 67 (1998), 3031–3037. https://doi.org/10.1143/JPSJ.67.3031.
  9. N.H. Abdel-All, M.T. Al-dossary, Motion of Curves Specified by Acceleration Field in R n , Appl. Math. Sci. 7 (2013), 3403–3418. https://doi.org/10.12988/ams.2013.33170.
  10. N.H. Abdel-All, R.A. Hussien, T. Youssef, Evolution of Curves via the Velocities of the Moving Frame, J. Math. Comput. Sci.2 (2012), 1170–1185.
  11. N.H. Abdel-All, M.A. Abdel-Razek, H.S. Abdel-Aziz, A.A. Khalil, Geometry of Evolving Plane Curves Problem via Lie Group Analysis, Stud. Math. Sci. 2 (2011), 51–62.
  12. N.H. Abdel-All, H.N. Abd-Ellah, H.S. Abdel-Aziz, M.A. Abdel-Razek, A.A. Khalil, Evolution of a Helix Curve by Observing Its Velocity, Life Sci. J. 11 (2014), 41–47.
  13. N. Gürbüz, Inextensible Flows of Spacelike, Timelike and Null Curves, Int. J. Contemp. Math. Sci. 4 (2009), 1599–1604.
  14. D.Y. Kwon, F.C. Park, Evolution of Inelastic Plane Curves, Appl. Math. Lett. 12 (1999), 115–119. https://doi.org/10.1016/S0893-9659(99)00088-9.
  15. D. Latifi, Inextensible Flows of Curves in Minkowskian Space, Adv. Stud. Theor. Phys. 16 (2008), 761–768.
  16. D.W. Yoon, Inelastic Flows of Curves According to Equiform in Galilean Space, J. Chungcheong Math. Soc. 24 (2011), 665–673.
  17. B. O’Neill, Semi-Riemannian Geometry, Academic Press, New York, 1983.
  18. H.S. Abdel-Aziz, M.K. Saad, A.A. Abdel-Salam, On Involute-Evolute Curve Couple in the Hyperbolic and de Sitter Spaces, J. Egypt. Math. Soc. 27 (2019), 25. https://doi.org/10.1186/s42787-019-0023-z.
  19. A.A. Abdel-Salam, M.K. Saad, Classification of Evolutoids and Pedaloids in Minkowski Space-Time Plane, WSEAS Trans. Math. 20 (2021), 97–105. https://doi.org/10.37394/23206.2021.20.10.
  20. M.K. Saad, H.S. Abdel-Aziz, A.A. Abdel-Salam, Evolutes of Fronts in de Sitter and Hyperbolic Spheres, Int. J. Anal. Appl. 20 (2022), 47. https://doi.org/10.28924/2291-8639-20-2022-47.
  21. H.S. Abdel-Aziz, H. Serry, M.K. Saad, Evolution Equations of Pseudo Spherical Images for Timelike Curves in Minkowski 3-Space, Math. Stat. 10 (2022), 884–893. https://doi.org/10.13189/ms.2022.100420.
  22. M.K. Saad, Geometrical Analysis of Spacelike and Timelike Rectifying Curves and Their Applications, Int. J. Anal. Appl. 22 (2024), 108. https://doi.org/10.28924/2291-8639-22-2024-3303.
  23. A.A. Abdel-Salam, M.I. Elashiry, M.K. Saad, On the Equiform Geometry of Special Curves in Hyperbolic and de Sitter Planes, AIMS Math. 8 (2023), 18435–18454. https://doi.org/10.3934/math.2023937.
  24. M.K. Saad, H.S. Abdel-Aziz, H.A. Ali, Geometry of Admissible Curves of Constant-Ratio in Pseudo-Galilean Space, Int. J. Anal. Appl. 21 (2023), 102. https://doi.org/10.28924/2291-8639-21-2023-102.