The Notes on Slant Helices According to Equiform Frame on Symplectic Space

Year 2024, Volume: 7 Issue: 6, 1241 - 1245, 15.11.2024

Esra Çiçek Çetin

https://doi.org/10.34248/bsengineering.1499614

Abstract

In this paper, first of all, we define basic definitions, some characterizations and theorems of symplectic space we calculated equiform frame in 4-dimensional symplectic space. Then, we obtain Frenet vectors and curvatures of a symplectic curve due to equiform frame. We have dealed with the properties of k-type slant helix according to equiform frame. It is seen that there exist k-type slant helices for all cases. In addition, we express some characterizations for k-type slant helix according to equiform frame geometry in symplectic regular curves. Finally, we give an example about symplectic space on graphics with symplectic frame on 4-dimensional symplectic space.

Keywords

Frenet vectors, Slant helix, Equiform frame

References

• Abdel-Aziz HS, Saad, MK, Abdel-Salam, AA. 2015. Equiform differential geometry of curves in Minkowski space-time. arXiv.org/math/ arXiv, 1501: 02283.
• Ali A, Lopez R, Turgut M. 2012. K-type partially null and pseudo null slant helices in Minkowski 4-space. Math Commun, 17: 93-103.
• Ali A, Lopez R. 2011. Slant helices in Minkowski space E₁³. J Korean Math Soc, 48: 159167.MR2778006.
• Ali AT, Turgut M. 2010. Some characterizations of slant helices in Euclidean space En, Hacet J Math Stat, 39(3): 327-336.
• Bulut F, Bektaş M. 2020. Special helices on equiform differential
• Bulut F, Eker A. 2023. Lorentz-Darboux çatısına göre k ve (k,m)−tip Slant Helisler, Iğdır Üniv Fen Bil Enst Derg, 13(2): 1237-1246. https://doi.org/10.21597/jist.1205226
• Bulut F, Tartık F. 2021. (k,m)-type Slant Helices according to parallel transport frame in Euclidean 4-Space. Turkish J Math Comput Sci, 13(2): 261-269. https://doi.org/10.47000/tjmcs.858489
• Bulut F. 2021a. Special helices on equiform differential geometry of timelike curves in E_1^4, Cumhuriyet Sci J, 42(4): 906-915. https://doi.org/10.17776/csj.962785
• Bulut F. 2021b. Slant Helices of (k,m)-type according to the ED-Frame in Minkowski 4-sSpace. Symmetry, 13(11): 2185-2201. https://doi.org/10.3390/sym13112185
• Bulut F. 2023, Darboux vector-based non-linear differential equations. Prespacetime J, 14(5): 533-543.
• Chern SS, Wang HC. 1947. Differential geometry in Symplectic spaces. Sci Rep Nat Tsing Hua, 1947: 57.
• Çiçek.Çetin E, Bektaş M. 2019. The characterizations of affine symplectic curves in R⁴. Mathemat, 7(1): 110
• Çiçek.Çetin E, Bektaş M. 2020a. K-type slant helices for symplectic curve in 4-dimensional symplectic space. Facta Univ Series, Math Inform, 2020: 641-646.
• Çiçek.Çetin E, Bektaş M. 2020b. Some new characterizations of symplectic curve in 4-dimensional symplectic space. Commun Adv Math Sci, 2(4): 331-334.
• Ferrandez A, Gimenĕz A, Lucas P. 2002. Null generalized helices in Lorentz-Minkowski space. J Phys A: Math Gen, 35: 8243-8251.
• Izumiya S, Takeuchi N. 2004. New special curves and developable surfaces. Turk J Math, 28: 153-163.
• Kamran N, Olver P. 2009. K. Tenenblat. Local symplectic invariants for curves. Commun Contemp Math, 11(2): 165-183.
• Kula L, Yaylı Y. 2005. On slant helix and its spherical indicatrix. App Math Comput, 169: 600-607.
• Önder M, Kazaz M, Kocayiğit H, Kılıç O. 2008.B₂-slant helix in Euclidean 4-space E⁴. Int J Cont Mat Sci, 3:1443-1440
• Struik DJ. 1988. Lectures on classical differential geometry. Dover, New York, US, pp: 143.
• Valiquette F. 2012. Geometric affine symplectic curve flows in R⁴. Diff Geo Appl, 30(6): 631-641.
• Yılmaz M, Bektaş M. 2018. Slant helices of (k,m) -type in E⁴. Acta Univ Sapientiae Math, 10(2): 395-401.
• Yılmaz M, Bektaş M. 2020. K, m-type slant helices for partially null and pseudo null curves in Minkowski space E₁⁴. Appl Math Nonlinear Sci, 5(1): 515-520.