3-Boyutlu Minkowski Uzayında Bishop Çatısına Göre Sabit Eğrilikli Null Olmayan Regle Yüzeyler

Year 2020, Volume: 10 Issue: 3, 1997 - 2008, 01.09.2020

Cansu Tekin İsmail Aydemir

https://doi.org/10.21597/jist.702398

Abstract

Bu çalışmada, 3-boyutlu Minkowski uzayında Bishop çatısına göre null olmayan regle yüzeylerin Gauss ve ortalama eğrilikleri hesaplanmış ve regle yüzeylerin sabit Gauss ve ortalama eğriliğe sahip olmaları için gerekli koşullar elde edilmiştir. Ayrıca regle yüzeylerin açılabilir ve minimal olması için denklemlerinin hangi formda olması gerektiği gösterilmiştir. Buna ilaveten teğet vektör alanı tarafından üretilen sabit ortalama eğrilikli null olmayan regle yüzeylerin dayanak eğrisinin Bishop slant helis olduğu tespit edilmiştir.

Keywords

Regle Yüzey, Gauss Eğriliği, Ortalama Eğrilik, Bishop Çatısı, Minkowski uzayı

Non-null Ruled Surfaces with Constant Curvatures According to Bishop Frame in Minkowski 3-space

Abstract

In this study, the Gaussian and mean curvatures of non-null ruled surfaces according to Bishop frame in 3-dimensional Minkowski space are calculated and obtained the necessary conditions for their curvatures to be constant. Moreover, in case the ruled surfaces to be developable and minimal, we find parametric forms. In addition, the base curve of non-null surfaces with constant mean curvares is specified to be Bishop slant helix.
In this study, the Gaussian and mean curvatures of non-null ruled surfaces according to Bishop frame in 3-dimensional Minkowski space are calculated and obtained the necessary conditions for their curvatures to be constant. Moreover, in case the ruled surfaces to be developable and minimal, we find parametric forms. In addition, the base curve of non-null surfaces with constant mean curvares is specified to be Bishop slant helix.

Keywords

Ruled Surface, Gaussian Curvature, Mean Curvature, Bishop Frame, Minkowski space

References

• Abdel-All NH, Abdel-Baky RA, Hamdon FM, 2004. Ruled Surfaces with Timelike Rulings. Applied Mathematics and Computation, 147: 241-253.
• Ali AT, Aziz HSA, Sorour AH, 2013. Ruled Surfaces Generated by Some Special Curves in Euclidean 3-Space, 21: 285-294.
• Ali AT, 2017. Non-lightlike Ruled Surfaces with Constant Curvatures in Minkowski 3-Space. International Journal of Geometric Methods in Modern Physics, 15 (4): 1850068.
• Karacan MK, Bükçü B, 2008. "Bishop Frame Of The Timelike Curve İn Minkowski 3 Space", S.D.U. Fen-Edebiyat Fak. Fen Dergisi (E-Dergi), 3(1), 80-90.
• Bükçü B, Karacan MK, 2008. On the Slant Helices According to Bishop Frame of the Timelike Curve in Lorentzian Space. Tamkang Journal of Mathematics, 39 (3): 255-262.
• Hacısalihoğlu HH, 1983. Diferensiyel Geometri. İnönü Üniversitesi Fen Edeb. Fak. Yayınları, Malatya-Türkiye.
• Masal M, Azak AZ, 2017.3-Boyutlu Öklid Uzayında Bertnard Eğriler ve Bishop Çatısı. Sakarya Üniversitesi Fen Bilimleri Dergisi, , 21 (6): 1140-1145.
• O'neill B, 1983. Semi-Riemannian Geometry with Applications to Relativity, Academic press.
• Orbay K, Aydemir İ, 2010. The Ruled Surfaces Generated by Frenet Vectors of a Curve in R_1^3. Celal Bayar Üniversitesi Fen Bilimleri Dergisi, 6 (2): 155-160.
• Önder M, Uğurlu HH, 2013. Frenet Frames and İnvariants of Timelike Ruled Surfaces. Ain Shams Engineering Journal, 4: 507-513.
• Walrave, J. 1995. Curves and surfaces in Minkowski space.Doctoral Dissertation, Katholieke Universiteit Leuven Faculteit der Wetenschappen, 147, Belgium.
• Yerlikaya F, 2019. Minkowski Uzayında Eğrilerin Elde Edilmesi. Ondokuz Mayıs Üniversitesi, Fen Bilimleri Enstitüsü, Doktora Tezi (Basılmış).
• Yerlikaya F, Aydemir İ, 2020. Integrability for the Derivative Formulas of Rotation Minimizing Frame in Euclidean 3-Space and its Applications. International Electronic Journal of Geometry, 13:116-128.
• Yüksel N, 2013. The Ruled Surfaces According to Bishop Frame in Minkowski 3-Space, Abstract and Applied Analysis: 1-5.