Prolonged Kovaryant Türevi Kullanarak Weyl uzayındaki Bir Eğrinin Tip-1 Bishop Çatısına Göre Küresel Resimlerinin İncelenmesi

Öz

Bu çalışmada, Weyl uzayındaki bir eğrinin tip-1 Bishop çatısına göre küresel resimlerini inceledik. Ayrıca, Frenet-Serret ve tip-1 Bishop çatı aparatları arasındaki bağıntıları ifade ettik. Prolonged kovaryant türevi kullanarak, Weyl uzayında genel helis, slant helis, küresel eğri ve ayrıca çember kavramlarını tanımladık. Daha sonra, bu küresel resimlerin yukarıdaki tanımları sağlaması halinde, elde edilen şartlar birinci ve ikinci eğrilikler ve dolayısıyla Bishop eğrilikleri cinsinden ifade edildi. Bunlara ek olarak, bir eğrinin 𝑛1 ve 𝑛2 Bishop küresel resimlerinin Frenet-Serret vektör alanlarının oluşturduğu şebekenin, birinci cins Chebyshev şebekesi olma şartı ele alındı .

Anahtar Kelimeler

• Weyl uzayı
• Genel helis
• Slant helis
• Küresel eğri
• Küresel resim.

Investigating Spherical Images of a Curve According to Type-1 Bishop Frame in Weyl Space Using Prolonged Covariant Derivative

Abstract

In this study, we investigated spherical images of a curve according to type-1 Bishop frame in three dimensional Weyl space. Further, we expressed the relations among Frenet-Serret and type-1 Bishop frame apparatus. We defined the concepts of general helix, slant helix, spherical curve and also circle by using prolonged covariant derivative in Weyl space. Later, provided that these spherical images satisfy the above definitions, the conditions obtained were expressed in terms of first and second curvatures and hence Bishop curvatures. Additionally, the condition of being Chebyshev net of the first kind of the net which is occurred with Frenet-Serret vector fields of the 𝑛1 and 𝑛2 Bishop spherical images of a curve was discussed.

Keywords

• Weyl space
• General helix
• Slant helix
• Spherical curve
• Spherical image.

References

1. Bishop, R. L. (1975). There is more than one way to frame a curve. The American Mathematical Monthly, 82(3), 246-251.
2. Bükçü, B., & Karacan, M. K. (2008a). Special Bishop motion and Bishop Darboux rotation axis of the space curve. Journal of Dynamical Systems and Geometric Theories, 6(1), 27-34.
3. Bükçü, B., & Karacan, M. K. (2009). The slant helices according to Bishop frame. International Journal of Computational and Mathematical Sciences, 3(2), 67-70.
4. Yılmaz, S., Özyılmaz, E., & Turgut, M. (2010). New spherical indicatrices and their characterizations. Analele Ştiinţifice ale Universităţii Ovidius, 18(2), 337-354.
5. Bükçü, B., & Karacan, M. K. (2008b). On the slant helices according to Bishop frame of the timelike curve in Lorentzian space. Tamkang Journal of Mathematics, 39(3), 255-262.
6. Bükçü, B., & Karacan, M. K. (2007). The Bishop Darboux rotation axis of the spacelike curve in Minkowski 3-space. Ege University Journal of Faculty of Science, 3(1), 1-5.
7. Karacan, M. K., & Bükçü, B. (2008). Bishop frame of the timelike curve in Minkowski 3-space. Suleyman Demirel University Journal of Science, 3(1), 80-90.
8. Yılmaz, S. (2009). Position vectors of some special spacelikecurves according to Bishop frame in Minkowski space 𝑬𝟏𝟑. Scientia Magna, 5, 47-49.

9. Kofoğlu, N. (2020). Slant Helices According to Type-1 Bishop Frame in Weyl Space. International Mathematical Forum 15(4), 163-171.
10. Tsareva, B., & Zlatanov, G. (1990). On the geometry of the nets in the n-dimensional space of Weyl. Journal of Geometry, 38(1-2), 182-197.
11. Şemin, F. (1983). Differential Geometry I. İstanbul University.
12. Izumiya, S., & Takeuchi, N. (2004). New special curves and developable surfaces. Turkish Journal of Mathematics, 28(2), 153-164.
13. Nomizu, K. & Yano, K. (1974). On circles and spheres in Riemannian Geometry. Mathematische Annalen, 134, 163-170.