L^3 de Spacelike Odak Eğrisinin Harmonik Eğrilikleri Üzerine Bir Çalışma

Yıl 2025, Cilt: 6 Sayı: 2, 210 - 218, 31.12.2025

Ezgi Yağci , Esen İyigün

Öz

Bu makalede, [6]’da verilen, indeksi 1 olan Lorentz 3-uzay L^3’deki spacelike odak eğrisinin Frenet denklemlerini inceliyoruz. Eğrinin sırasıyla teğet, esas normal ve binormal Frenet vektörleri olan T, N ve W’yi ve Frenet denklemlerini kullanarak, spacelike odak eğrisi için bazı önemli tanımlar, teoremler ve sonuçlar veriyoruz. Ek olarak, indeksi 1 olan üç boyutlu Lorentz uzayındaki spacelike odak eğrisini, harmonik eğrilikler açısından inceledik. Son olarak, bir örnekle L^3’ deki spacelike odak eğrisini ve Lorentz dairesel helisini elde ettik.

Anahtar Kelimeler

lorentz uzayı , harmonik eğriler , odak eğrisi

A Study on Harmonic Curvatures of a Spacelike Focal Curve in L^3

Öz

In this paper,we study with Frenet equations which are given in [6] of a spacelike focal curve in Lorentz 3-space L^3 with index 1. By using the T, N and W are the tangent, the principal normal and the binormal Frenet vectors of the curve respectively and Frenet equations, we give some important definitions, theorems and results for spacelike focal curve. Addtionally, we examined a spacelike focal curve in the three-dimensional Lorentz space with index 1 in terms of harmonic curvatures. Finally, with a example obtained a spacelike focal curve and the Lorentzian circular helix, of a spacelike curve in L^3.

Anahtar Kelimeler

lorentzian space , harmonic curvates , focal curve

Kaynakça

• [1] İyigün, E. and Arslan K. (2005). On harmonic curvatures of curves in Lorentzian n-space. Communications Faculty of Sciences University Ankara Series A1 Mathematics and Statistcs, 54(1), 29-34.
• [2] Karacan, M. K., Bukcu, B. (2007). An alternative moving frame for tubular surfaces around spacelike curves with a spacelike binormal in the Minkowski 3-space, Mathematica Moravica, 11, 47-54.
• [3] O.Neill, B. (1983). Semi-Riemannian geometry with applications to relativity, Academic Press, New York.
• [4] Petrovic-Torgasev, M. and Sucurovic E. (2000). Some characterizations of the spacelike, the timelike and the null curves on the pseudohyperbolic space H_0^2 in E_1^3, Kragujevac Journal of Mathematics, 22, 71-82.
• [5] Yıldırım, A. (2016). Tubular surfaces around a timelike focal curve in Minkowski 3-space, International Electronic Journal of Pure and Applied Mathematics, 10(2), 103-113.
• [6] Yıldırım, A. (2018). Canal surface around a spacelike focal curve in Lorentzian 3-space, Süleyman Demirel University Journal of Natural and Applied Sciences, 22(2), 608-614.
• [7] Özdemir, M. (2004). On the focal curvatures of non-lightlike curves in minkowski (m+1)-space, Fırat Üniversitesi Fen ve Mühendislik Bilimleri Dergisi, 16(3), 401-409.
• [8] İlarslan, K. (2005). Spacelike Normal Curves in Minkowski Space E31. Turkish Journal of Mathematics, 29(1): 53-63.
• [9] İyigün, E. (2019). On timelike and null curves in R_1^3. Pioneer Journal of Advances in Applied Mathematics, 25(1-2):9-20.
• [10] Bonnor, W.B. (1985). Curves with null normals in Minkowski space-time. Wiley Easten Limited, 33-47.