TB_1 Lorentz Hybrid Partner Ruled Surfaces in R^3_1

Yıl 2025, Cilt: 7 Sayı: 2, 1 - 15, 29.12.2025

Müge Cengiz , Nevin Ertuğ Gürbüz

Öz

In this paper, we derive the first and second fundamental forms, as well as the Gaussian and mean curvatures, of $TB_{1}$ Lorentz hybrid partner ruled surfaces in three distinct cases. We establish the necessary and sufficient conditions for these surfaces to be developable and minimal with respect to the Lorentz hybrid frame in Minkowski 3-space. Moreover, we demonstrate that the $\sigma $-parameter curves are asymptotic but not geodesic, while the $v$-parameter curves are simultaneously geodesic and asymptotic. We investigate the necessary and sufficient conditions for the $\sigma $- and $v$-parameter curves of $TB_{1}$ Lorentz hybrid partner ruled surfaces to be simultaneously lines of curvature for three cases.

Anahtar Kelimeler

Partner ruled surfaces , Minimal surface , Minkowski 3 space , Hybrid frame

R^3_1' de TB_1 Lorentz Hibrit Regle Yüzeyler

Öz

Bu makalede, üç farklı durumda TB 1 Lorentz hibrit ortak regle yüzeylerinin birinci ve ikinci temel formlarını, ayrıca Gauss ve ortalama eğriliklerini çıkarıyoruz. Bu yüzeylerin Minkowski 3-uzayında Lorentz hibrit çatısına göre açılabilir ve minimal olmaları için gerekli ve yeterli koşulları belirliyoruz. Dahası, σ-parametre eğrilerinin asimptotik ancak jeodezik olmadığını, v-parametre eğrilerinin ise aynı anda jeodezik ve asimptotik olduğunu gösteriyoruz. Üç durum için TB 1 Lorentz hibrit ortak regle yüzeylerinin σ- ve v-parametre eğrilerinin aynı anda eğrilik çizgileri olmaları için gerekli ve yeter koşulları inceliyoruz.

Anahtar Kelimeler

Ortak regle yüzeyler , minimal yüzey , Minkowski 3-uzay , Hibrit çatı

Kaynakça

• Kreyszig, E. (1991). Differential Geometry. Dover Publications.
• Sentürk, G. Y., & Yüce, S. (2015). Characteristic proprerties of the ruled surface with Darboux frame. Kuwait Journal of Science, 42(2), 14–33.
• Turgut, A., & Hacısalihoğlu, H. (1998). Timelike ruled surfaces in the Minkowski 3-space-II. Turkish Journal of Mathematics, 22 (1), 33–46.
• Bishop, L. R. (1975). There is more than one way to frame a curve. The American Mathematical Monthly, 82(3), 246–251.
• Karacan, M. K., & Bükcü, B. (2008). Bishop frame the timelike curve in Minkowski 3-space. Suleyman Demirel University Journal of Science, 3(1), 80–90.
• Mofarreh, F., Abdel-Baky, R., & Naghi, M. F. (2021). Developable surfaces through spacelike sweeping surfaces in Minkowski 3-space, Appl. Math. Inf. Sci., 15(3), 263–270.
• Ouarab, S. (2021). Simultaneous developability of partner-ruled surfaces according to Darboux frame in E3. Abstract and Applied Analysis, 2021 (1), Article Id: 3151501, 1–9.
• Li, Y., Eren, K., Ayvacı, K. H., & Ersoy, S. (2022). Simultaneous characterizations of partner ruled surfaces using Flc frame. AIMS Mathematics, 7 (11), 20213–20229.
• Li, Y., Eren, K., & Ersoy, S. (2023). On simultaneous characterizations of partner-ruled surfaces in Minkowski 3-space. AIMS Mathematics, 8(9), 22256–22273.
• Eren, K., Ersoy, S., & Khan, M. N. I. (2025). Simultaneous characterizations of alternative partner-ruled surfaces. AIMS Mathematics, 10(4), 8891–8906.
• Arbind, A., Reddy, J.N., & Srinivasa, A. R. (2018). A one-dimensional model of 3-D structure for large deformation: a general higher-order rod theory. Acta Mechanica, 229, 1803–1831.
• Körpinar, T., Körpinar, Z.& Demirkol, R.C. (2023). Solitonic hybrid magnetic parallel transportation and energy distribution flows in Minkowski space. Optical and Quantum Electronics, 55(1146), 1–22.
• Ertuğ Gürbüz, N., Myrzakulov, R. & Myrzakulova, Z. (2022). Three anholonomy densities for three formulations with anholonomic coordinates with hybrid frame in R^3_1. Optik, 261, Article Id: 169161.
• Cengiz, M. (2025). On partner ruled surfaces (in Turkish). Master’s Thesis, Eskisehir Osmangazi University, Eskisehir.
• [15] O’Neill, B. (1983). Semi-Riemannian geometry with applications to relativity. Department of Mathematics University of California Los Angeles, Academic press, California.