3-Boyutlu Galilean Uzayında Tüp Yüzeyler Üzerine Bir Araştırma

Öz

Bu çalışmada, Galilean 3-uzayında tanımlanan eğri tarafından üretilen tüp yüzeyleri incelenmiş ve yüzeyler üzerindeki jeodeziklerin açıklanmasının bazı sonuçları da verilmiştir. Ayrıca, tüp yüzeyde jeodezik olma koşulları, spesifik enerjiyi oluşturmak için Clairaut's teoremi yardımıyla elde edildi. Spesifik enerjinin ve açısal momentumun fiziksel anlamı elbette ki fiziksel anlamın kendisiyle ilişkilidir. Sonuçlarımız tüp yüzeylerde elde edilen spesifik enerjinin ve açısal momentumun Galilean uzayında keyfi jeodezik eğri kullanılarak ifade edilebildiğini göstermektedir. Ayrıca, elde edilen ortalama ve Gauss eğrilikleri elde edilerek, bu yüzeyler için bazı karakterizasyonlar verildi.

Anahtar Kelimeler

• Galilean uzay
• tüp yüzeyleri
• jeodezik eğri
• spesifik kinetik enerji
• spesifik açısal momentum

A Survey on Tube Surfaces in Galilean 3-Space

Öz

In this study, the tube surfaces generated by the curve defined in Galilean 3-space are examined and some certain results of describing the geodesics on the surfaces are also given. Furthermore, the conditions of being geodesic on the tubular surface are obtained with the help of Clairaut’s theorem, which allows us to constitute the specific energy. The physical meaning of the specific energy and the angular momentum is of course related with the physical meaning itself. Our results show that the specific energy and the angular momentum obtained on tubular surfaces can be expressed using arbitrary geodesic curve in Galilean space. In addition, some characterizations are given for these surfaces, with the obtained mean and Gaussian curvatures.

Anahtar Kelimeler

• Galilean space
• tube surfaces
• geodesic curve
• specific kinetic energy
• specific angular momentum

Kaynakça

1. [1] Almaz F., Külahci M.A., “The notes on rotational surfaces in Galilean space,” International J. Geometric Methods in Modern Physics, 18(2): 2150017 (15 pages) , (2020).
2. [2] Ali A.T., “Position vectors of curves in the Galilean space, ” Matematicki Vesnik., 64(3): 200-210, (2012).
3. [3] Aminova A.V., “Pseudo-Riemannian manifolds with common geodesics,” Uspekhi Mat. Nauk., 48: 107-164, (1993).
4. [4] Dede M., “Tubular surfaces in Galilean space,” Math. Commun., 18: 209-217, (2013).
5. [5] Dede M., Ekici C., Çöken A.C., “On the Parallel Surfaces in Galilean Space,” Hacettepe J. Math. Statistics, 42(6): 605-615, (2013).
6. [6] Ekici C., Dede M., “On the Darboux vector of ruled surfaces in pseudo-Galilean space,” Mathematical and Computational Applications, 16(4): 830-838, (2011).
7. [7] Kasap E., Akyildiz F.T., “Surfaces with a Common Geodesic in Minkowski 3-Space,” App. Math. and Comp., 177: 260-270, (2006).
8. [8] Karacan M.K., Yayli Y., “On the geodesics of tubular surfaces in Minkowski 3-Space,” Bull. Malays. Math. Sci. Soc., 31: 1-10, (2008).

9. [9] Kazan A., Karadağ H.B., “Twisted Surfaces in the Pseudo-Galilean Space,” New Trends in Mathematical Sciences, 5(4): 72-79, (2017).
10. [10] Kazan A., Karadağ H.B., “Rotation Surfaces in 4-Dimensional Pseudo-Euclidean Spaces,” Orbit, 2(2): 2347-9051, (2014).
11. [11] Kazan A., Karadağ H.B., “Weighted Minimal and Weighted Flat Surfaces of Revolution in Galilean 3- Space with Density,” International J. Anal. and Apply., 16(3): 414-426, (2018).
12. [12] Kuhnel W., “Differential Geometry Curves-Surfaces and Manifolds,” Second Edition, American Math. Soc., Providence, RI, United States, (2005).
13. [13] Li C.Y., Wang R.H., Zhu C.G., “Parametric representation of a surface pencil with a common line of curvature,” Comput. Aid. Design, 43: 1110-1117, (2011).
14. [14] Milin-Šipuš Z., Divjak B., “Surfaces of constant curvature in the pseudo-Galilean space,” Int. J. Math. Math. Sci., 2012(2012), Art. ID 375264
15. [15] Ogrenmis A.O., Ergut M., Bektas M., “On the helices in the Galilean space, ” Iran. J. Sci. Technol. Trans. A Sci., 31(2): 177-181, (2007).
16. [16] Oztekin H.B., Tatlipinar S., “On some curves in Galilean plane and 3-dimensional Galilean space,” J. Dyn. Syst. Geom. Theor., 10(2): 189-196, (2012).
17. [17] Pressley A., “Elementary Differential Geometry,” second edition, Springer-Verlag London Limited, London, UK., (2010).
18. [18] Röschel O., “Die Geometrie des Galileischen Raumes,” Forschungszentrum Graz ResearchCentre, Austria, (1986).
19. [19] Röschel O., “Die Geometrie des Galileischen Raumes,” Bericht der Mathematisch Statistischen Sektion in der Forschungs-Gesellschaft Joanneum, Bericht Nr. 256, Habilitationsschrift, Leoben, (1984).
20. [20] Saad A., Low R.J., “A generalized Clairaut's theorem in Minkowski space,” J. Geometry and Symmetry in Phys., 35: 103-111, (2014).
21. [21] Walecka J.D., “Introduction to General Relativity,” World Scientific, Singapore, (2007)
22. [22] Walecka J.D., “Topics in Modern Physics: Theoretical Foundations,” World Scientific, (2013).
23. [23] Wang G.J., Tang K., Tai C.L, “Parametric Representation of a Surface Pencil with a Common Spatial Geodesic,” Comput. Aided Des., 36: 447-459, (2004).
24. [24] Yilmaz S., “Construction of the Frenet-Serret frame of a curve in 4D Galilean space and some applications,” Int. J. Phys. Sci., 5(5): 1284-1289, (2010).
25. [25] Yoon D.W., “Surfaces of Revolution in the three Dimensional Pseudo-Galilean Space,” Glasnik Math., 48(68): 415-428, (2013).