Pseudo Quaterniyonik Lorentzian Evolute and İnvolute Eğriler İçin Yeni Bir Yaklaşım

Year 2022, Volume: 25 Issue: 3, 1369 - 1373, 01.10.2022

Müge Karadağ

https://doi.org/10.2339/politeknik.936090

Cited By: 1

Abstract

Bu çalışmada asıl amaç LQ^3 üzerindeki pseudo kuaterniyonik Lorentzian evolut-involut eğrilerini elde etmektir. Burada yapılması istenen, kuaterniyonları kullanarak 𝐿𝑄^4 deki kuaterniyonik time-like Lorentzian evolut-involut eğrilerini oluşturarak ve bu eğriler arasındaki bazı ilişkileri ortaya çıkarmaktır. Burada LQ^4 4-boyutlu Kuaterniyonik Lorenz Uzayını göstermektedir.

Keywords

Kuaterniyon uzayı, Lorentz uzayı, İkili işlem, Simerik bilineer form

A New Approach for the Pseudo-Quaternionic Lorentzian Evolute and Involute Curves

Abstract

In this study, the main purpose is to obtain pseudo quaternionic Lorentzian evolute-involute curves on L_Q^4. In fact the pseudo quaternion is a quaternion which has 3-dimension and its temporal part is ziro. So it has only vectoral part of the quaternions. Thus, if defined a pseudo-quaternionic curve on L_Q^4, then its quaternionic part is pseudo quaternion. What is meant to do here, to write out Lorentzian evolute-involute curves of the quaternionic time-like curves in L_Q^4 using quaternions and to reveal some relationships between these curves.

Keywords

Quaternion space, Lorentzian space, binary operation, symmetric bilinear form.

References

• [1]Hacısalihoğlu, H. H., "Diferensiyel Geometri", İnönü Üniv. Fen-Edebiyat Fakültesi Yayınları, No: 2; Malatya, 895s, (1983).
• [2]Hacısalihoğlu, H. H., "Hareket Geometrisi ve Kuaterniyonlar Teorisi", Hacısalihoğlu Yayıncılık, 1. Baskı / 338 Syf., (1983).
• [3]Bharathi, K. and Nagaraj, M., "Geometry of quaternionic and Pseudo-quaternionic multiplications", Ind. J. P. App. Math., 16(7): 741-756, (1985).
• [4]Bharathi, K. and Nagaraj, M., "Quaternion valued function of a real variable Serret-Frenet formulae", Ind. J. P. App. Math., 18(6): 507-511, (1987).
• [5]Bilici, M. and Çalışkan, "Some new notes on the involutes of the time-like curves in Minkowski 3-space", International Journal of Contemporary Mathematical Sciences, 6(41), 2019-2030, (2011).
• [6]Bilici, M. and Çalışkan, M., "On the involutes of space like curvewith a time like binormal in Minkowski 3-space", International Mathematical Forum, 5(31), 1997-1509, (2013).
• [7]Bükçü, B. and Karacan, M. K., "On the involute and evolute curves of spacelike curves with a spacelike binormal in Minkowski 3-space", International Journal of Mathematical Sciences, 2(5), 6(41), 221-232, (2007).
• [8]Erişir, T. and Güngör, M. A., "On the quaternionic curves in the semi-Euclidean space E_2^4", Caspian Journal of Mathematical Sciences, 1667, (2017).
• [9]Kalkan, Ö. B., Öztürk, H. and Zeybek, D., "T*N*B-Samarandache curves of involute-evolute curves in Minkowski 3-space", AKU J. Sci. Eng. 19, 011301, 71-78, (2019).
• [10]Altın, M., Kazan, A. and Karadağ, H.B. ‘’Nonnull Curves with Constant Weighted Curvature in Lorentz-Minkowski Plane with Density’’, Turkish Journal of Mathematics, 44(2),588-610,(2020).
• [11]Karadağ, H.B. and Karadağ, M. ‘’Null Generalized Slant Helices in Lorentzian Space’’ Differential Geometry-Dynamical System 10,178-185 (2008).
• [12]Karadağ, M. and Sivridağ, A. İ., "Kuaterniyonik Lorentz eğrileri üzerine bir çalışma", Politeknik Dergisi, 21(4): 937-940, (2018).
• [13]Karadağ, M. and Sivridağ, A. İ., "Some characterizations for a quaternion-valued and dual variable curve", Symmetry 11(2): 12, (2019).
• [14]O’Neill, B., “Semi-Riemannian Geometry with Applications to Relativity”, Academic Press,London, (1983).
• [15]Ozturk, U., Ozturk, E. B. K., Ilarslan, K., On the Involute-Evolute of the Pseudo null Curve in Minkowski 3-Space, Hindawi Publishing Corporation Journal of App. Math., V.,6,(2013).
• [16]Almaz, F. And Külahçı,A.M.,Involute Evolute D-Curves in Minkowski 3-space E31, Bol. Soc.Paran.Mat,V(39)1, 147-156,(2021).
• [17]Erişir, T. and Güngör, M. A.,On the Qaternionic Evolute-Involute Curves,4 November (2013).
• [18]Lone M.S., Es H., Karacan M.K., Bükcü B., On some curves with modified orthogonal frame in Euclidean 3-space, Iranian Journal of Science and Technology, Transactions A:Science,43(4):1905-1916, (2019).
• [19]Millman, R. S. and Parker , G. D., Elements of Differential Geometry, Prentice-Hall, Englewood Cliffs, NJ, USA, (1977).