Minkowski Uzay-Zamanda Timelike Eğriler Arasındaki Bäcklund Dönüşümü Üzerine

Bu çalışmanın amacı, Minkowski uzay-zamanda timelike eğriler arasında Bäcklund dönüşümünü tanımlamaktır. Bu amaç doğrultusunda, timelike Bäcklund eğrilerin Frenet çatıları arasında ilişkiyi ortaya koyan dönme matrisinin seçimine bağlı olarak dönüşümü inceledik. İkisi spacelike hiperdüzlemde küresel dönme ve biri ise timelike hiperdüzlemde hiperbolik dönme olmak üzere üç farklı dönme matrisi durumu söz konusudur. Her durum için, timelike Bäcklund eğrilerinin eğrilik fonksiyonları arasındaki ilişki ortaya konmuştur. Bu arada, işaret farkı gözeterek timelike Bäcklund eğrilerin eşit ikinci burulma fonksiyonuna sahip olması gerektiği ispatlanmıştır. Bu aynı zamanda; Bäcklund dönüşümün bir sabit ikinci burulmaya sahip timelike eğriyi bir başka sabit ikinci burulmaya sahip timelike eğriye taşıyan dönüşüm olduğu anlamıma gelir.

Anahtar Kelimeler:

Minkowski uzay-zaman, Timelike eğriler, Bäcklund dönüşümü

On Bäcklund Transformation Between Timelike Curves in Minkowski Space-Time

The aim of this paper is to define Bäcklund transformation between two timelike curves in four dimensional Minkowski space. For this purpose, we examine the transformation depending on the choice of rotation matrix which determines the relations between Frenet frames of timelike Bäcklund curves. There are three different cases for rotation matrix; two of them are spherical rotations on the spacelike hyperplane and one of them is hyperbolical rotation on the timelike hyperplane. For each case, we get the relations between curvature functions of timelike Bäcklund curves. By the way, we prove that timelike Bäcklund curves must have equal constant second torsion functions up to sign. This also means that Bäcklund transformation is a transformation which maps a timelike curve with constant second torsion to another timelike curve with constant second torsion.

Keywords:

Minkowski space-time, Timelike curves, Bäcklund transformation,

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[1] Abdel-baky, R.A. The Bäcklund's theorem in Minkowski 3-space R_1^3. App. Math. and Comp. 160 (2005), 41-55.
[2] Rogers, C. and Schief, W.K. Bäcklund and Darboux Transformations, Geometry and Modern Applications in Soliton Theory. Cambridge Univ. Press, Cambridge. (2002).
[3] Rogers, C. and Shadwick, W.K.: Backlund Transformations and Their Applications (1st Edition). Academic Press. ISBN 0-12-592850-5.
[4] Schief, W.K. and Rogers, C.: Binormal motion of curves of constant curvature and torsion. Generation of soliton surfaces. Proc. R. Soc. Lond. 455, 3163-3188 (1999).
[5] Özdemir, M., Erdoğdu, M., Şimşek, H., Ergin,A.A. Bäcklund transformation for spacelike curves in the Minkowski space-time. Kuwait Journal of Science and Engineering. 41 (2014), 21-34.
[6] Şimşek, H. and Özdemir, M. Bäcklund's Theorem for n Dimensional Lorentzian Submanifold in Minkowski n space E_1^(2n-1). Results in Math. 69 (2016), 201-223.
[7] Gürses, M. Motion of curves on two-dimensional surfaces and soliton equations. Physics Letters A. 241(1998), 329-334.
[8] Deng, S-F. Bäcklund transformation and soliton solutions for KP equation. Chaos, Solitons & Fractals. 25 (2005), 475-480.
[9] Özdemir, M. and Erdoğdu, M. On the Rotation Matrix in Minkowski space-time. Reports on Mathematical Physics. 74(2014), 27-38.
[10] Gu, C.H., Hu H.S., Inoguchi, J.I. On time-like surfaces of positive constant Gaussian curvature and imaginary principal curvatures. Journal of Geometry and Physics. 41 (2002), 296-311.
[11] Tian, C. Bäcklund transformations for surfaces with K=-1 in R^2,1. Journal of Geometry and Physics 22 (1997), 212-218.
[12] Nemeth, S.Z. Bäcklund Transformations of n -dimensional constant torsion curves. Publi-cationes Mathematicae. 53(1998), 271-279.
[13] Nemeth, S.Z. Bäcklund transformations of constant torsion curves in 3-dimensional constant curvature spaces. Italian Journal of Pure and Applied Mathematics. 7(2000), 125-138.
[14] Aminov, Y. and Sym, A. On Bianchi and Bäcklund transformations of two-dimensional surfaces inE_1^4. Math. Phys. Anal. Geom. 3(2000), 75-89.
[15] Özdemir, M. and Çöken, A.C. Bäcklund transformation for non-lightlike curves in Minkowski 3-space. Chaos, Solitons & Fractals. 42 (2009), 2540-2545.
[16] Grbovic, M. and Nesovic, E. On Bäcklund transformation and vortex filament equation for null Cartan curve in Minkowski 3-space. Mathematical Physics, Analysis and Geometry. 19:23 (2016), 1-15.
[17] Grbovic, M. and Nesovic, E. On Bäcklund transformation and vortex filament equation for pseudo null curves in Minkowski 3-space. International Journal of Geometric Methods in Modern Physics 13 (2016), 1-14.
[18] Erdoğdu, M. and Özdemir, M. Reflections and Rotation in Minkowski 3-Space. Journal of Geometry, Symmetry and Physics. 39 (2015), 1-16.
[19] Erdoğdu, M., Özdemir, M. Cayley Formula in Minkowski Space-time. International Journal of Geometric Methods in Modern Physics. 12 (2015).
[20] O'Neill, B. Semi-Riemannian Geometry with Applications to Relativity. Academic Press Inc., London, 1983.