Conformal Weyl-Euler-Lagrangian equations on 4-Walker manifolds

Yıl 2016, Cilt: 4 Sayı: 2, 11 - 22, 01.03.2016

Zeki Kasap

Öz

The main purpose of the present paper is to study almost complex structures conformalWeyl-Euler-Lagrangian equations on 4-imensionalWalker manifolds for (conservative) dynamical systems. In this study, routes of objects moving in space will be modeled mathematically on 4-imensional Walker manifolds that these are time-dependent partial differential equations. A Walker n-manifold is a semi-Riemannian n-manifold, which admits a field of parallel null r-planes, with r <= n/2 . It is well-known that semi-Riemannian geometry has an important tool to describe spacetime events. Therefore, solutions of some structures about 4-Walker manifold can be used to explain spacetime singularities. Then, here we present complex analogues of Lagrangian mechanical systems on 4-Walker manifold. Also, the geometrical-physical results related to complex mechanical systems are also discussed for conformal Weyl-Euler-Lagrangian equations for (conservative) dynamical systems and solution of the motion equations using Maple Algebra software will be made.

Anahtar Kelimeler

Walker Manifolds, Weyl theory, holomorphic, symplectic geometry, conformal geometry, Lagrangian, mechanical system, Riemannian manifold

Kaynakça

• J. Klein, Escapes Variationnels Et M´ecanique, Ann. Inst. Fourier, Grenoble, 12, (1962).
• M. De Leon, P.R. Rodrigues, Methods of Differential Geometry in Analytical Mechanics, Elsevier Sc. Pub. Com. Inc., Amsterdam, (1989); 263-397.
• R. Abraham and J. E. Marsden, T. Ratiu, Manifolds, Tensor Analysis and Applications, Springer, (2001); 483-542.
• Z. Kasap and M. Tekkoyun, Mechanical Systems on Almost Para/Pseudo-K¨ahler–Weyl Manifolds, IJGMMP, Vol. 10, No. 5, (2013); 1-8.
• Z. Kasap, Weyl-Euler-Lagrange Equations of Motion on Flat Manifold, Advances in Mathematical Physics, (2015), 1-11.
• A.G. Walker, Canonical Form for A Rimannian Space with A Paralel Field of Null Planes, Quart. J. Math. Oxford, Vol. 1, No. 2, (1950); 69-79.
• A. Salimov, M. Iscan and K. Akbulut, Notes on Para-Norden–Walker 4-Manifolds, IJGMMP, Vol. 07, No. 08, (2010), 1331-1347.
• Y. Matsushita, Walker 4-Manifolds with Proper Almost Complex Structures, JGP, Vol. 55, (2005); 385-398.
• E. Garcia-Rio, S. Haze, N. Katayama and Y. Matsushita, Symplectic, Hermitian and K¨ahler Structures on Walker 4-Manifolds, J. Geom., 90, (2008); 56–65.
• W. Batat, G. Calvaruso and B. De Leo, On the Geometry of Four-DimensionalWalker Manifolds, Rendiconti di Matematica, Serie VII, Vol. 29, (2008); 163–173.
• M. Nadjafikhah and M. Jafari, Some General New Einstein Walker Manifolds, arXiv:1206.3730v1, (2012); 1-14.
• A. A. Salimov and M. Iscan, On Norden-Walker 4-manifolds, Note Di Matematica, Note Mat., 30, (2010); 111–128.
• M. Brozos-Vazquez, E. Garcia–Rio, P. Gilkey and R. Vazquez-Lorenzo, Tsankov–Videv Theory forWalker Manıfolds of Signature (2,2), SIGMA 3 (2007); 1-13.
• J. Davidova, J.C. Diaz-Ramosd, E. Garcia-Riob, Y. Matsushitac, O. Muskarova and R. Vazquez-Lorenzob, Almost K¨ahler Walker 4-Manifolds, JGP, 57, (2007); 1075–1088.
• M. Tekkoyun, A Survey on Geometric Dynamics of 4-Walker Manifold, JMP, 2, (2011); 1318-1323.
• P.R. Law, A Spinor Approach to Walker Geometry, arXiv:math/0612804v4, (2011); 1-10.
• Y. Matsushita, Four-Dimensional Walker Metrics and Symplectic Structures, Journal of Geometry and Physics, Vol. 52, (2004); 89-99.
• D. McDu and D. Salamon, J-holomorphic Curves and Quantum Cohomology, (1995).
• https://en.wikipedia.org/wiki/Conformal geometry.
• W.O. Straub, Simple Derivation of the Weyl Conformal Tensor, Pasadena, California, 2006.
• G.B. Folland, Weyl Manifolds, J. Differential Geometry, 4, (1970), 145-153.
• L. Kadosh, Topics in Weyl Geometry, Dissertationial, University of California, (1996):
• P. Gilkey, S. Nikˇcevi´c and U. Simon, Geometric Realizations, Curvature Decompositions, and Weyl Manifolds. JGP, 61, (2011),270–275.
• http://en.wikipedia.org/wiki/Conformally flat manifold.
• P. Gilkey, S. Nikˇcevi´c, K¨ahler-Weyl manifolds of Dimension 4, (2010); 1-11.
• https://en.wikipedia.org/wiki/Weyl tensor.
• https://en.wikipedia.org/wiki/Weyl transformation.
• M. Brozos-Vazquez, S. Nikcevic, P. Gilkey, E. Garcia-Rio and R. Vazquez-Lorenzo, The Geometry of Walker Manifolds, Vol. 2, No. 1, (2009), 1-179.
• H. Weyl, Space-Time-Matter, Dover Publ. (1952):
• B. Thid´e, Electromagnetic Field Theory, http://www.physics.irfu.se/CED/Book/EMFT Book.pdf, (2012).