A Survey on Geometric Dynamics of 4-Walker Manifold

Author(s)
Mehmet Tekkoyun

ABSTRACT

A Walker n-manifold is a semi-Riemannian n-manifold, which admits a field of parallel null r-planes, with r ≤ 2/n . 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 and paracomplex analogues of Lagrangian and Hamiltonian mechanical systems on 4-Walker manifold. Finally, the geometrical-physical results related to complex (paracomplex) mechanical systems are also discussed.

A Walker n-manifold is a semi-Riemannian n-manifold, which admits a field of parallel null r-planes, with r ≤ 2/n . 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 and paracomplex analogues of Lagrangian and Hamiltonian mechanical systems on 4-Walker manifold. Finally, the geometrical-physical results related to complex (paracomplex) mechanical systems are also discussed.

Cite this paper

nullM. Tekkoyun, "A Survey on Geometric Dynamics of 4-Walker Manifold,"*Journal of Modern Physics*, Vol. 2 No. 11, 2011, pp. 1318-1323. doi: 10.4236/jmp.2011.211163.

nullM. Tekkoyun, "A Survey on Geometric Dynamics of 4-Walker Manifold,"

References

[1] M. De Leon and P. R. Rodrigues, “Methods of Differential Geometry in Analytical Mechanics,” North-Holland Mathematics Studies, Elsevier, Amsterdam, 1989.

[2] M. De Leon and P. R. Rodrigues, “Second-Order Differential Equations and Non-Conservative Lagrangian Mechanics,” Journal of Physics A: Mathematical and General, Vol. 20, 1987, pp. 393-5396.

[3] M. Zambine, “Hamiltonian Perspective on Generalized Complex Structure,” Communications in Mathematical Physics, Vol. 263, 2006, pp. 711-722. doi:10.1007/s00220-005-1512-5

[4] M. Tekkoyun and M. Gorgülü, “Higher Order Complex Lagrangian and Hamiltonian Mechanics Systems,” Physical Letters A, Vol. 357, 2006, pp. 261-269. doi:10.1016/j.physleta.2006.04.049

[5] M. Tekkoyun, “On Para-Euler-Lagrange and Para-Hamiltonian Equations,” Physical Letters A, Vol. 340, No. 1-4, 2005, pp. 7-11.

[6] A. G. Walker, “Canonical Form for a Rimannian Space with a Paralel Field of Null Planes,” Quarterly of Applied Mathematics Oxford, Vol. 1, No. 2, 1950, pp. 69-79.

[7] R. Ghanam and G. Thompson, “The Holonomy Lie Algebras of Neutral Metrics in Dimension Four,” Journal of Mathematical Physics, Vol. 42, 2001, pp. 2266-2284. doi:10.1063/1.1362284

[8] Y. Matsushita, “Four-Dimensional Walker Metrics and Symplectic Structures,” Journal of Geometry and Physics, Vol. 52, 2004, pp. 89-99. doi:10.1016/j.geomphys.2004.02.009

[9] J. Carlos Diaz-Ramos, E. Garcia-Rio and R. Vazquez- Lorenzo, “Osserman Metrics on Walker 4-Manifolds Equipped with a Para-Hermitian Structure,” Matematica Contemporanea, Vol. 30, 2006, pp. 91-108.

[10] M. Chaichi, E. Garca-Ro and Y. Matsushita, “Curvature Properties of Four-Dimensional Walker Metrics,” Classical and Quantum Gravity, Vol. 22, No. 3, 2005, pp. 559-577. doi:10.1088/0264-9381/22/3/008

[11] J. Davidov and O. Mukarov, “Self-DualWalker Metrics with Two-Step Nilpotent Ricci Operator,” Journal of Geometry and Physics, Vol. 57, 2006, pp. 157-165. doi:10.1016/j.geomphys.2006.02.007

[12] J. C. Diaz-Ramos, E. Garca-Ro and R. Vazquez-Lorenzo, “New Examples of Osserman Metrics with Nondiagonalizable Jacobi Operators,” Differential Geometry and Its Applications, Vol. 24, 2006, pp. 433-442. doi:10.1016/j.difgeo.2006.02.006

[13] E. Garca-Ro, Z. Raki and M. E. Vazquez-Abal, “Four- Dimensional Indefinite Kahler Osserman Manifolds,” Journal of Mathematical Physics, Vol. 46, 2005, p. 073505 (11 Pages).

[14] Y. Matsushita, “Walker 4-Manifolds with Proper Almost Complex Structures,” Journal of Geometry and Physics, Vol. 55, 2005, pp. 385-398. doi:10.1016/j.geomphys.2004.12.014

[15] E. Garca-Ro, S. Haze, N. Katayama and Y. Matsushita, “Symplectic, Hermitian and K?hler Structures on Walker 4-Manifolds,” Journal of Geometry, Vol. 90, 2008, pp. 56-65. doi:10.1007/s00022-008-1999-y

[1] M. De Leon and P. R. Rodrigues, “Methods of Differential Geometry in Analytical Mechanics,” North-Holland Mathematics Studies, Elsevier, Amsterdam, 1989.

[2] M. De Leon and P. R. Rodrigues, “Second-Order Differential Equations and Non-Conservative Lagrangian Mechanics,” Journal of Physics A: Mathematical and General, Vol. 20, 1987, pp. 393-5396.

[3] M. Zambine, “Hamiltonian Perspective on Generalized Complex Structure,” Communications in Mathematical Physics, Vol. 263, 2006, pp. 711-722. doi:10.1007/s00220-005-1512-5

[4] M. Tekkoyun and M. Gorgülü, “Higher Order Complex Lagrangian and Hamiltonian Mechanics Systems,” Physical Letters A, Vol. 357, 2006, pp. 261-269. doi:10.1016/j.physleta.2006.04.049

[5] M. Tekkoyun, “On Para-Euler-Lagrange and Para-Hamiltonian Equations,” Physical Letters A, Vol. 340, No. 1-4, 2005, pp. 7-11.

[6] A. G. Walker, “Canonical Form for a Rimannian Space with a Paralel Field of Null Planes,” Quarterly of Applied Mathematics Oxford, Vol. 1, No. 2, 1950, pp. 69-79.

[7] R. Ghanam and G. Thompson, “The Holonomy Lie Algebras of Neutral Metrics in Dimension Four,” Journal of Mathematical Physics, Vol. 42, 2001, pp. 2266-2284. doi:10.1063/1.1362284

[8] Y. Matsushita, “Four-Dimensional Walker Metrics and Symplectic Structures,” Journal of Geometry and Physics, Vol. 52, 2004, pp. 89-99. doi:10.1016/j.geomphys.2004.02.009

[9] J. Carlos Diaz-Ramos, E. Garcia-Rio and R. Vazquez- Lorenzo, “Osserman Metrics on Walker 4-Manifolds Equipped with a Para-Hermitian Structure,” Matematica Contemporanea, Vol. 30, 2006, pp. 91-108.

[10] M. Chaichi, E. Garca-Ro and Y. Matsushita, “Curvature Properties of Four-Dimensional Walker Metrics,” Classical and Quantum Gravity, Vol. 22, No. 3, 2005, pp. 559-577. doi:10.1088/0264-9381/22/3/008

[11] J. Davidov and O. Mukarov, “Self-DualWalker Metrics with Two-Step Nilpotent Ricci Operator,” Journal of Geometry and Physics, Vol. 57, 2006, pp. 157-165. doi:10.1016/j.geomphys.2006.02.007

[12] J. C. Diaz-Ramos, E. Garca-Ro and R. Vazquez-Lorenzo, “New Examples of Osserman Metrics with Nondiagonalizable Jacobi Operators,” Differential Geometry and Its Applications, Vol. 24, 2006, pp. 433-442. doi:10.1016/j.difgeo.2006.02.006

[13] E. Garca-Ro, Z. Raki and M. E. Vazquez-Abal, “Four- Dimensional Indefinite Kahler Osserman Manifolds,” Journal of Mathematical Physics, Vol. 46, 2005, p. 073505 (11 Pages).

[14] Y. Matsushita, “Walker 4-Manifolds with Proper Almost Complex Structures,” Journal of Geometry and Physics, Vol. 55, 2005, pp. 385-398. doi:10.1016/j.geomphys.2004.12.014

[15] E. Garca-Ro, S. Haze, N. Katayama and Y. Matsushita, “Symplectic, Hermitian and K?hler Structures on Walker 4-Manifolds,” Journal of Geometry, Vol. 90, 2008, pp. 56-65. doi:10.1007/s00022-008-1999-y