Magnetic curves on flat para-K¨ahler manifolds
Magnetic curves on flat para-K¨ahler manifolds
In this paper we prove that spacelike and timelike magnetic trajectories corresponding to the para-K¨ahler 2-form on a para-K¨ahler manifold (M, P, g) are circles on M . We then classify all para-K¨ahler magnetic curves in pseudo-Euclidean spaces E 2n n .
___
- Adachi T. K¨ahler magnetic fields on a complex projective space. Proc Japan Acad 1994; 70A: 1213.
- Adachi T. K¨ahler Magnetic flow for a manifold of constant holomorphic sectional curvature. Tokyo J Math 1995; 18: 473483.
- Barros M, Cabrerizo JL, Fern´andez M, Romero A. Magnetic vortex filament flows. J Math Phys 2007; 48: 082904.
- Barros M, Romero A. Magnetic vortices. EPL 2007; 77: 34002.
- Barros M, Romero A, Cabrerizo JL, Fern´andez M. The Gauss-Landau-Hall problem on Riemannian surfaces. J Math Phys 2005; 46: 112905.
- Cabrerizo JL, Fern´andez M, G´omez JS. On the existence of almost contact structure and the contact magnetic field. Acta Math Hung 2009; 125: 191199.
- Cabrerizo JL, Fern´andez M, G´omez JS. The contact magnetic flow in 3D Sasakian manifolds. J Phys A Math Theor 2009; 42: 195201.
- Chen BY. Pseudo-Riemannian Geometry, δ -Invariants and Applications. Hackensack, NJ, USA: Word Scientific, 2011.
- Comtet A. On the Landau levels on the hyperbolic plane. Ann Phys 1987; 173: 185209.
- Cort´es V. The special geometry of Euclidean supersymmetry: a survey. Rev Union Mat Argent 2006; 47: 2934.
- Cort´es V, Lawn MA, Sch¨afer L. Affine hyperspheres associated to special para-K¨ahler manifolds. Int J Geom Methods M 2006; 3: 9951009.
- Cruceanu V, Fortuny P, Gadea PM. A survey on paracomplex geometry. Rocky Mt J Math 1996; 26: 83115.
- Drut¸˘a-Romaniuc SL, Inoguchi J, Munteanu MI, Nistor AI. Magnetic curves in Sasakian manifolds. J Nonlinear Math Phy 2015; 22: 428447.
- Drut¸˘a-Romaniuc SL, Inoguchi J, Munteanu MI, Nistor AI. Magnetic curves in cosymplectic manifolds (submitted manuscript).
- Drut¸˘a-Romaniuc SL, Munteanu MI. Magnetic curves corresponding to Killing magnetic fields in E 3 . J Math Phys 2011; 52: 113506.
- Drut¸˘a-Romaniuc SL, Munteanu MI. Killing magnetic curves in a Minkowski space. Nonlinear Anal Real 2013; 14: 383396.
- Etayo F, Santamar´ıa R, Tr´ıas UR. The geometry of a bi-Lagrangian manifold. Diff Geom Appl 2006; 24: 3359.
- Gadea PM, Montesinos Amilibia A. Spaces of constant paraholomorphic sectional curvature. Pacific J Math 1989; 136: 85101.
- Ikawa O. Motion of charged particles in Sasakian manifolds. SUT J Math 2007; 43: 263266.
- Inoguchi J, Munteanu MI. Periodic magnetic curves in Berger spheres. Tohoku Math J (in press).
- Kalinin D. Trajectories of charged particles in K¨ahler magnetic fields. Rep Math Phys 1997; 39: 299309.
- Libermann P. Sur le probl`eme d´equivalence de certains structures infinit´esimales. Ann Mat Pura Appl 1954; 36: 27120 (in French).
- Munteanu MI. Magnetic curves in the Euclidean space: one example, several approaches. Publ I Math-Beograd 2013; 94: 141150.
- Munteanu MI, Nistor AI. The classification of Killing magnetic curves in S 2 × R. J Geom Phys 2012; 62: 170182.
- Rashevskij PK. The scalar field in a stratified space. Trudy Sem Vektor Tenzor Anal 1948; 6: 225248.
- Rozenfeld BA. On unitary and stratified spaces. Trudy Sem Vektor Tenzor Anal 1949; 7: 260275.
- Ruse HS. On parallel fields of planes in a Riemannian manifold. Quart J Math Oxford Ser 1949; 20: 218234.
- Sunada T. Magnetic flows on a Riemann surface. In: Choi UJ, Kwak DY, Yim JW, editors. Proceedings of the KAIST Mathematics Workshop: Analysis and Geometry; 36 August 1993; Taejeon, Korea. Daejeon, Korea: KAIST, 1993, pp. 93108.