Killing magnetic curves in three-dimensional almost paracontact manifolds

Abstract

For an arbitrary three-dimensional normal paracontact metric structure equipped with a Killing characteristic vector field, we obtain a complete classification of the magnetic curves of the corresponding magnetic field. In particular, this yields to a complete description of magnetic curves for the characteristic vector field of threedimensional paraSasakian and paracosymplectic manifolds. Explicit examples are described for the hyperbolic Heisenberg group and a paracosymplectic model.

Hopf magnetic curves in the anti-de Sitter space 𝔿13

“…If the ambient is a contact manifold, the fundamental two-form defines the so-called contact magnetic field. Interesting results are obtained when the manifold is Sasakian, namely, the angle between the velocity of a normal magnetic curve and the Reeb vector field is constant, and for their analogues of Lorentzian signature, that is, paraSasakian three-manifolds [11]. Moreover, an explicit description for normal flowlines of the contact magnetic field on a three-dimensional Sasakian manifold is known [13,14] (see also [16,21]).…”

On Pseudo-Hermitian Magnetic Curves in Sasakian Manifolds

“…Moreover, in [18], they studied magnetic curves on S 2n+1 . 3-dimensional normal para-contact metric manifolds and their magnetic curves of a Killing vector field were investigated in [5], by Calvaruso, Munteanu and Perrone. In [20], the present authors studied slant curves in contact Riemannian 3-manifolds with pseudo-Hermitian proper mean curvature vector field and pseudo-Hermitian harmonic mean curvature vector field for the Tanaka-Webster connection in the tangent and normal bundles, respectively.…”

The Equations and Characteristics of the Magnetic Curves in the Sphere Space

“…The magnetic curves on the Riemannian manifolds are trajectories of charged particles moving on under the magnetic field. Meanwhile, the different magnetic fields were extended to different ambient spaces [1][2][3][4][5][6][7][8][9][10][11][12]. Corresponding to parallel Lorentz forces, the magnetic trajectories are obtained on some 2-dimensional space [1,2].…”

“…In [3,4], the authors had researched the magnetic fields in complex space, which are called Kḧler form, and in Sasakian 3manifold. The classification of the magnetic curves in 3dimensional Minkowski space with Killing magnetic field and in three-dimensional almost paracontact manifolds was given in [5,6]. The authors obtained the magnetic trajectories as solutions of a variational problem that neither involves any local potential nor constraints the topology given a magnetic field in 3D [7].…”