The contact magnetic flow in 3D Sasakian manifolds

Abstract

We first present a geometrical approach to magnetic fields in three-dimensional Riemannian manifolds, because this particular dimension allows one to easily tie vector fields and 2-forms. When the vector field is divergence free, it defines a magnetic field on the manifold whose Lorentz force equation presents a simple and useful form. In particular, for any three-dimensional Sasakian manifold the contact magnetic field is studied, and the normal magnetics trajectories are determined. As an application, we consider the three-dimensional unit sphere, where we prove the existence of closed magnetic trajectories of the contact magnetic field, and that this magnetic flow is quantized in the set of rational numbers.

Magnetic curves in Sasakian manifolds

“…There exist no non-geodesic circles as magnetic curves corresponding to the contact magnetic field F q for 0 < |q| ≤ 1, on M 2n+1 . Now, let us emphasize that the study of magnetic curves in 3-dimensional Sasakian manifolds was done by Cabrerizo et al in [15], obtaining the next result:…”

“…These have a very particular unitary Killing vector field, called the Reeb vector field, which naturally defines a magnetic field, called contact magnetic field. Contact magnetic fields on Sasakian 3-manifolds are investigated in [14,15]. Generally speaking, Sasakian manifolds may be regarded as the odd dimensional analogue of a Kähler manifold.…”

Some Properties of Special Magnetic Curves

“…The study of magnetic curves was extended to other ambient spaces, such as complex space forms [4,5], Sasakian 3-manifold [6,7], and so on. Very recent results of classification for the Killing magnetic trajectories on two special 3-dimensional manifolds, namely E 3 and S 2 × R, were obtained in [8,9], respectively.…”

Killing magnetic curves in three-dimensional almost paracontact manifolds

“…In fact, given a nontrivial magnetic field on a Riemannian manifold, there exists no linear connection, whose geodesics coincide with the magnetic curves of F [2, Prop. In recent years, magnetic curves have been intensively studied (see for example [1][2][3][4]6,8,10] and the references therein), since they are a natural object of investigation under several points of view. In fact, besides their interpretation as a generalization of geodesics, they also encode further geometrical and physical meanings.…”

“…The contact magnetic field determined by the characteristic vector field of any Sasakian three-manifold was studied in [3], determining the normal magnetic trajectories and proving that they are helices with axis the characteristic vector field itself. In dimension three, paracontact structures are the Lorentzian counterpart to contact Riemannian structures.…”