Magnetic curves in Sasakian manifolds

Abstract

In this paper we classify the magnetic trajectories corresponding to contact magnetic fields in Sasakian manifolds of arbitrary dimension. Moreover, when the ambient is a Sasakian space form, we prove that the codimension of the curve may be reduced to 2. This means that the magnetic curve lies on a 3-dimensional Sasakian space form, embedded as a totally geodesic submanifold of the Sasakian space form of dimension (2n + 1).

C-parallel and C-proper slant curves of S-manifolds

“…Then we have ε = 1, Remark. The order of a slant magnetic curve in an S-manifold is still r ≤ 3, as in the case of a magnetic curve of a Sasakian manifold, which was considered in [10]. Now, let us remove the slant condition from the hypothesis and show that the osculating order is still r ≤ 3.…”

“…where h α are arbitrary constants for α = 1, ..., s. Since γ is unit-speed, from (4.2), we have c 2 1 + ... + c 2 2n = 4 1 − s cos 2 θ . To sum up, we give the following Theorem: In particular, if s = 1, we obtain Theorem 3.5 in [10].…”

“…If q = 0 then the magnetic curves are geodesics of M 2n+s . Because of this reason we shall consider q = 0 (see [6] and [10]). From (1.1) and (2.5), the Lorentz force Φ associated to the contact magnetic field F q can be written as Φ q = −qϕ.…”

“…Magnetic trajectories have constant speed. If the speed of the magnetic curve γ is equal to 1, then it is called a normal magnetic curve [10].…”

On Slant Magnetic Curves in S-manifolds

“…Magnetic trajectories have constant speed. If the speed of the magnetic curve γ is equal to 1, then it is called a normal magnetic curve [6]. For fundamentals of almost contact metric manifolds, we refer to Blair's book [4].…”