Abstract

In this paper, we study natural paracontact magnetic trajectories in the unit tangent bundle, i.e., those that are associated to g-natural paracontact metric structures. We characterize slant natural paracontact magnetic trajectories as those satisfying a certain conservation law. Restricting to two-dimensional base manifolds of constant Gaussian curvature and to Kaluza–Klein type metrics on their unit tangent bundles, we give a full classification of natural paracontact slant magnetic trajectories (and geodesics).

Highlights

  • Introduction and Main ResultsMagnetic curves represent, in physics, the trajectories of charged particles moving on a Riemannian manifold under the action of magnetic fields

  • In physics, the trajectories of charged particles moving on a Riemannian manifold under the action of magnetic fields

  • For the particular case of the velocity vector field ċ of a unit-speed curve c of M, we prove that Ċ is not a natural paracontact magnetic trajectory unless c is a Riemannian circle and the metric Gon T1 M is of Kaluza–Klein type (Theorem 5)

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Summary

Introduction and Main Results

In physics, the trajectories of charged particles moving on a Riemannian manifold under the action of magnetic fields. For the particular case of the velocity vector field ċ of a unit-speed curve c of M (which is a curve of T1 M), we prove that Ċ is not a natural paracontact magnetic trajectory unless c is a Riemannian circle and the metric Gon T1 M is of Kaluza–Klein type (Theorem 5). We restrict to manifolds M of constant sectional curvature k and to pseudo-Riemannian g-natural metrics of Kaluza–Klein type on T1 M, and we characterize natural paracontact magnetic trajectories, which are slant, i.e., of constant contact angle. 4ak and to give a geometric insight to the second type of paracontact normal magnetic trajectories in Theorem 3, we will draw some pictures of slant magnetic curves along Riemannian circles on the unit tangent bundle of the hyperbolic plane of constant Gaussian curvature.

Natural Paracontact Metric Structures on Unit Tangent Bundles
Natural Paracontact Magnetic Curves in Unit Tangent Bundles
Contact Angle
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