Abstract
In the theory of curves, a magnetic field generates a magnetic flow whose trajectories are curves called magnetic curves. This paper aims at studying some properties for these curves which corresponding to the Killing magnetic fields in the 3-dimensional Euclidean space. We investigate the trajectories of the magnetic fields called $T$-magnetic and $e$-magnetic curves, also we give some characterizations of these curves. In addition, we determine all magnetic curves for new spherical images of a spherical curve and finally, we defray some examples to confirm our main results.
Highlights
The magnetic curves on a Riemannian manifold (M, g) are trajectories of charged particles moving on M under the action of a magnetic field F
We determine all magnetic curves for new spherical images of a spherical curve and we defray some examples to confirm our main results
Each trajectory γ may be found by solving the Lorentz equation ∇γ γ = φ(γ ), where φ is the Lorentz force corresponding to F and ∇ is the Levi Civita connection of g
Summary
The magnetic curves on a Riemannian manifold (M, g) are trajectories of charged particles moving on M under the action of a magnetic field F. Let γ : I → S2 ⊂ M 3 be a spherical curve in 3D oriented Riemannian space, (M 3, g) and F be a magnetic field on M. (Main result) Let γ be a spherical T-magnetic curve and V be a Killing vector field on a connected space form (M 3(C), g). The unit speed e-magnetic trajectories of (M 3(C), g, V ) are curves with curvature satisfying κg = 0, ρ = 1, C = 1, where C is the curvature of the Riemannian space M 3. Let γt be the et-magnetic curve and Vt be a Killing vector field on a connected space form (M 3(K), g).
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