Narrow Results

In this paper, we introduce the notion of magnetic maps between Riemannian manifolds. They are generalizations of both magnetic curves and harmonic maps. We provide some fundamental examples of magnetic maps. Furthermore, we study some classes of magnetic surfaces in Euclidean 3-space.

We are interested in defining new energy functionals and solving them by using the variational approach method and Darboux equations. That is, we aim to define a new class of elastic curves on the regular surface $\mathrm{\Lambda }$. We further improve an alternative method to find critical points of the bending energy functionals acting on a class of magnetic curves on $\mathrm{\Lambda }$. As a result, we classify these critical curves as elastic magnetic curves of the Darboux vector family.

In this paper, we examine curves drawn by a timelike or spacelike curve$\alpha$ on the pseudo-hyperbolic sphere $H_0^2$ and the unit Lorentzian sphere $S_1^2$in three-dimensional Minkowski space ${ℝ}_1^3.$ We present Lorentz forces and magnetic curves produced by the Frenet frame $\{T_T,N_T,B_T\}$ on spherical indicatricies in a magnetic field of the curve $\alpha$. We calculate magnetic vector fields of magnetic curves for $T_T,N_T,B_T$. Furthermore, we define magnetic flux surfaces constructed by magnetic vector fields along magnetic spherical indicatricies. We obtain developability conditions for these surfaces. Finally, we illustrate magnetic flux surfaces with some interesting examples.

In this paper, our aim is to research the change in the magnetic field using the curvature theory of ruled surfaces. For this, we examine magnetic curves of the curve whose existence is guaranteed from the derivative formulas of the rotation frame. The Killing vector fields and Lorentz forces of these magnetic curves are calculated. Additionally, correlations between curvatures are obtained. Flux ruled surfaces are obtained and the conditions for developable are achieved. Finally, examples of the magnetic flux developable ruled surfaces are given.

In this paper we give an affirmative answer to sustain the conjecture about the order of a magnetic curve in a quasi-Sasakian manifold. More precisely, we show that the magnetic curves in a quasi-Sasakian manifold obtained as the product of a Sasakian and a Kähler manifold have maximum order 5. Moreover, we obtain the explicit parametrizations, the periodicity conditions and examples in the study of magnetic curves in 𝕊³ × 𝕊².