## Periodic magnetic curves in Berger spheres.(English)Zbl 1367.53040

A magnetic field $$F$$ on a Riemannian manifold $$(M,g)$$ is a closed $$2$$-form $$F$$, and the Lorentz force associated to $$F$$ is a tensor field $$\varphi$$ of type $$(1,1)$$ such that $$F(X,Y)=g(\varphi X,Y)$$. A curve $$\gamma$$ on $$M$$ that satisfies the Lorentz equation $$\nabla_{\dot\gamma}\dot\gamma=\varphi(\dot\gamma)$$ is called a magnetic trajectory of $$F$$, or simply a magnetic curve. When the magnetic curve $$\gamma(s)$$ is arc length parametrized, it is called a normal magnetic curve. A Riemannan manifold $$(M,g)$$ with an almost contact structure $$(\varphi,\xi,\eta)$$ satisfying $$\varphi^2=-\roman{id}+\eta\otimes\xi$$, $$\eta(\xi)=1$$, $$g(\varphi X,\varphi Y)=g(X,Y)-\eta(X)\eta(Y)$$, and $$\eta(X)=g(\xi,X)$$ is called an almost contact metric manifold. A $$2$$-form $$\Omega$$ defined as $$\Omega(X,Y)=g(\varphi X,Y)$$ is called the fundamental $$2$$-form on $$M$$. If $$\Omega=d\eta$$, then $$M$$ is called contact metric manifold, and if $$N_{\varphi}+2d\eta\otimes\xi=0$$, where $$N_{\varphi}$$ is the Nijenhuis tensor, then $$M$$ is called normal. A Sasakian manifold is defined as a normal contact manifold and is characterized by $$\nabla_X\xi=\varphi X$$. A plane section $$\Pi$$ at $$p\in M$$ is called a $$\varphi$$-section if it is invariant under $$\varphi$$. The sectional curvature $$k(\Pi)$$ of a $$\varphi$$-section is called the $$\varphi$$-sectional curvature of $$M$$ at $$p$$. A Sasakian manifold $$M$$ is said to be a Sasakian space form if it has constant $$\varphi$$-sectional curvature. A Sasakian space form $$M(c)$$ of constant $$\varphi$$-sectional curvature $$c$$ is said to be elliptic, parabolic, or hyperbolic if $$c>-3$$, $$c=-3$$, or $$c<-3$$, respectively. A $$2$$-form $$F_q$$ on a Sasakian manifold $$M$$ defined by $$F_q(X,Y)=q\Omega(X,Y)$$ is called the contact magnetic field with the strength $$q$$. The Lorentz force is defined as $$\varphi_q=q\varphi$$, and the Lorentz equation is $$\nabla_{\gamma'}\gamma'=q\varphi\gamma'$$, where $$\gamma:I\subset\mathbb R\to M$$ is a smooth curve parametrized by arc length.
In this paper, the authors investigate periodic magnetic curves in elliptic Sasakian space forms and obtain a quantization principle for periodic magnetic flowlines on Berger spheres. They also give a criterion for periodicity of magnetic curves on the unit sphere $$\mathbb S^3$$. It is proven that if $$\mathcal{M}^3(c)$$ is the $$3$$-dimensional Berger sphere equipped with a natural Sasakian structure of constant $$\varphi$$-sectional curvature $$c>-3$$, then the set of all periodic magnetic curves of arbitrary strength on the Sasakian space form $$\mathcal{M}^3(c)$$ can be quantized in the set of rational numbers. Also, the authors show that if $$\gamma$$ is a normal magnetic curve on the unit sphere $$\mathbb S^3$$, then $$\gamma$$ is periodic if and only if $$\frac{q}{\sqrt{q^2-4q\cos\theta+4}}\in\mathbb Q$$, where $$q$$ is the strength and $$\theta$$ is the constant contact angle of $$\gamma$$.

### MSC:

 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C30 Differential geometry of homogeneous manifolds 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010) 53C80 Applications of global differential geometry to the sciences
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