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 \(\gamma\).


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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