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

Inoguchi, Jun-Ichi; Munteanu, Marian Ioan

Tohoku Math. J. (2) 69, No. 1, 113-128 (2017).

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 \(\gamma\).

Reviewer: Andrew Bucki (Edmond)

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

Keywords:

magnetic field; Berger sphere; elliptic Sasakian space form; Hopf torus; periodic curve