Kähler magnetic flows for a product of complex space forms
Let \((M,\langle, \rangle)\) be a Riemannian manifold and \(\pi:TM\to M\) the natural bundle projection. The metric \(g\) determines an isomorphism of \(TM\) with the cotangent bundle. Let \(\omega_0\) be the symplectic 2-form on \(TM\) formed by pulling back the canonical symplectic form \(\omega_{\text{can}}\) on \(T^\ast M\) by this isomorphism. Associated to a closed two-form \(B\) (magnetic field) on \(M\) one has also a symplectic form \(\omega=\omega_0+\pi^\ast B\) on \(TM\). Define \(H:TM\to \mathbb{R}\) by \(H(x,v)=\frac{1}{2}\langle v, v\rangle_x\). The Hamiltonian flow of \(H\) with respect to \(\omega\) is called the \textit{magnetic flow} of the pair \((\langle,\rangle, B)\), denoted by \(\phi=\{\phi^t\}\). It describes the motion of a unit charged particle on the Riemannian manifold \((M,\langle, \rangle)\) in the magnetic field \( B\). Consider the bundle map \(\Omega_B\colon TM\to TM\) given by \(B(u,v)=\langle u,\Omega_B(v)\rangle\) for all tangent vectors \(u,v \in T_x M\) at an arbitrary \(x\in M\). It is easily seen that for \(V\in TM\) the curve \(t\mapsto \phi^t(V)=(\gamma(t),\dot\gamma(t))\) is characterized by the fact that \(\nabla_{\dot\gamma}\dot\gamma=\Omega_B(\dot\gamma)\), where \(\nabla_{\dot\gamma}\) denotes covariant differentiation along \(\gamma\) with respect to the Riemannian connection \(\nabla\) on \(M\). There exist two methods to study the existence of periodic orbits of this magnetic flow. One is the variational method of multivalued functionals [cf. \textit{S. P. Novikov}, Sov. Math., Dokl. 24, 222--226 (1981); translation from Dokl. Akad. Nauk SSSR 260, 31--35 (1981; Zbl 0505.58011); and \textit{I. A. Taimanov}, Russ. Math. Surv. 47, 163--211 (1992); translation from Usp. Mat. Nauk 47, No.~2(284), 143--185 (1992; Zbl 0791.58026)]; another is the method of symplectic geometry [cf. \textit{V. I. Arnold}, Usp. Mat. Nauk 41, No. 6, 3--18 (1986; Zbl 0618.58021) and \textit{V. L. Ginzburg}, Prog. Math. 232, 139--172 (2005)]. In the paper under review the author considers a class of uniform magnetic fields (\(\nabla\Omega_B=0\)) given by a constant multiple of the Kähler form on a Kähler manifold \((M,\langle,\rangle,J)\) with complex structure \(J\) and investigates the semi-conjugacy of the induced magnetic flows.
scientific article; zbMATH DE number 2136474
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