Kähler magnetic flows for a product of complex space forms
type: Item
instance of: scholarly article
title: Kähler magnetic flows for a product of complex space forms
zbMATH Open document ID: 1085.53075
author: Toshiaki Adachi
published in: Topology and its Applications
publication date: 2005-02-22T00:00:00Z
review text: 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 …
reviewed by: Guangcun Lu
Mathematics Subject Classification ID: 53D25
Mathematics Subject Classification ID: 37C15
Mathematics Subject Classification ID: 53D35
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 2136474
zbMATH Keywords: Kähler magnetic fields
zbMATH Keywords: complex space form
zbMATH Keywords: magnetic flows
zbMATH Keywords: conjugate
MaRDI profile type: Publication
full work available at URL: j.topol.2003.05.006
OpenAlex ID: W2049978779
cites work: Kähler magnetic flows for a manifold of constant holomorphic sectional curvature
cites work: A comparison theorem on magnetic jacobi fields
cites work: Circles in a complex projective space
cites work: Rigidity for periodic magnetic fields
cites work: The theorem of E. Hopf under uniform magnetic fields
DOI: 10.1016/J.TOPOL.2003.05.006
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description: scientific article; zbMATH DE number 2136474