### On symplectic manifolds with aspherical symplectic form

DOI: http://dx.doi.org/10.12775/TMNA.1999.038

#### Abstract

We consider closed symplectically aspherical manifolds, i.e. closed

symplectic manifolds $(M,\omega)$ satisfying the condition

$[\omega]|_{\pi_2M}=0$. Rudyak and Opre[< i> On the Lustrnik–Schnirelmann category of symplectic

manifolds and the Arnold conjecture< /i> , Math. Z.

< b> 230< /b> (1999), 673–678] remarked that such

manifolds have nice and controllable homotopy properties. Now it is

clear that these properties are mostly determined by the fact that the

strict category weight of $[\omega]$ equals 2. We apply the theory of

strict category weight to the problem of estimating the number of

closed orbits of charged particles in symplectic magnetic fields. In

case of symplectically aspherical manifolds our theory enables us to

improve some known estimations.

symplectic manifolds $(M,\omega)$ satisfying the condition

$[\omega]|_{\pi_2M}=0$. Rudyak and Opre[< i> On the Lustrnik–Schnirelmann category of symplectic

manifolds and the Arnold conjecture< /i> , Math. Z.

< b> 230< /b> (1999), 673–678] remarked that such

manifolds have nice and controllable homotopy properties. Now it is

clear that these properties are mostly determined by the fact that the

strict category weight of $[\omega]$ equals 2. We apply the theory of

strict category weight to the problem of estimating the number of

closed orbits of charged particles in symplectic magnetic fields. In

case of symplectically aspherical manifolds our theory enables us to

improve some known estimations.

#### Keywords

Lusternik-Schnirelmann category; sympletic manifolds

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