## Topological Methods in Nonlinear Analysis

### On symplectic manifolds with aspherical symplectic form

#### Abstract

We consider closed symplectically aspherical manifolds, i.e. closed symplectic manifolds $(M,\omega)$ satisfying the condition $[\omega]|_{\pi_2M}=0$. Rudyak and Oprea [On the Lustrnik–Schnirelmann category of symplectic manifolds and the Arnold conjecture, Math. Z. 230 (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.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 14, Number 2 (1999), 353-362.

Dates
First available in Project Euclid: 29 September 2016

https://projecteuclid.org/euclid.tmna/1475179849

Mathematical Reviews number (MathSciNet)
MR1766181

Zentralblatt MATH identifier
0973.55004

#### Citation

Rudyak, Yuli; Tralle, Aleksy. On symplectic manifolds with aspherical symplectic form. Topol. Methods Nonlinear Anal. 14 (1999), no. 2, 353--362. https://projecteuclid.org/euclid.tmna/1475179849

#### References

• \ref\key9 E. Kerman, Periodic orbits of Hamiltonian flows near symplectic critical submanifolds , Preprint, 1999, Math. DG/9903100 \ref\key10 L. A. Lusternik and L. G. Schnirelmann, Methodes Topologiques dáns le Problèmes Variationels, Hermann, Paris (1934) \ref\key11 Yu. Rudyak, On category weight and its applications , Topology, 38 (1999), 37–55 \ref\key12 ––––, On analytical applications of stable homotopy $($the Arnold conjecture, critical points$)$ , Math. Z., 230 (1999), 659–672 \ref\key13 ––––, Category weight: new ideas concerning Lusternik–Schnirelmann category , Homotopy and Geometry, Banach Center Publications