Topological Methods in Nonlinear Analysis

On symplectic manifolds with aspherical symplectic form

Yuli Rudyak and Aleksy Tralle

Full-text: Open access


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

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

First available in Project Euclid: 29 September 2016

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Rudyak, Yuli; Tralle, Aleksy. On symplectic manifolds with aspherical symplectic form. Topol. Methods Nonlinear Anal. 14 (1999), no. 2, 353--362.

Export citation


  • \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