New generalizations of Poincaré's geometric theorem (Q1110154)

Let \(S^ 1\to M\to^{p}B\) be an orientable bundle over a closed orientable manifold. Let \({\mathcal E}_ g(p)\) denote the set of all functions on M with the following properties: 1) their critical manifolds are smooth curves; 2) the projections of the critical manifolds on the base are homologous to zero. Let \({\mathcal E}_ a(p)\) denote the subset of \({\mathcal E}_ g(p)\), consisting of the functions f whose critical manifolds are non-degenerate in the sense of Bott. Denote by \(K_ g(p)\) (respectively, \(K_ a(p))\) the geometric (algebraic) criticality of the bundle p [i.e., the minimal number of critical manifolds of a function in \({\mathcal E}_ g(p)\) (respectively, \({\mathcal E}_ a(p))]\). Then the main result of this paper can be stated as follows: Let \(S^ 1\to M\to^{p}B\) be an orientable bundle over a closed, orientable, even- dimensional manifold B. Let \(\omega\) be a closed nondegenerate CGP 2-form on M such that the field of directions Ker \(\omega\) is \(C^ 1\)-close to the vertical. Then Ker \(\omega\) has at least \(K_ g(p)\)- and counting multiplicities, at least \(K_ a(p)\)- closed trajectories.