New generalizations of Poincar�'s geometric theorem

Abstract

In this work we show that the number of closed trajectories of the field of kernels of a closed, nondegenerate, and center of gravity preserving 2-form on the total space of an oriented fiber bundle with fiber the circle over an orientable compact two-dimensional base is not less than the minimal number of critical points of a smooth function on the basis, under the assumption that the field of kernels is C1-close to a vertical field. Counting multiplicities, this number is not smaller than the minimal number of critical points of a Morse function on the base. We also give a lower estimate for the number of closed trajectories in the case of a higher-dimensional base. A form preserves.the center of gravity if its cohomology class is the lift of a class from the base. As an application we prove that the number of closed trajectories of a particle on a surface under the action of a strong and little varying magnetic field perpendicular to the surface is not smaller than the minimal number of critical points of a function on the given surface. The author is grateful to V. I. Arnol'd for formulating the problem on the closed trajectories of fields of kernels of differential forms preserving the center of gravity and for helping in work. I. Condition of Preservation of the Center of Gravity. Let S I + M ~ B be an orientable bundle over a closed (compact and boundaryless) orientable manifold B.