On the existence of closed magnetic geodesics via symplectic reduction

Abstract

Let (M, g) be a closed Riemannian manifold and σ be a closed 2-form on M representing an integer cohomology class. In this paper, using symplectic reduction, we show how the problem of existence of closed magnetic geodesics for the magnetic flow of the pair (g, σ) can be interpreted as a critical point problem for a Rabinowitz-type action functional defined on the cotangent bundle T * E of a suitable S 1 -bundle E over M or, equivalently, as a critical point problem for a Lagrangian-type action functional defined on the free loopspace of E. We then study the relation between the stability property of energy hypersurfaces in (T * M, dp ∧ dq + π * σ) and of the corresponding codimension 2 coisotropic submanifolds in (T * E, dp∧dq) arising via symplectic reduction. Finally, we reprove the main result of [9] in this setting.

On geodesic flows with symmetries and closed magnetic geodesics on orbifolds

“…The next lemma shows that maximal flow-lines of X k that are defined on a finite interval have to approach vertical elements. The proof is analogous to the one of [11,Lemma 4.9] and will be omitted. Lemma 5.14.…”

“…The lack of such a compactness property poses therefore major difficulties and one is forced to look for additional informations in order to prove existence of critical points. An evidence of this is represented precisely by the functional S k : M → R. Indeed, in the case of S 1 -actions treated in [11], the Palais-Smale condition for S k does not hold on M, but rather on subsets M [T * ,T * ] ⊂ M of triples (γ, X, T ) with 0 < T * ≤ T ≤ T * . As it turns out, this is enough to show existence of critical points of S k -for almost every k -by means of a clever monotonicity argument, better known as the Struwe monotonicity argument [36] (for other applications we refer e.g.…”

“…We will then show that such a minimax function yields critical points of S k for almost every k > 1 2 . The proof for Theorem 1.1 follows closely [11]; however, some extra care is needed in the whole line of argument. Indeed, on the one hand the functional S k satisfies the Palais-Smale condition on M [T * ,T * ] only up to gauge transformations and, on the other hand, the construction of the minimax class requires techniques coming from rational homotopy theory and orbifold theoretical homotopy theory.…”

“…For that we will need the piece of additional information given by the following lemma. For the proof we refer to [11,Lemma 5.3]. Lemma 6.3.…”

“…The study of the geodesic flow of such a Riemannian manifold (Q, g Q ) is made particularly interesting by the fact that from the existence of geodesics which are closed up to G-action and satisfy some additional constraint (roughly speaking, the angle that such geodesics form with fundamental vector fields is in any point constant and equal to some a priori fixed angle), constrained G-closed geodesics for short, one obtains the existence of closed magnetic geodesics with given energy on the quotient orbifold Q/G. This approach has been already implemented in [11] in the particular case of free principal S 1 -bundle and yielded an alternative proof of the existence of one closed magnetic geodesic for almost evergy energy level on every closed nonaspherical Riemannian manifold equipped with a closed two-form representing an integer cohomology class.…”