Periodic geodesics on compact riemannian manifolds

Abstract

The interest in periodic geodesies arose at a very early stage of differential geometry, and has grown rapidly since then. It is a basic general problem to estimate the number of distinct periodic geodesies c: R -• Λί, c(t + 1) = c(t) 9 on a complete riemannian manifold M in terms of topological invariants. Here periodic geodesies are always understood to be non-constant, and two such curves c l9 c 2 will be said to be distinct if they are geometrically different, c x (R) Φ c£R).If M is non-compact, then it is even difficult to find reasonable conditions for the existence of at least one periodic geodesic, see §4. However, in the compact case many results are available. It is classical and rather elementary that any non-trivial conjugacy class of the fundamental group n x {M) gives rise to periodic geodesies, and very often the existence of a larger number of distinct periodic geodesies can be deduced from further properties of π x (M) compare [4, p. 240], [7], and also §4. When M is simply connected* the problem is getting much more delicate. Here the first result was obtained in 1905 by Poincare, who proved that there exists a periodic geodesic on every surface analytically equivalent to the euclidean sphere S2 . Yet rather late, in 1952, Fet and Lusternik established the theorem that on any compact riemannian manifold M at least one geodesic is periodic. Several authors have proved the existence of certain finite numbers of distinct periodic geodesies for special topological types of manifolds, partly under restrictive metric conditions. We mention the work of Lusternik, Schnirelmann, Morse, Fet, Alber, and Klingenberg; for references see [11].In §4 of this paper we shall prove: There always exist infinitely many distinct periodic geodesies on an arbitrary compact manifold, provided some weak topological condition holds. In fact, it seems that our condition will be satisfied except in comparatively few cases. Until now it was not even possible to decide whether there is some compact simply connected differentiable manifold M with infinitely many distinct periodic geodesies for all riemannian structures on M.As yet the mast natural ami successful way of dealing with periodic geodesies

Local Floer homology and the action gap

“…The analogy with the Shub-Sullivan theorem and with the results of Gromoll and Meyer,[GrMe2], suggests a number of applications of Theorem 1.1 to the existence problem for periodic points of Hamiltonian diffeomorphisms. Namely, for some Hamiltonian diffeomorphisms of non-compact manifolds or symplectomorphisms arising in classical Hamiltonian dynamics, the rank of (filtered) Floer homology appears to grow with the order of iteration, and then the Hamiltonian Shub-Sullivan theorem implies the existence of infinitely many periodic orbits.…”

Book Review: Lectures on closed geodesics

“…By astute application of Bott's formulas for the indices of multiply covered geodesies and a careful examination of the degenerate case, GromoU and Meyer [4] were able to provide a surprising answer to the question of how many closed geodesies must exist on a manifold: If the sequence of Betti numbers oî /\ M is unbounded, there are infinitely many prime closed geodesies on M. In one direction this remarkable result is not a generalization of Lyusternik-Schnirelmann, for spheres are among the few manifolds which do not satisfy the topological hypothesis. Nor does the Gromoll-Meyer theorem say whether these geodesies can be found without self-intersections.…”

On the existence of infinitely many invariant Reeb orbits

“…In 1951 Lyusternik and Fet [LF51] proved that every closed Riemannian manifold (Q, g) has at least one closed geodesic. In 1969 Gromoll and Meyer [GM69] proved the following remarkable extension: if Q is a closed simply connected manifold with the property that the Betti numbers of the free loop space Λ(Q) are asymptotically unbounded, then every Riemannian metric g on Q has infinitely many embedded closed geodesics.…”