The Problem Of Closed Orbits Of Real Projective Spaces And Hamiltonian Systems

In this paper, we consider the problem of closed geodesics on the real projective spaces and the minimal period problem of the second order autonomous Hamiltonian systems with even po-tentials.In the first part, based on the works of Westerland [1] and [2] we use the theory of Leray-Serre spectral sequence and the work of Chas and Sullivan on the Batalin-Vilkovisky algebraic structure of the homology of the free loop space to obtain the following result:For the non-simply connected space M=RP2n-1, let LgM denote the space consisting of the non-contractible loops on M. Then, either dimkS1(LgM; Z2) is unbounded, or For the first case, we can use the theory of Gromoll and Meyer to prove that there are infinite different non-contractible closed geodesics on LgM. While if it is the second case, we can establish a mean index identity for closed geodesics on LgM. Notice that a similar mean index identity for closed geodesics on Sn is one of the foundation stones for a series of works in search of the second geodesic on the spheres by many authors such as Bangert, Rademacher, Y. Long, W. Wang and H. Duan, so we may anticipate that combining with Long’s precise iteration index formulae the mean index identity in this paper can be used to prove that there are at least two different non-contractible closed geodesics on M.By the same way, we have also proved that under the Cohen-Jones isomorphism, the two Batalin-Vilkovisky algebras for Sk with even k≥4are isomorphic, that is,(H*(LSk;Z2),·,â–³)(?)(HH*(H*(Sk),H*(Sk),Z2),U,â–³). Combining with the results for odd k≥3in [3], we get that the above two B-V algebras are isomorphic for every k≥3.While for S2, the above two algebras are not isomorphic due to Menichi[3]. Nevertheless, we can prove for any sphere Sk with k≥2that the Leray-Serre spectral sequence for the fibration LSkâ†'ES1×S1LSkâ†'BS1collapses at the third page.In the second part, we study the co-index iteration formulae for symplectic paths with ω=-1. Combing with the Maslov index theory, which can be seen as the case of co-index with ω=1, we obtain the precise iteration formulae of Morse indices for unorientable closed geodesics. We hope that the formulaes in this paper will be used in study of the problem of close geodesies on unorientable manifolds such as RP2n.In the third part, we consider the second order autonomous Hamiltonian system x+V’(x)=0,(?)x∈Rn, and have proved that if V(x)=V(-x) and there exists a constant θ>1such that0<θV’(x)·x≤V"(x)x-x,(?)x∈Rn\{0}, then for any T>0the above equation has at least a periodic solution with the minimal period T.