Multiple closed geodesics on 3-spheres
Long and Duan call a closed geodesic rational if its linearized Poincaré map does not turn any invariant plane by an angle which is an irrational multiple of \(\pi\). They prove that if an irreversible Finsler sphere has a prime rational geodesic, then it has more than one. They prove that an irreversible Finsler sphere of dimension 3 has at least two distinct prime closed geodesics. They prove that on any compact simply connected 3-manifold with any Finsler metric, there are at least two geometrically distinct closed geodesics. The proof is based on examining the sublevel sets of the energy function on the loop space. They consider the natural inclusion of one sublevel set, bounded by one range of energies, into another one, bounded by a larger range of energies. This inclusion mapping, they prove, yields an isomorphism on homologies, in the presence of a prime closed geodesic, under certain hypotheses. This provides a new homological method to relate the alternating sum of Betti numbers of these sublevel sets to various geometric invariants of a closed geodesic.
scientific article; zbMATH DE number 5572198
scientific article