The Relation of the Morse Index of Closed Geodesics with the Maslov–type Index of Symplectic Paths

Abstract

In this paper, we consider the relation of the Morse index of a closed geodesic with the Maslov–type index of a path in a symplectic group. More precisely, for a closed geodesic c on a Riemannian manifold M with its linear Poincare map P (a symplectic matrix), we construct a symplectic path γ(t) starting from identity I and ending at P, such that the Morse index of the closed geodesic c equals the Maslov–type index of γ. As an application of this result, we study the parity of the Morse index of any closed geodesic.