The Relation of the Morse Index of Closed Geodesics with the Maslovtype Index of Symplectic Paths

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