A Review of the Index Method in Closed Geodesic Problem

In this paper we review and systematize the index method in closed geodesic problem.As we know,the closed geodesic problem on compact Riemannian or Finsler manifold is a famous problem,and has far from been resolved.In recent years,the Maslov-type index theory for symplectic path has been applied to studying the closed geodesic problem,and has induced a great number of results on the multiplicity and stability of closed geodesics.We will systematically introduce these progresses in this review.