Abstract: In this paper we are concerned with a sufficient condition for a Riemannian metric on a compact simply–connected manifold to have infinitely many geometrically distinct closed geodesics. 1969 Gromoll–Meyer proved in [GM] that for every Riemannian metric on a compact simply–connected manifold M there are infinitely many geometrically distinct closed geodesics if the sequence (bi(ΛM ;F ))i of Betti numbers of the free loop space ΛM is unbounded for some field F . Using the theory of minimal models Vigue-Poirrier/Sullivan proved that the rational cohomology algebra H∗(M ; I Q) of M is generated by a single element if and only if the sequence (bi(ΛM ; I Q))i of Betti numbers of the free loop space ΛM of M is bounded. It is a conjecture that the same statement holds for all fields of prime characteristic, partial results are due to McCleary–Ziller [MZ] and Halperin/Vigue-Poirrier [HV]. Now we turn to the manifolds for which the hypothesis of Gromoll– Meyer’s theorem does not hold, for example spheres and projective spaces. Then stability properties of the closed geodesics become important. A closed geodesic is hyperbolic if the linearized Poincare map has no eigenvalue of norm 1. From the bumpy metrics theorem due to Abraham [Ab] and Anosov [An2] and from a pertubation result due to Klingenberg–Takens [KT] one can conclude: A C4–generic metric on a compact manifold has either a non–hyperbolic closed geodesic of twist type or all closed geodesics are hyperbolic. In the first case there are infinitely many geometrically distinct closed geodesics in every tubular neighborhood of the closed geodesic of twist type due to a theorem by Moser [Mo]. In the second case there are infinitely many closed geodesics if M is simply–connected due to results by Hingston [Hi] and the author [Ra1]. Hence it follows from these results that a C4–generic metric on a compact simply–connected manifold has infinitely many geometrically distinct closed geodesics. We remark that it is an open question whether there is a Riemannian metric on a simply–connected compact manifold all of whose closed geodesics are hyperbolic. In [Ra2] we show that the examples of metrics on the 2–