Let be a closed, simply connected, smooth manifold. Let be the finite field with elements, where is a prime integer. Suppose that is an –elliptic space in the sense of Félix, Halperin and Thomas (1991). We prove that if the cohomology algebra cannot be generated (as an algebra) by one element, then any Riemannian metric on has an infinite number of geometrically distinct closed geodesics. The starting point is a classical theorem of Gromoll and Meyer (1969). The proof uses string homology, in particular the spectral sequence of Cohen, Jones and Yan (2004), the main theorem of McCleary (1987), and the structure theorem for elliptic Hopf algebras over from Félix, Halperin and Thomas (1991).
"String homology, and closed geodesics on manifolds which are elliptic spaces." Algebr. Geom. Topol. 16 (5) 2677 - 2690, 2016. https://doi.org/10.2140/agt.2016.16.2677