  We enumerate a necessary condition for the existence of infinitely many
geometrically distinct, non-constant, prime closed geodesics on an arbitrary
closed Riemannian manifold $M$. That is, we show that any Riemannian metric on
$M$ admits infinitely many prime closed geodesics such that the energy
functional $E:\Lambda M\to\mathbb{R}$ has infinitely many non-degenerate
critical points on the free loop space $\Lambda M$ of Sobolev class
$H^1=W^{1,2}$. This result is obtained by invoking a handle decomposition of
free loop space and using methods of cellular homology to study its topological
invariants.
