  In this paper, we use Chas-Sullivan theory on loop homology and Leray-Serre
spectral sequence to investigate the topological structure of the
non-contractible component of the free loop space on the real projective spaces
with odd dimensions. Then we apply the result to get the resonance identity of
non-contractible homologically visible prime closed geodesics on such spaces
provided the total number of distinct prime closed geodesics is finite.
