On the Length Spectrum for Compact, Odd-dimensional, Real Hyperbolic Spaces
Abstract: We derive a prime geodesic theorem for compact, odd-dimensional, real hyperbolic spaces. The obtained result corresponds to the best known result obtained in the compact, even-dimensional case, as well as to the best known result obtained in the case of non-compact, real hyperbolic manifolds with cusps. The result derived in this paper follows from the fact that the prime geodesic theorem gives a growth asymptotic for the number of closed geodesics counted by their lengths, and the fact that free homotopy classes of closed paths on compact locally symmetric Riemannian manifold with negative sectional curvature are in natural one-to-one correspondence with the set of conjugacy classes of the corresponding discrete, co-compact, torsion-free group. The current article is dedicated to quotients of the real hyperbolic space.