We examine the Erdos distinct distance problem on compact Riemannian \(2\)-manifolds. We prove that on a compact Riemannian manifold, a set of \(n\) points determines at least \(C_M\sqrt{n}\) distinct distances for some constant \(C_M\) depending only on \(M\) and for \(n\) large enough. In the process, we prove several results about geodesic existence and regularity, along with results about perturbations of Euclidean metrics. We finally describe a possible application of the Erdos problem to spectral sets on Riemannian manifolds similar to that in the Euclidean case.