On the Homogeneity Conjecture

Consider a connected homogeneous Riemannian manifold $(M,d{s}^2)$ and a Riemannian covering $(M,d{s}^2)\to \mathrm{\Gamma }\mathrm{\backslash }(M,d{s}^2)$. If $\mathrm{\Gamma }\mathrm{\backslash }(M,d{s}^2)$ is homogeneous then every $\gamma \in \mathrm{\Gamma }$ is an isometry of constant displacement. The Homogeneity Conjecture suggests the converse: if every $\gamma \in \mathrm{\Gamma }$ is an isometry of constant displacement on $(M,d{s}^2)$ then $\mathrm{\Gamma }\mathrm{\backslash }(M,d{s}^2)$ is homogeneous. We survey the cases in which the Homogeneity Conjecture has been verified, including some new results, and suggest some related open problems.