Is it true that a constant curvature metric on a manifold $M$ is equivalent to a presentation of $M$ as a locally symmetric space $ M \cong \Gamma \backslash G/H $ where $G/H$ is a symmetric space and $ \Gamma $ a discrete subgroup of $G$?
I believe this is true based on the response of Jason DeVito to my question:
Is every surface locally symmetric?
And for a topological manifold M homeomorphic to a locally symmetric space is there any sense in which the triple $ (\Gamma, G, H) $ is unique? Or can the triple be picked to be minimal in some way? For example pick H connected and G simple or semisimple or simply connected or of minimal dimension etc...