Characterization of Constant-Curvature Metrics

Theorem

Let $M$ be a complete connected $n$-dimensional Riemannian manifold.

Let $M$ have constant sectional curvature.

Let $\tilde M$ be the Euclidean space, sphere, or the hyperbolic space with the constant sectional curvature.

Let $\Gamma$ be a discrete group of isometries of $\tilde M$ that acts freely on $\tilde M$.

Then, up to isometry, $M$ is the Riemannian quotient of the form $\tilde M / \Gamma$.