Here is the statement of Killing Hopf Theorem in Wikipedia:
Complete connected Riemannian manifolds of constant curvature are isometric to a quotient of a sphere, Euclidean space, or hyperbolic space by a group acting freely and properly discontinuously
I'm a little confused about what this means in practice, perhaps due to a confusion about quotient spaces. Is this just saying that if I start with a manifold of constant curvature then I can apply an isometry and get one of the three space forms?