Killing–Hopf theorem

In geometry, the Killing–Hopf theorem states that 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. These manifolds are called space forms. The Killing–Hopf theorem was proved by Killing (1891) and Hopf (1926).

References

• Hopf, Heinz (1926), "Zum Clifford-Kleinschen Raumproblem", Mathematische Annalen, Springer Berlin / Heidelberg, 95: 313–339, doi:10.1007/BF01206614, ISSN 0025-5831
• Killing, Wilhelm (1891), "Ueber die Clifford-Klein'schen Raumformen", Mathematische Annalen, Springer Berlin / Heidelberg, 39: 257–278, doi:10.1007/BF01206655, ISSN 0025-5831