Killing–Hopf theorem

{{Short description|Characterizes complete connected Riemannian manifolds of constant curvature}}

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.{{cite book |last=Lee |first=John M. |author-link=John M. Lee |date=2018 |title=Introduction to Riemannian Manifolds |url= |location=New York |publisher=Springer-Verlag |page=348 |isbn=978-3-319-91754-2}} These manifolds are called space forms. The Killing–Hopf theorem was proved by {{harvs|txt|last=Killing|authorlink=Wilhelm Killing|year=1891}} and {{harvs|txt|last=Hopf|authorlink=Heinz Hopf|year=1926}}.