Killing–Hopf_theorem

Killing–Hopf theorem

Killing–Hopf theorem

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. 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, 95 (1): 313–339, doi:10.1007/BF01206614, ISSN 0025-5831
  • Killing, Wilhelm (1891), "Ueber die Clifford-Klein'schen Raumformen", Mathematische Annalen, 39 (2): 257–278, doi:10.1007/BF01206655, ISSN 0025-5831


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