Killing–Hopf theorem - Misplaced Pages

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

Lee, John M. (2018). Introduction to Riemannian Manifolds. New York: Springer-Verlag. p. 348. ISBN 978-3-319-91754-2.

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

Basic concepts

Main results (list)

Types of
manifolds

Vectors	Distribution Lie bracket Pushforward Tangent space bundle Torsion Vector field Vector flow
Covectors	Closed/Exact Covariant derivative Cotangent space bundle De Rham cohomology Differential form Vector-valued Exterior derivative Interior product Pullback Ricci curvature flow Riemann curvature tensor Tensor field density Volume form Wedge product
Bundles	Adjoint Affine Associated Cotangent Dual Fiber (Co) Fibration Jet Lie algebra (Stable) Normal Principal Spinor Subbundle Tangent Tensor Vector
Connections	Affine Cartan Ehresmann Form Generalized Koszul Levi-Civita Principal Vector Parallel transport

Generalizations

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