higher geometry / derived geometry
Ingredients
Concepts
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
Constructions
Examples
derived smooth geometry
Theorems
geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
A Clifford–Klein space form is a double coset space $\Gamma \backslash G/H$, where $G$ is a Lie group, $H$ a closed subgroup of $G$, and $\Gamma$ a discrete subgroup of $G$ that acts properly discontinuously and freely on the homogeneous space $G/H$.
The classical space forms are the cosets of the n-sphere $S^n$ or the Cartesian space $\mathbb{R}^n$ or the hyperbolic space $\mathbb{H}^n$ by discrete subgroups of their isometry group acting properly discontinuously (Carmo 92, chapt 8).
The properly discontinuous free quotients of n-spheres by discrete groups of isometries are also called spherical space forms. The classification of these was raised as an open problem by Killing 1891 and a complete solution was finally compiled by (Wolf 74). For more on this see at group actions on n-spheres.
synthetic geometry |
---|
Euclidean geometry |
hyperbolic geometry |
elliptic geometry |
Wilhelm Killing, Ueber die Clifford-Klein’schen Raumformen, Math. Ann. 39 (1891), 257–278 (doi:10.1007/BF01206655)
Heinz Hopf, Zum Clifford-Kleinschen Raumproblem, Math. Ann. 95, 313–339 (1926). (doi:10.1007/BF01206614, pdf)
Toshiyki Koyabashi, Discontinuous Groups and Clifford—Klein Forms of Pseudo-Riemannian Homogeneous Manifolds, (pdf)
Joseph Wolf, Spaces of constant curvature, Third ed.: Publish or Perish, Boston, 1974, Sixth edition: AMS Chelsea Publishing 2011 (doi:10.1090/chel/372)
M. P. do Carmo, Riemannian geometry, Mathematics: Theory & Applications, Birkhaeuser Boston Inc., Boston, MA, 1992, Translated from the second Portuguese edition by Francis Flaherty.
Ian Hambleton, Topological spherical space forms, Handbook of Group Actions (Vol. II), ALM 32 (2014), 151-172. International Press, Beijing-Boston (arXiv:1412.8187)
Last revised on October 27, 2021 at 11:17:21. See the history of this page for a list of all contributions to it.