Editing Clifford–Klein form

In [[mathematics]], a '''Clifford–Klein form''' is a [[double coset]] space

:{{math|Γ\''G''/''H''}},

where ''G'' is a [[Semisimple Lie group|reductive]] [[Lie group]], ''H'' a closed subgroup of ''G'', and Γ a [[discrete subgroup]] of G that acts [[properly discontinuously]] on the [[homogeneous space]] ''G''/''H''.  A suitable discrete subgroup Γ may or may not exist, for a given ''G'' and ''H''. If Γ exists, there is the question of whether {{math|Γ\''G''/''H''}} can be taken to be a [[compact space]], called a '''compact Clifford–Klein form'''.

When ''H'' is  itself compact, classical results show that a compact Clifford–Klein form exists. Otherwise it may not, and there are a number of negative results.

==History==
According to [[Moritz Epple]], the Clifford-Klein forms began when [[W. K. Clifford]] used [[quaternion]]s to ''twist'' their space. "Every twist possessed a space-filling family of invariant lines", the [[Clifford parallel]]s. They formed "a particular structure embedded in elliptic 3-space", the [[Clifford surface]], which demonstrated that "the same local geometry may be tied to spaces that are globally different." [[Wilhelm Killing]] thought that for free mobility of rigid bodies there are four spaces: Euclidean, hyperbolic, elliptic and spherical. They are spaces of [[constant curvature]] but constant curvature differs from free mobility: it is local, the other is both local and global. Killing's contribution to Clifford-Klein space forms involved formulation in terms of [[group (mathematics)|group]]s, finding new classes of examples, and consideration of the scientific relevance of spaces of constant curvature. He took  up  the task to develop physical theories of CK space forms. [[Karl Schwarzchild]] wrote “The admissible measure of the curvature of space”, and noted in an appendix that physical space may actually be a non-standard space of constant curvature.

==See also==
* [[Killing-Hopf theorem]]
* [[Space form]]

==References==
* [[Moritz Epple]] (2003) [https://archive.org/details/arxiv-math0305023 From Quaternions to Cosmology: Spaces  of Constant Curvature ca. 1873 &mdash; 1925], invited address to International Congress of Mathematicians
* {{Cite journal|author=Killing, W.|year=1891|title=Ueber die Clifford-Klein'schen Raumformen|journal=Mathematische Annalen|volume=39|issue=2|pages=257–278|url= http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN00225333X|doi=10.1007/bf01206655|s2cid=119473479|url-access=subscription}}
* {{Citation|last1=Hopf|first1=Heinz|author1-link=Heinz Hopf|title=Zum Clifford-Kleinschen Raumproblem|doi=10.1007/BF01206614|year=1926|journal=[[Mathematische Annalen]]|issn=0025-5831|volume=95 |issue=1|pages=313–339}}


{{DEFAULTSORT:Clifford-Klein Form}}
[[Category:Lie groups]]
[[Category:Homogeneous spaces]]