Editing Homogeneous space (section)

== Homogeneous spaces in physics ==
Given the [[Poincaré group]] ''G'' and its subgroup the [[Lorentz group]] ''H'', the space of [[coset]]s {{nowrap|''G'' / ''H''}} is the [[Minkowski space]].<ref>[[Robert Hermann (mathematician)|Robert Hermann]] (1966) ''Lie Groups for Physicists'', page 4, [[W. A. Benjamin]]</ref> Together with [[de Sitter space]] and [[Anti-de Sitter space]] these are the maximally symmetric lorentzian spacetimes. There are also homogeneous spaces of relevance in physics that are non-lorentzian, for example Galilean, Carrollian or Aristotelian spacetimes.<ref name=":0" />

[[Physical cosmology]] using the [[general theory of relativity]] makes use of the [[Bianchi classification]] system. Homogeneous spaces in relativity represent the [[space (physics)|space part]] of background [[Metric (mathematics)|metrics]] for some [[Physical cosmology|cosmological model]]s; for example, the three cases of the [[Friedmann–Lemaître–Robertson–Walker metric]] may be represented by subsets of the Bianchi I (flat), V (open), VII (flat or open) and IX (closed) types, while the [[Mixmaster universe]] represents an [[isotropy|anisotropic]] example of a Bianchi IX cosmology.<ref>{{citation |title=Course of Theoretical Physics vol. 2: The Classical Theory of Fields |author=[[Lev Landau]] and [[Evgeny Lifshitz]] |isbn=978-0-7506-2768-9 |year=1980 |publisher=Butterworth-Heinemann }}</ref>

A homogeneous space of ''N'' dimensions admits a set of {{nowrap|{{sfrac|1|2}}''N''(''N'' + 1)}} [[Killing vectors]].<ref>{{citation |title=Gravitation and Cosmology |author=[[Steven Weinberg]] |publisher=John Wiley and Sons |year=1972 }}</ref> For three dimensions, this gives a total of six linearly independent Killing vector fields; homogeneous 3-spaces have the property that one may use linear combinations of these to find three everywhere non-vanishing Killing vector fields ''ξ''{{su|lh=1|p=(''a'')|b=''i''}},
: <math>\xi^{(a)}_{[i;k]}=C^a_{\ bc}\xi^{(b)}_i \xi^{(c)}_k ,</math>
where the object ''C''<sup>''a''</sup><sub>''bc''</sub>, the "[[structure constants]]", form a [[constant (mathematics)|constant]] [[tensor|order-three tensor]] [[antisymmetric tensor|antisymmetric]] in its lower two indices (on the left-hand side, the brackets denote antisymmetrisation and ";" represents the [[covariant derivative|covariant differential operator]]). In the case of a [[Lambda-CDM|flat isotropic universe]], one possibility is {{nowrap|1=''C''<sup>''a''</sup><sub>''bc''</sub> = 0}} (type I), but in the case of a closed FLRW universe, {{nowrap|1=''C''<sup>''a''</sup><sub>''bc''</sub> = ''ε''<sup>''a''</sup><sub>''bc''</sub>}}, where ''ε''<sup>''a''</sup><sub>''bc''</sub>is the [[Levi-Civita symbol]].