Editing Homogeneous space (section)

== Formal definition ==
Let ''X'' be a non-empty set and ''G'' a group. Then ''X'' is called a ''G''-space if it is equipped with an action of ''G'' on ''X''.<ref>We assume that the action is on the ''left''. The distinction is only important in the description of ''X'' as a coset space.</ref> Note that automatically ''G'' acts by automorphisms (bijections) on the set. If ''X'' in addition belongs to some [[category (mathematics)|category]], then the elements of ''G'' are assumed to act as [[automorphism]]s in the same category. That is, the maps on ''X'' coming from elements of ''G'' preserve the structure associated with the category (for example, if ''X'' is an object in '''Diff''' then the action is required to be by [[diffeomorphism]]s). A homogeneous space is a ''G''-space on which ''G'' acts transitively.

If ''X'' is an object of the category '''C''', then the structure of a ''G''-space is a [[homomorphism]]:
: <math>\rho : G \to \mathrm{Aut}_{\mathbf{C}}(X)</math>
into the group of [[automorphism]]s of the object ''X'' in the category '''C'''. The pair {{nowrap|(''X'', ''ρ'')}} defines a homogeneous space provided ''ρ''(''G'') is a transitive group of symmetries of the underlying set of&nbsp;''X''.

=== Examples ===
For example, if ''X'' is a [[topological space]], then group elements are assumed to act as [[homeomorphism]]s on ''X''. The structure of a ''G''-space is a group homomorphism ''ρ''&nbsp;:&nbsp;''G''&nbsp;→&nbsp;Homeo(''X'') into the [[homeomorphism group]] of&nbsp;''X''.

Similarly, if ''X'' is a [[differentiable manifold]], then the group elements are [[diffeomorphism]]s. The structure of a ''G''-space is a group homomorphism {{nowrap|''ρ'' : ''G'' → Diffeo(''X'')}} into the diffeomorphism group of&nbsp;''X''.

[[Riemannian symmetric space]]s are an important class of homogeneous spaces, and include many of the examples listed below.

Concrete examples include:
{| class="wikitable"
|+ Examples of homogeneous spaces
|-
 ! space ''X'' !! group ''G'' !! stabilizer ''H''
|-
 | spherical space ''S''<sup>''n''−1</sup> || O(''n'') || O(''n'' − 1)
|-
 | oriented ''S''<sup>''n''−1</sup> || SO(''n'') || SO(''n'' − 1)
|-
 | projective space P'''R'''<sup>''n''−1</sup>|| PO(''n'') || PO(''n'' − 1)
|-
 | Euclidean space E<sup>''n''</sup>|| E(''n'') || O(''n'')
|-
 | oriented E<sup>''n''</sup> || E<sup>+</sup>(''n'') || SO(''n'')
|-
 | hyperbolic space H<sup>''n''</sup>|| O<sup>+</sup>(1, ''n'') || O(''n'')
|-
 | oriented H<sup>''n''</sup> || SO<sup>+</sup>(1, ''n'') || SO(''n'')
|-
 | anti-de Sitter space AdS<sub>''n''+1</sub> || O(2, ''n'') || O(1, ''n'')
|-
 | Grassmannian Gr(''r'', ''n'') || O(''n'') || O(''r'') × O(''n'' − ''r'')
|-
 | affine space A(''n'', ''K'') || Aff(''n'', ''K'') || GL(''n'', ''K'')
|}

; Isometry groups
* Positive curvature:
*# Sphere ([[orthogonal group]]): {{nowrap|''S''<sup>''n''−1</sup> ≅ O(''n'') / O(''n''−1)}}. This is true because of the following observations: First, ''S''<sup>''n''−1</sup> is the set of vectors in '''R'''<sup>''n''</sup> with norm 1. If we consider one of these vectors as a base vector, then any other vector can be constructed using an orthogonal transformation. If we consider the span of this vector as a one dimensional subspace of '''R'''<sup>''n''</sup>, then the complement is an {{nowrap|(''n'' − 1)}}-dimensional vector space that is invariant under an orthogonal transformation from {{nowrap|O(''n'' − 1)}}. This shows us why we can construct ''S''<sup>''n''−1</sup> as a homogeneous space.
*# Oriented sphere ([[special orthogonal group]]): {{nowrap|''S''<sup>''n''−1</sup> ≅ SO(''n'') / SO(''n'' − 1)}}
*# Projective space ([[projective orthogonal group]]): {{nowrap|P<sup>''n''−1</sup> ≅ PO(''n'') / PO(''n'' − 1)}}
* Flat (zero curvature):
*# Euclidean space ([[Euclidean group]], point stabilizer is orthogonal group): {{nowrap|E<sup>''n''</sup> ≅ E(''n'') / O(''n'')}}
* Negative curvature:
*# Hyperbolic space ([[orthochronous Lorentz group]], point stabilizer orthogonal group, corresponding to [[hyperboloid model]]): {{nowrap|H<sup>''n''</sup> ≅ O<sup>+</sup>(1, ''n'') / O(''n'')}}
*# Oriented hyperbolic space: {{nowrap|SO<sup>+</sup>(1, ''n'') / SO(''n'')}}
*# [[Anti-de Sitter space]]: {{nowrap|1=AdS<sub>''n''+1</sub> = O(2, ''n'') / O(1, ''n'')}}

; Others
* [[Affine space]] over [[field (mathematics)|field]] ''K'' (for [[affine group]], point stabilizer [[general linear group]]): {{nowrap|1=A<sup>''n''</sup> = Aff(''n'', ''K'') / GL(''n'', ''K'')}}.
* [[Grassmannian]]: {{nowrap|1=Gr(''r'', ''n'') = O(''n'') / (O(''r'') × O(''n'' − ''r''))}}
* [[Topological vector space]]s (in the sense of topology)
* There are other interesting homogeneous spaces, in particular with relevance in physics: This includes [[Minkowski space]] {{nowrap|M<sup>''n''</sup> ≅ ISO(''n-1,1'') / SO(''n,1'')}} or Galilean and Carrollian spaces.<ref name=":0">{{Cite journal |last1=Figueroa-O’Farrill |first1=José |last2=Prohazka |first2=Stefan |date=2019-01-31 |title=Spatially isotropic homogeneous spacetimes |url=https://doi.org/10.1007/JHEP01(2019)229 |journal=Journal of High Energy Physics |language=en |volume=2019 |issue=1 |pages=229 |doi=10.1007/JHEP01(2019)229 |issn=1029-8479|arxiv=1809.01224 |bibcode=2019JHEP...01..229F }}</ref>