Editing Homogeneous space

{{Short description|Topological space in group theory}}
[[File:Torus.png|300px|thumb|A [[torus]]. The standard torus is homogeneous under its [[diffeomorphism]] and [[homeomorphism]] groups, and the [[flat torus]] is homogeneous under its diffeomorphism, homeomorphism, and [[isometry group]]s. ]]

In [[mathematics]], a '''homogeneous space''' is, very informally, a space that looks the same everywhere, as you move through it, with movement given by the [[Group action (mathematics)|action]] of a [[Group (mathematics)|group]]. Homogeneous spaces occur in the theories of [[Lie group]]s, [[algebraic group]]s and [[topological group]]s. More precisely, a homogeneous space for a [[Group (mathematics)|group]] ''G'' is a [[Empty set|non-empty]] [[manifold]] or [[topological space]] ''X'' on which ''G'' [[Group action (mathematics)|acts]] [[Group action (mathematics)#Transitivity properties|transitively]].  The elements of ''G'' are called the '''symmetries''' of ''X''. A special case of this is when the group ''G'' in question is the [[automorphism group]] of the space ''X'' – here "automorphism group" can mean [[isometry group]], [[diffeomorphism group]], or [[homeomorphism group]]. In this case, ''X'' is homogeneous if intuitively ''X'' looks locally the same at each point, either in the sense of isometry (rigid geometry), diffeomorphism ([[differential geometry]]), or homeomorphism ([[topology]]). Some authors insist that the action of ''G'' be [[effective group action|faithful]] (non-identity elements act non-trivially), although the present article does not. Thus there is a [[Group action (mathematics)|group action]] of ''G'' on ''X'' that can be thought of as preserving some "geometric structure" on ''X'', and making ''X'' into a single [[orbit (group theory)|''G''-orbit]].

== 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>

== Geometry ==
From the point of view of the [[Erlangen program]], one may understand that "all points are the same", in the [[geometry]] of ''X''. This was true of essentially all geometries proposed before [[Riemannian geometry]], in the middle of the nineteenth century.

Thus, for example, [[Euclidean space]], [[affine space]] and [[projective space]] are all in natural ways homogeneous spaces for their respective [[symmetry group]]s. The same is true of the models found of [[non-Euclidean geometry]] of constant [[curvature]], such as [[hyperbolic space]].

A further classical example is the space of lines in projective space of three dimensions (equivalently, the space of two-dimensional subspaces of a four-dimensional [[vector space]]). It is simple [[linear algebra]] to show that GL<sub>4</sub> acts transitively on those. We can parameterize them by ''line co-ordinates'': these are the 2×2 [[minor (linear algebra)|minors]] of the 4×2 matrix with columns two basis vectors for the subspace. The geometry of the resulting homogeneous space is the [[line geometry]] of [[Julius Plücker]].

== Homogeneous spaces as coset spaces ==
In general, if ''X'' is a homogeneous space of ''G'', and ''H''<sub>''o''</sub> is the [[stabilizer (group theory)|stabilizer]] of some marked point ''o'' in ''X'' (a choice of [[origin (mathematics)|origin]]), the points of ''X'' correspond to the left [[coset]]s ''G''/''H''<sub>''o''</sub>, and the marked point ''o'' corresponds to the coset of the identity. Conversely, given a coset space ''G''/''H'', it is a homogeneous space for ''G'' with a distinguished point, namely the coset of the identity. Thus a homogeneous space can be thought of as a coset space without a choice of origin.

For example, if ''H'' is the identity subgroup {{mset|''e''}}, then ''X'' is the [[principal homogeneous space|''G''-torsor]], which explains why ''G''-torsors are often described intuitively as "''G'' with forgotten identity".

In general, a different choice of origin ''o'' will lead to a quotient of&nbsp;''G'' by a different subgroup ''H<sub>o′</sub>'' that is related to ''H<sub>o</sub>'' by an [[inner automorphism]] of&nbsp;''G''.  Specifically,
{{NumBlk||<math display="block">H_{o'} = gH_og^{-1}</math>|{{EquationRef|1}}}}
where ''g'' is any element of ''G'' for which {{nowrap|1=''go'' = ''o''′}}. Note that the inner automorphism (1) does not depend on which such ''g'' is selected; it depends only on ''g'' modulo&nbsp;''H''<sub>''o''</sub>.

If the action of ''G'' on ''X'' is [[continuous map|continuous]] and ''X'' is [[Hausdorff space|Hausdorff]], then ''H'' is a [[closed subgroup]] of ''G''. In particular, if ''G'' is a [[Lie group]], then ''H'' is a [[Lie subgroup]] by [[Closed subgroup theorem|Cartan's theorem]].  Hence {{nowrap|''G'' / ''H''}} is a [[smooth manifold]] and so ''X'' carries a unique [[smooth structure]] compatible with the group action.

One can go further to [[double coset|''double'' coset]] spaces, notably [[Clifford–Klein form]]s Γ\''G''/''H'', where Γ is a discrete subgroup (of ''G'') acting [[properly discontinuously]].

== Example ==
For example, in the line geometry case, we can identify ''H'' as a 12-dimensional subgroup of the 16-dimensional [[general linear group]], GL(4), defined by conditions on the matrix entries
: ''h''<sub>13</sub> = ''h''<sub>14</sub> = ''h''<sub>23</sub> = ''h''<sub>24</sub> = 0,
by looking for the stabilizer of the subspace spanned by the first two standard basis vectors. That shows that ''X'' has dimension 4.

Since the [[homogeneous coordinates]] given by the minors are 6 in number, this means that the latter are not independent of each other. In fact, a single quadratic relation holds between the six minors, as was known to nineteenth-century geometers.

This example was the first known example of a [[Grassmannian]], other than a projective space. There are many further homogeneous spaces of the classical linear groups in common use in mathematics.

== Prehomogeneous vector spaces ==
The idea of a [[prehomogeneous vector space]] was introduced by [[Mikio Sato]].

It is a finite-dimensional [[vector space]] ''V'' with a [[Group action (mathematics)|group action]] of an [[algebraic group]] ''G'', such that there is an orbit of ''G'' that is open for the [[Zariski topology]] (and so, dense). An example is GL(1) acting on a one-dimensional space.

The definition is more restrictive than it initially appears: such spaces have remarkable properties, and there is a classification of irreducible prehomogeneous vector spaces, up to a transformation known as "castling".

== 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]].

== See also ==
* [[Erlangen program]]
* [[Klein geometry]]
* [[Heap (mathematics)]]
* [[Homogeneous variety]]

== Notes ==
{{reflist|30em}}

== References ==
{{refbegin|30em}}
* [[John Milnor]] & [[James D. Stasheff]] (1974) ''Characteristic Classes'', [[Princeton University Press]] {{ISBN|0-691-08122-0}}
* Takashi Koda [http://webbuild.knu.ac.kr/~yjsuh/proceedings/13th/%5B10%5D09Prowork_Koda.pdf An Introduction to the Geometry of Homogeneous Spaces] from [[Kyungpook National University]]
* Menelaos Zikidis [https://www.mathi.uni-heidelberg.de/~lee/MenelaosSS16.pdf Homogeneous Spaces] from [[Heidelberg University]]
* [[Shoshichi Kobayashi]], [[Katsumi Nomizu]] (1969) ''[[Foundations of Differential Geometry]]'', volume 2, chapter X, (Wiley Classics Library)
{{refend}}

[[Category:Topological groups]]
[[Category:Lie groups]]
[[Category:Homogeneous spaces| ]]