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<title>Homogeneous space</title>

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Homogeneous space

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In <a href="Mathematics" title="Mathematics">mathematics</a>, a homogeneous space is, very informally, a space that looks the same everywhere, as you move through it, with movement given by the <a href="Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">action</a> of a <a href="Group_(mathematics)" title="Group (mathematics)">group</a>. Homogeneous spaces occur in the theories of <a href="Lie_group" title="Lie group">Lie groups</a>, <a href="Algebraic_group" title="Algebraic group">algebraic groups</a> and <a href="Topological_group" title="Topological group">topological groups</a>. More precisely, a homogeneous space for a <a href="Group_(mathematics)" title="Group (mathematics)">group</a> G is a <a href="Empty_set" title="Empty set">non-empty</a> <a href="Manifold" title="Manifold">manifold</a> or <a href="Topological_space" title="Topological space">topological space</a> X on which G <a href="Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">acts</a> <a href="Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">transitively</a>. The elements of G are called the symmetries of X. A special case of this is when the group G in question is the <a href="Automorphism_group" title="Automorphism group">automorphism group</a> of the space X – here "automorphism group" can mean <a href="Isometry_group" title="Isometry group">isometry group</a>, <a href="Diffeomorphism_group" class="mw-redirect" title="Diffeomorphism group">diffeomorphism group</a>, or <a href="Homeomorphism_group" title="Homeomorphism group">homeomorphism group</a>. 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 (<a href="Differential_geometry" title="Differential geometry">differential geometry</a>), or homeomorphism (<a href="Topology" title="Topology">topology</a>). Some authors insist that the action of G be <a href="Effective_group_action" class="mw-redirect" title="Effective group action">faithful</a> (non-identity elements act non-trivially), although the present article does not. Thus there is a <a href="Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">group action</a> of G on X that can be thought of as preserving some "geometric structure" on X, and making X into a single <a href="Orbit_(group_theory)" class="mw-redirect" title="Orbit (group theory)">G-orbit</a>.

<div class="mw-heading mw-heading2"><h2 id="Formal_definition">Formal definition</h2></div>

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.<a href="#cite_note-1">[1]</a> Note that automatically G acts by automorphisms (bijections) on the set. If X in addition belongs to some <a href="Category_(mathematics)" title="Category (mathematics)">category</a>, then the elements of G are assumed to act as <a href="Automorphism" title="Automorphism">automorphisms</a> 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 <a href="Diffeomorphism" title="Diffeomorphism">diffeomorphisms</a>). 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 <a href="Homomorphism" title="Homomorphism">homomorphism</a>:

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<annotation encoding="application/x-tex">{\displaystyle \rho :G\to \mathrm {Aut} _{\mathbf {C} }(X)}</annotation>

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into the group of <a href="Automorphism" title="Automorphism">automorphisms</a> of the object X in the category C. The pair (X, ρ) defines a homogeneous space provided ρ(G) is a transitive group of symmetries of the underlying set of X.

<div class="mw-heading mw-heading3"><h3 id="Examples">Examples</h3></div>

For example, if X is a <a href="Topological_space" title="Topological space">topological space</a>, then group elements are assumed to act as <a href="Homeomorphism" title="Homeomorphism">homeomorphisms</a> on X. The structure of a G-space is a group homomorphism ρ : G → Homeo(X) into the <a href="Homeomorphism_group" title="Homeomorphism group">homeomorphism group</a> of X.

Similarly, if X is a <a href="Differentiable_manifold" title="Differentiable manifold">differentiable manifold</a>, then the group elements are <a href="Diffeomorphism" title="Diffeomorphism">diffeomorphisms</a>. The structure of a G-space is a group homomorphism ρ : G → Diffeo(X) into the diffeomorphism group of X.

<a href="Riemannian_symmetric_space" class="mw-redirect" title="Riemannian symmetric space">Riemannian symmetric spaces</a> are an important class of homogeneous spaces, and include many of the examples listed below.

Concrete examples include:

<caption>Examples of homogeneous spaces

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<th>space X</th>

<th>group G</th>

<th>stabilizer H

</th></tr>

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<td>spherical space Sn−1</td>

<td>O(n − 1)

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<td>oriented Sn−1</td>

<td>SO(n − 1)

</td></tr>

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<td>projective space PRn−1</td>

<td>PO(n − 1)

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<td>Euclidean space En</td>

<td>O(n)

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<td>oriented En</td>

<td>SO(n)

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<td>hyperbolic space Hn</td>

<td>O(n)

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<td>oriented Hn</td>

<td>SO(n)

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<td>anti-de Sitter space AdSn+1</td>

<td>O(1, n)

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<td>Grassmannian Gr(r, n)</td>

<td>O(r) × O(n − r)

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<td>affine space A(n, K)</td>

<td>GL(n, K)

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<dl><dt>Isometry groups</dt></dl>

<ul><li>Positive curvature:

<ol><li>Sphere (<a href="Orthogonal_group" title="Orthogonal group">orthogonal group</a>): Sn−1 ≅ O(n) / O(n−1). This is true because of the following observations: First, Sn−1 is the set of vectors in Rn 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 Rn, then the complement is an (n − 1)-dimensional vector space that is invariant under an orthogonal transformation from O(n − 1). This shows us why we can construct Sn−1 as a homogeneous space.</li>

<li>Oriented sphere (<a href="Special_orthogonal_group" class="mw-redirect" title="Special orthogonal group">special orthogonal group</a>): Sn−1 ≅ SO(n) / SO(n − 1)</li>

<li>Projective space (<a href="Projective_orthogonal_group" title="Projective orthogonal group">projective orthogonal group</a>): Pn−1 ≅ PO(n) / PO(n − 1)</li></ol></li>

<li>Flat (zero curvature):

<ol><li>Euclidean space (<a href="Euclidean_group" title="Euclidean group">Euclidean group</a>, point stabilizer is orthogonal group): En ≅ E(n) / O(n)</li></ol></li>

<li>Negative curvature:

<ol><li>Hyperbolic space (<a href="Orthochronous_Lorentz_group" class="mw-redirect" title="Orthochronous Lorentz group">orthochronous Lorentz group</a>, point stabilizer orthogonal group, corresponding to <a href="Hyperboloid_model" title="Hyperboloid model">hyperboloid model</a>): Hn ≅ O+(1, n) / O(n)</li>

<li>Oriented hyperbolic space: SO+(1, n) / SO(n)</li>

<li><a href="Anti-de_Sitter_space" title="Anti-de Sitter space">Anti-de Sitter space</a>: AdSn+1 = O(2, n) / O(1, n)</li></ol></li></ul>

<dl><dt>Others</dt></dl>

<ul><li><a href="Affine_space" title="Affine space">Affine space</a> over <a href="Field_(mathematics)" title="Field (mathematics)">field</a> K (for <a href="Affine_group" title="Affine group">affine group</a>, point stabilizer <a href="General_linear_group" title="General linear group">general linear group</a>): An = Aff(n, K) / GL(n, K).</li>

<li><a href="Grassmannian" title="Grassmannian">Grassmannian</a>: Gr(r, n) = O(n) / (O(r) × O(n − r))</li>

<li><a href="Topological_vector_space" title="Topological vector space">Topological vector spaces</a> (in the sense of topology)</li>

<li>There are other interesting homogeneous spaces, in particular with relevance in physics: This includes <a href="Minkowski_space" title="Minkowski space">Minkowski space</a> Mn ≅ ISO(n-1,1) / SO(n,1) or Galilean and Carrollian spaces.<a href="#cite_note-:0-2">[2]</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Geometry">Geometry</h2></div>

From the point of view of the <a href="Erlangen_program" title="Erlangen program">Erlangen program</a>, one may understand that "all points are the same", in the <a href="Geometry" title="Geometry">geometry</a> of X. This was true of essentially all geometries proposed before <a href="Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a>, in the middle of the nineteenth century.

Thus, for example, <a href="Euclidean_space" title="Euclidean space">Euclidean space</a>, <a href="Affine_space" title="Affine space">affine space</a> and <a href="Projective_space" title="Projective space">projective space</a> are all in natural ways homogeneous spaces for their respective <a href="Symmetry_group" title="Symmetry group">symmetry groups</a>. The same is true of the models found of <a href="Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean geometry</a> of constant <a href="Curvature" title="Curvature">curvature</a>, such as <a href="Hyperbolic_space" title="Hyperbolic space">hyperbolic space</a>.

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 <a href="Vector_space" title="Vector space">vector space</a>). It is simple <a href="Linear_algebra" title="Linear algebra">linear algebra</a> to show that GL4 acts transitively on those. We can parameterize them by line co-ordinates: these are the 2×2 <a href="Minor_(linear_algebra)" title="Minor (linear algebra)">minors</a> of the 4×2 matrix with columns two basis vectors for the subspace. The geometry of the resulting homogeneous space is the <a href="Line_geometry" class="mw-redirect" title="Line geometry">line geometry</a> of <a href="Julius_Pl%C3%BCcker" title="Julius Plücker">Julius Plücker</a>.

<div class="mw-heading mw-heading2"><h2 id="Homogeneous_spaces_as_coset_spaces">Homogeneous spaces as coset spaces</h2></div>

In general, if X is a homogeneous space of G, and Ho is the <a href="Stabilizer_(group_theory)" class="mw-redirect" title="Stabilizer (group theory)">stabilizer</a> of some marked point o in X (a choice of <a href="Origin_(mathematics)" title="Origin (mathematics)">origin</a>), the points of X correspond to the left <a href="Coset" title="Coset">cosets</a> G/Ho, 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 {e}, then X is the <a href="Principal_homogeneous_space" title="Principal homogeneous space">G-torsor</a>, 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 G by a different subgroup Ho′ that is related to Ho by an <a href="Inner_automorphism" title="Inner automorphism">inner automorphism</a> of G. Specifically,

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</style><table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{o'}=gH_{o}g^{-1}}">

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<annotation encoding="application/x-tex">{\displaystyle H_{o'}=gH_{o}g^{-1}}</annotation>

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where g is any element of G for which go = o′. Note that the inner automorphism (1) does not depend on which such g is selected; it depends only on g modulo Ho.

If the action of G on X is <a href="Continuous_map" class="mw-redirect" title="Continuous map">continuous</a> and X is <a href="Hausdorff_space" title="Hausdorff space">Hausdorff</a>, then H is a <a href="Closed_subgroup" class="mw-redirect" title="Closed subgroup">closed subgroup</a> of G. In particular, if G is a <a href="Lie_group" title="Lie group">Lie group</a>, then H is a <a href="Lie_subgroup" class="mw-redirect" title="Lie subgroup">Lie subgroup</a> by <a href="Closed_subgroup_theorem" class="mw-redirect" title="Closed subgroup theorem">Cartan's theorem</a>. Hence G / H is a <a href="Smooth_manifold" class="mw-redirect" title="Smooth manifold">smooth manifold</a> and so X carries a unique <a href="Smooth_structure" title="Smooth structure">smooth structure</a> compatible with the group action.

One can go further to <a href="Double_coset" title="Double coset">double coset</a> spaces, notably <a href="Clifford%E2%80%93Klein_form" title="Clifford–Klein form">Clifford–Klein forms</a> Γ\G/H, where Γ is a discrete subgroup (of G) acting <a href="Properly_discontinuously" class="mw-redirect" title="Properly discontinuously">properly discontinuously</a>.

<div class="mw-heading mw-heading2"><h2 id="Example">Example</h2></div>

For example, in the line geometry case, we can identify H as a 12-dimensional subgroup of the 16-dimensional <a href="General_linear_group" title="General linear group">general linear group</a>, GL(4), defined by conditions on the matrix entries

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 <a href="Homogeneous_coordinates" title="Homogeneous coordinates">homogeneous coordinates</a> 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 <a href="Grassmannian" title="Grassmannian">Grassmannian</a>, other than a projective space. There are many further homogeneous spaces of the classical linear groups in common use in mathematics.

<div class="mw-heading mw-heading2"><h2 id="Prehomogeneous_vector_spaces">Prehomogeneous vector spaces</h2></div>

The idea of a <a href="Prehomogeneous_vector_space" title="Prehomogeneous vector space">prehomogeneous vector space</a> was introduced by <a href="Mikio_Sato" title="Mikio Sato">Mikio Sato</a>.

It is a finite-dimensional <a href="Vector_space" title="Vector space">vector space</a> V with a <a href="Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">group action</a> of an <a href="Algebraic_group" title="Algebraic group">algebraic group</a> G, such that there is an orbit of G that is open for the <a href="Zariski_topology" title="Zariski topology">Zariski topology</a> (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".

<div class="mw-heading mw-heading2"><h2 id="Homogeneous_spaces_in_physics">Homogeneous spaces in physics</h2></div>

Given the <a href="Poincar%C3%A9_group" title="Poincaré group">Poincaré group</a> G and its subgroup the <a href="Lorentz_group" title="Lorentz group">Lorentz group</a> H, the space of <a href="Coset" title="Coset">cosets</a> G / H is the <a href="Minkowski_space" title="Minkowski space">Minkowski space</a>.<a href="#cite_note-3">[3]</a> Together with <a href="De_Sitter_space" title="De Sitter space">de Sitter space</a> and <a href="Anti-de_Sitter_space" title="Anti-de Sitter space">anti-de Sitter space</a> these are the maximally symmetric <a href="Lorentzian_metric" class="mw-redirect" title="Lorentzian metric">lorentzian</a> spacetimes. There are also homogeneous spaces of relevance in physics that are non-lorentzian, for example Galilean, Carrollian or Aristotelian spacetimes.<a href="#cite_note-:0-2">[2]</a>

<a href="Physical_cosmology" title="Physical cosmology">Physical cosmology</a> using the <a href="General_theory_of_relativity" class="mw-redirect" title="General theory of relativity">general theory of relativity</a> makes use of the <a href="Bianchi_classification" title="Bianchi classification">Bianchi classification</a> system. Homogeneous spaces in relativity represent the <a href="Space_(physics)" class="mw-redirect" title="Space (physics)">space part</a> of background <a href="Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">metrics</a> for some <a href="Physical_cosmology" title="Physical cosmology">cosmological models</a>; for example, the three cases of the <a href="Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric" title="Friedmann–Lemaître–Robertson–Walker metric">Friedmann–Lemaître–Robertson–Walker metric</a> may be represented by subsets of the Bianchi I (flat), V (open), VII (flat or open) and IX (closed) types, while the <a href="Mixmaster_universe" title="Mixmaster universe">Mixmaster universe</a> represents an <a href="Isotropy" title="Isotropy">anisotropic</a> example of a Bianchi IX cosmology.<a href="#cite_note-4">[4]</a>

A homogeneous space of N dimensions admits a set of <style data-mw-deduplicate="TemplateStyles:r1214402035">

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</style>⁠1/2⁠N(N + 1) <a href="Killing_vectors" class="mw-redirect" title="Killing vectors">Killing vectors</a>.<a href="#cite_note-5">[5]</a> 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(a) i,

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<annotation encoding="application/x-tex">{\displaystyle \xi _{[i;k]}^{(a)}=C_{\ bc}^{a}\xi _{i}^{(b)}\xi _{k}^{(c)},}</annotation>

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where the object Cabc, the "<a href="Structure_constants" title="Structure constants">structure constants</a>", form a <a href="Constant_(mathematics)" title="Constant (mathematics)">constant</a> <a href="Tensor" title="Tensor">order-three tensor</a> <a href="Antisymmetric_tensor" title="Antisymmetric tensor">antisymmetric</a> in its lower two indices (on the left-hand side, the brackets denote antisymmetrisation and ";" represents the <a href="Covariant_derivative" title="Covariant derivative">covariant differential operator</a>). In the case of a <a href="Lambda-CDM" class="mw-redirect" title="Lambda-CDM">flat isotropic universe</a>, one possibility is Cabc = 0 (type I), but in the case of a closed FLRW universe, Cabc = εabc, where εabc is the <a href="Levi-Civita_symbol" title="Levi-Civita symbol">Levi-Civita symbol</a>.

<ul><li><a href="Erlangen_program" title="Erlangen program">Erlangen program</a></li>

<li><a href="Klein_geometry" title="Klein geometry">Klein geometry</a></li>

<li><a href="Heap_(mathematics)" title="Heap (mathematics)">Heap (mathematics)</a></li>

<li><a href="Homogeneous_variety" title="Homogeneous variety">Homogeneous variety</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-1"><a href="#cite_ref-1">^</a> We assume that the action is on the left. The distinction is only important in the description of X as a coset space.

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</style><cite id="CITEREFFigueroa-O’FarrillProhazka2019" class="citation journal cs1">Figueroa-O’Farrill, José; Prohazka, Stefan (2019-01-31). <a rel="nofollow" class="external text" href="https://doi.org/10.1007/JHEP01(2019)229">"Spatially isotropic homogeneous spacetimes"</a>. Journal of High Energy Physics. 2019 (1): 229. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1809.01224">1809.01224</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2019JHEP...01..229F">2019JHEP...01..229F</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FJHEP01%282019%29229">10.1007/JHEP01(2019)229</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1029-8479">1029-8479</a>.</cite>

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<li id="cite_note-3"><a href="#cite_ref-3">^</a> <a href="Robert_Hermann_(mathematician)" title="Robert Hermann (mathematician)">Robert Hermann</a> (1966) Lie Groups for Physicists, page 4, <a href="W._A._Benjamin" class="mw-redirect" title="W. A. Benjamin">W. A. Benjamin</a>

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<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFLev_Landau_and_Evgeny_Lifshitz1980" class="citation cs2"><a href="Lev_Landau" title="Lev Landau">Lev Landau</a> and <a href="Evgeny_Lifshitz" title="Evgeny Lifshitz">Evgeny Lifshitz</a> (1980), Course of Theoretical Physics vol. 2: The Classical Theory of Fields, Butterworth-Heinemann, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-7506-2768-9</bdi></cite>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFSteven_Weinberg1972" class="citation cs2"><a href="Steven_Weinberg" title="Steven Weinberg">Steven Weinberg</a> (1972), Gravitation and Cosmology, John Wiley and Sons</cite>

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</ol></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em">

<ul><li><a href="John_Milnor" title="John Milnor">John Milnor</a> & <a href="James_D._Stasheff" class="mw-redirect" title="James D. Stasheff">James D. Stasheff</a> (1974) Characteristic Classes, <a href="Princeton_University_Press" title="Princeton University Press">Princeton University Press</a> <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-691-08122-0</bdi></li>

<li>Takashi Koda <a rel="nofollow" class="external text" href="http://webbuild.knu.ac.kr/~yjsuh/proceedings/13th/%5B10%5D09Prowork_Koda.pdf">An Introduction to the Geometry of Homogeneous Spaces</a> from <a href="Kyungpook_National_University" title="Kyungpook National University">Kyungpook National University</a></li>

<li>Menelaos Zikidis <a rel="nofollow" class="external text" href="https://www.mathi.uni-heidelberg.de/~lee/MenelaosSS16.pdf">Homogeneous Spaces</a> from <a href="Heidelberg_University" title="Heidelberg University">Heidelberg University</a></li>

<li><a href="Shoshichi_Kobayashi" title="Shoshichi Kobayashi">Shoshichi Kobayashi</a>, <a href="Katsumi_Nomizu" title="Katsumi Nomizu">Katsumi Nomizu</a> (1969) <a href="Foundations_of_Differential_Geometry" title="Foundations of Differential Geometry">Foundations of Differential Geometry</a>, volume 2, chapter X, (Wiley Classics Library)</li></ul>

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