Homogeneous space

Inmathematics,ahomogeneous spaceis, very informally, a space that looks the same everywhere, as you move through it, with movement given by theactionof agroup.Homogeneous spaces occur in the theories ofLie groups,algebraic groupsandtopological groups.More precisely, a homogeneous space for agroupGis anon-emptymanifoldortopological spaceXon whichGactstransitively.The elements ofGare called thesymmetriesofX.A special case of this is when the groupGin question is theautomorphism groupof the spaceX– here "automorphism group" can meanisometry group,diffeomorphism group,orhomeomorphism group.In this case,Xis homogeneous if intuitivelyXlooks 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 ofGbefaithful(non-identity elements act non-trivially), although the present article does not. Thus there is agroup actionofGonXthat can be thought of as preserving some "geometric structure" onX,and makingXinto a singleG-orbit.

Atorus.The standard torus is homogeneous under itsdiffeomorphismandhomeomorphismgroups, and theflat torusis homogeneous under its diffeomorphism, homeomorphism, andisometry groups.

Formal definition

LetXbe a non-empty set andGa group. ThenXis called aG-space if it is equipped with an action ofGonX.Note that automaticallyGacts by automorphisms (bijections) on the set. IfXin addition belongs to somecategory,then the elements ofGare assumed to act asautomorphismsin the same category. That is, the maps onXcoming from elements ofGpreserve the structure associated with the category (for example, ifXis an object inDiffthen the action is required to be bydiffeomorphisms). A homogeneous space is aG-space on whichGacts transitively.

IfXis an object of the categoryC,then the structure of aG-space is ahomomorphism:

$\rho:G\to \mathrm {Aut} _{\mathbf {C} }(X)$

into the group ofautomorphismsof the objectXin the categoryC.The pair(X,ρ)defines a homogeneous space providedρ(G) is a transitive group of symmetries of the underlying set ofX.

Examples

For example, ifXis atopological space,then group elements are assumed to act ashomeomorphismsonX.The structure of aG-space is a group homomorphismρ:G→ Homeo(X) into thehomeomorphism groupofX.

Similarly, ifXis adifferentiable manifold,then the group elements arediffeomorphisms.The structure of aG-space is a group homomorphismρ:G→ Diffeo(X)into the diffeomorphism group ofX.

Riemannian symmetric spacesare an important class of homogeneous spaces, and include many of the examples listed below.

Concrete examples include:

Examples of homogeneous spaces

spaceX	groupG	stabilizerH
spherical spaceSn−1	O(n)	O(n− 1)
orientedSn−1	SO(n)	SO(n− 1)
projective space PRn−1	PO(n)	PO(n− 1)
Euclidean space En	E(n)	O(n)
oriented En	E+(n)	SO(n)
hyperbolic space Hn	O+(1,n)	O(n)
oriented Hn	SO+(1,n)	SO(n)
anti-de Sitter space AdSn+1	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:
  1. Sphere (orthogonal group):Sⁿ⁻¹≅ O(n) / O(n−1).This is true because of the following observations: First,Sⁿ⁻¹is the set of vectors inRⁿwith 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 ofRⁿ,then the complement is an(n− 1)-dimensional vector space that is invariant under an orthogonal transformation fromO(n− 1).This shows us why we can constructSⁿ⁻¹as a homogeneous space.
  2. Oriented sphere (special orthogonal group):Sⁿ⁻¹≅ SO(n) / SO(n− 1)
  3. Projective space (projective orthogonal group):Pⁿ⁻¹≅ PO(n) / PO(n− 1)
• Flat (zero curvature):
  1. Euclidean space (Euclidean group,point stabilizer is orthogonal group):Eⁿ≅ E(n) / O(n)
• Negative curvature:
  1. Hyperbolic space (orthochronous Lorentz group,point stabilizer orthogonal group, corresponding tohyperboloid model):Hⁿ≅ O⁺(1,n) / O(n)
  2. Oriented hyperbolic space:SO⁺(1,n) / SO(n)
  3. Anti-de Sitter space:AdS_n+1= O(2,n) / O(1,n)

Others

• Affine spaceoverfieldK(foraffine group,point stabilizergeneral linear group):Aⁿ= Aff(n,K) / GL(n,K).
• Grassmannian:Gr(r,n) = O(n) / (O(r) × O(n−r))
• Topological vector spaces(in the sense of topology)
• There are other interesting homogeneous spaces, in particular with relevance in physics: This includesMinkowski spaceMⁿ≅ ISO(n-1,1) / SO(n,1)or Galilean and Carrollian spaces.

Geometry

From the point of view of theErlangen program,one may understand that "all points are the same", in thegeometryofX.This was true of essentially all geometries proposed beforeRiemannian geometry,in the middle of the nineteenth century.

Thus, for example,Euclidean space,affine spaceandprojective spaceare all in natural ways homogeneous spaces for their respectivesymmetry groups.The same is true of the models found ofnon-Euclidean geometryof constantcurvature,such ashyperbolic 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-dimensionalvector space). It is simple linear algebra to show that GL₄acts transitively on those. We can parameterize them byline co-ordinates:these are the 2×2minorsof the 4×2 matrix with columns two basis vectors for the subspace. The geometry of the resulting homogeneous space is theline geometryofJulius Plücker.

Homogeneous spaces as coset spaces

In general, ifXis a homogeneous space ofG,andH_ois thestabilizerof some marked pointoinX(a choice oforigin), the points ofXcorrespond to the leftcosetsG/H_o,and the marked pointocorresponds to the coset of the identity. Conversely, given a coset spaceG/H,it is a homogeneous space forGwith 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, ifHis the identity subgroup {e}, thenXis theG-torsor,which explains whyG-torsors are often described intuitively as "Gwith forgotten identity ".

In general, a different choice of originowill lead to a quotient ofGby a different subgroupH_o′that is related toH_oby aninner automorphismofG.Specifically,

$$H_{o'}=gH_{o}g^{-1}$$(1)

wheregis any element ofGfor whichgo=o′.Note that the inner automorphism (1) does not depend on which suchgis selected; it depends only ongmoduloH_o.

If the action ofGonXiscontinuousandXisHausdorff,thenHis aclosed subgroupofG.In particular, ifGis aLie group,thenHis aLie subgroupbyCartan's theorem.HenceG/His asmooth manifoldand soXcarries a uniquesmooth structurecompatible with the group action.

One can go further todoublecosetspaces, notablyClifford–Klein formsΓ\G/H,where Γ is a discrete subgroup (ofG) actingproperly discontinuously.

Example

For example, in the line geometry case, we can identifyHas a 12-dimensional subgroup of the 16-dimensionalgeneral linear group,GL(4), defined by conditions on the matrix entries

h₁₃=h₁₄=h₂₃=h₂₄= 0,

by looking for the stabilizer of the subspace spanned by the first two standard basis vectors. That shows thatXhas dimension 4.

Since thehomogeneous coordinatesgiven 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 aGrassmannian,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 aprehomogeneous vector spacewas introduced byMikio Sato.

It is a finite-dimensionalvector spaceVwith agroup actionof analgebraic groupG,such that there is an orbit ofGthat is open for theZariski 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 thePoincaré groupGand its subgroup theLorentz groupH,the space ofcosetsG/His theMinkowski space.Together withde Sitter spaceandAnti-de Sitter spacethese 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.

Physical cosmologyusing thegeneral theory of relativitymakes use of theBianchi classificationsystem. Homogeneous spaces in relativity represent thespace partof backgroundmetricsfor somecosmological models;for example, the three cases of theFriedmann–Lemaître–Robertson–Walker metricmay be represented by subsets of the Bianchi I (flat), V (open), VII (flat or open) and IX (closed) types, while theMixmaster universerepresents ananisotropicexample of a Bianchi IX cosmology.

A homogeneous space ofNdimensions admits a set of⁠1/2⁠N(N+ 1)Killing vectors.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 fieldsξ^(a)
_i,

$\xi _{[i;k]}^{(a)}=C_{\ bc}^{a}\xi _{i}^{(b)}\xi _{k}^{(c)},$

where the objectC^a_bc,the "structure constants", form aconstantorder-three tensorantisymmetricin its lower two indices (on the left-hand side, the brackets denote antisymmetrisation and ";" represents thecovariant differential operator). In the case of aflat isotropic universe,one possibility isC^a_bc= 0(type I), but in the case of a closed FLRW universe,C^a_bc=ε^a_bc,whereε^a_bcis theLevi-Civita symbol.

See also

• Erlangen program
• Klein geometry
• Heap (mathematics)
• Homogeneous variety

References

• John Milnor&James D. Stasheff(1974)Characteristic Classes,Princeton University PressISBN0-691-08122-0
• Takashi KodaAn Introduction to the Geometry of Homogeneous SpacesfromKyungpook National University
• Menelaos ZikidisHomogeneous SpacesfromHeidelberg University
• Shoshichi Kobayashi,Katsumi Nomizu(1969)Foundations of Differential Geometry,volume 2, chapter X, (Wiley Classics Library)