In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, particularly in the theories of
Lie groups,
algebraic groups and
topological groups, a homogeneous space for a
group ''G'' is a
non-empty manifold or
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
''X'' on which ''G''
acts 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 mathematics, particularly topology, the homeomorphism group of a topological space is the group consisting of all homeomorphisms from the space to itself with function composition as the group operation. Homeomorphism groups are very important ...
. 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
faithful (non-identity elements act non-trivially), although the present article does not. Thus there is a
group action of ''G'' on ''X'' which can be thought of as preserving some "geometric structure" on ''X'', and making ''X'' into a single
''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''. Note that automatically ''G'' acts by automorphisms (bijections) on the set. If ''X'' in addition belongs to some
category, then the elements of ''G'' are assumed to act as
automorphisms 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
diffeomorphisms). A homogeneous space is a ''G''-space on which ''G'' acts transitively.
Succinctly, if ''X'' is an object of the category C, then the structure of a ''G''-space is a
homomorphism
In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
:
:
into the group of
automorphisms 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''.
Examples
For example, if ''X'' is a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
, then group elements are assumed to act as
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isom ...
s on ''X''. The structure of a ''G''-space is a group homomorphism ''ρ'' : ''G'' → Homeo(''X'') into the
homeomorphism group In mathematics, particularly topology, the homeomorphism group of a topological space is the group consisting of all homeomorphisms from the space to itself with function composition as the group operation. Homeomorphism groups are very important ...
of ''X''.
Similarly, if ''X'' is a
differentiable manifold, then the group elements are
diffeomorphisms. The structure of a ''G''-space is a group homomorphism ''ρ'' : ''G'' → Diffeo(''X'') into the diffeomorphism group of ''X''.
Riemannian symmetric spaces are an important class of homogeneous spaces, and include many of the examples listed below.
Concrete examples include:
;Isometry groups
*Positive curvature:
# Sphere (
orthogonal group):
. This is true because of the following observations: First,
is the set of vectors in
with norm
. 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
, then the complement is an
-dimensional vector space which is invariant under an orthogonal transformation from
. This shows us why we can construct
as a homogeneous space.
# Oriented sphere (
special orthogonal group):
# Projective space (
projective orthogonal group In projective geometry and linear algebra, the projective orthogonal group PO is the induced action of the orthogonal group of a quadratic space ''V'' = (''V'',''Q'')A quadratic space is a vector space ''V'' together with a quadratic form ''Q'' ...
):
* Flat (zero curvature):
# Euclidean space (
Euclidean group, point stabilizer is orthogonal group): A
''n'' ≅ E(''n'')/O(''n'')
* Negative curvature:
# Hyperbolic space (
orthochronous Lorentz group
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch phys ...
, point stabilizer orthogonal group, corresponding to
hyperboloid model
In geometry, the hyperboloid model, also known as the Minkowski model after Hermann Minkowski, is a model of ''n''-dimensional hyperbolic geometry in which points are represented by points on the forward sheet ''S''+ of a two-sheeted hyperbo ...
): H
''n'' ≅ O
+(1, ''n'')/O(''n'')
# Oriented hyperbolic space: SO
+(1, ''n'')/SO(''n'')
#
Anti-de Sitter space: AdS
''n''+1 = O(2, ''n'')/O(1, ''n'')
;Others
*
Affine space over field ''K'' (for
affine group, point stabilizer
general linear group): A
''n'' = Aff(''n'', ''K'')/GL(''n'', ''K'').
*
Grassmannian:
*
Topological vector spaces (in the sense of topology)
Geometry
From the point of view of the
Erlangen program, one may understand that "all points are the same", in the
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
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
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
,
affine space and
projective space are all in natural ways homogeneous spaces for their respective
symmetry groups. 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
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
). It is simple linear algebra to show that GL
4 acts transitively on those. We can parameterize them by ''line co-ordinates'': these are the 2×2
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
In geometry, line coordinates are used to specify the position of a line just as point coordinates (or simply coordinates) are used to specify the position of a point.
Lines in the plane
There are several possible ways to specify the position of ...
of
Julius Plücker.
Homogeneous spaces as coset spaces
In general, if ''X'' is a homogeneous space of ''G'', and ''H''
''o'' is the
stabilizer of some marked point ''o'' in ''X'' (a choice of
origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* The Origin (Buffy comic), ''The Origin'' (Bu ...
), the points of ''X'' correspond to the left
cosets ''G''/''H''
''o'', 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 , then ''X'' is the
G-torsor, which explains why G-torsors are often described intuitively as "
with forgotten identity".
In general, a different choice of origin ''o'' will lead to a quotient of ''G'' by a different subgroup ''H
o′'' which is related to ''H
o'' by an
inner automorphism
In abstract algebra an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the ''conjugating element''. They can be realized via simple operations from within the group itse ...
of ''G''. Specifically,
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 ''H''
''o''.
If the action of ''G'' on ''X'' is continuous and ''X'' is Hausdorff, then ''H'' is a
closed subgroup
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two s ...
of ''G''. In particular, if ''G'' is a
Lie group, then ''H'' is a
Lie subgroup
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additi ...
by
Cartan's theorem. Hence ''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 spaces, notably
Clifford–Klein form In mathematics, a Clifford–Klein form is a double coset space
:Γ\''G''/''H'',
where ''G'' is a reductive Lie group, ''H'' a closed subgroup of ''G'', and Γ a discrete subgroup of G that acts properly discontinuously on the homogeneous spac ...
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''
13 = ''h''
14 = ''h''
23 = ''h''
24 = 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
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometr ...
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 In mathematics, a prehomogeneous vector space (PVS) is a finite-dimensional vector space ''V'' together with a subgroup ''G'' of the general linear group GL(''V'') such that ''G'' has an open dense orbit (group theory), orbit in ''V''. Prehomogeneou ...
was introduced by
Mikio Sato.
It is a finite-dimensional
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
''V'' with a
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
Physical cosmology using the
general theory of relativity makes use of the
Bianchi classification In mathematics, the Bianchi classification provides a list of all real 3-dimensional Lie algebras (up to isomorphism). The classification contains 11 classes, 9 of which contain a single Lie algebra and two of which contain a continuum-sized fam ...
system. Homogeneous spaces in relativity represent the
space part of background
metrics for some
cosmological models; 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
anisotropic example of a Bianchi IX cosmology.
A homogeneous space of ''N'' dimensions admits a set of
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 fields
,
:
where the object
, the "structure constants", form a
constant order-three tensor antisymmetric in its lower two indices (on the left-hand side, the brackets denote antisymmetrisation and ";" represents the
covariant differential operator). In the case of a
flat isotropic universe, one possibility is
(type I), but in the case of a closed FLRW universe,
where
is the
Levi-Civita symbol.
See also
*
Erlangen program
*
Klein geometry
*
Heap (mathematics)
In abstract algebra, a semiheap is an algebraic structure consisting of a non-empty set ''H'' with a ternary operation denoted ,y,z\in H that satisfies a modified associativity property:
\forall a,b,c,d,e \in H \ \ \ \ a,b,cd,e] = ,c,b.html"_;" ...
*
Homogeneous variety
Notes
References
*
John Milnor &
James D. Stasheff
James Dillon Stasheff (born January 15, 1936, New York City) is an American mathematician, a professor emeritus of mathematics at the University of North Carolina at Chapel Hill. He works in algebraic topology and algebra as well as their applicat ...
(1974) ''Characteristic Classes'',
Princeton University Press
Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large.
The press was founded by Whitney Darrow, with the financial ...
{{ISBN, 0-691-08122-0
* Takashi Kod
An Introduction to the Geometry of Homogeneous Spacesfrom
Kyungpook National University
Kyungpook National University (경북대학교, abbreviated as KNU or Kyungdae, 경대) is one of ten Flagship Korean National Universities representing Daegu Metropolitan City and Gyeongbuk Province in South Korea. It is located in the Dae ...
* Menelaos Zikidi
Homogeneous Spacesfrom
Heidelberg University
}
Heidelberg University, officially the Ruprecht Karl University of Heidelberg, (german: Ruprecht-Karls-Universität Heidelberg; la, Universitas Ruperto Carola Heidelbergensis) is a public research university in Heidelberg, Baden-Württemberg, ...
*
Shoshichi Kobayashi,
Katsumi Nomizu (1969) ''
Foundations of Differential Geometry'', volume 2, chapter X, (Wiley Classics Library)
Topological groups
Lie groups