In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, particularly in the theories of
Lie groupIn mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s,
algebraic groupIn algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...
s and
topological groupIn mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology...
s, a
homogeneous space for a
groupIn mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
G is a non-empty
manifoldIn mathematics, more specifically in differential geometry and topology, a manifold is a mathematical space that on a small enough scale resembles the Euclidean space of a certain dimension, called the dimension of the manifold....
or
topological spaceTopological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
X on which
G acts
continuouslyIn mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous. A continuous function with a continuous inverse function is called bicontinuous...
by symmetry in a
transitiveIn mathematics, the word transitive admits at least three distinct meanings:* A group G acts transitively on a set S if for any x, y ∈ S, there is some g ∈ G such that gx = y. See group action...
way. A special case of this is when the topological group,
G, in question is the
homeomorphism groupIn 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 the theory of topological spaces and in general...
of the space,
X. In this case
X is homogeneous if intuitively
X looks locally the same everywhere. Some authors insist that the action of
G be
effectiveIn algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
(i.e. faithful), although the present article does not. Thus there is a
group actionIn algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
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
categoryIn mathematics, a category is an algebraic structure consisting of a collection of "objects", linked together by a collection of "arrows" that have two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. Objects and arrows may...
, then the elements of
G are assumed to act as
automorphismIn mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism...
s in the same category. Thus the maps on
X effected by
G are structure preserving. 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
homomorphismIn abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...
into the group of
automorphismIn mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism...
s 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 spaceTopological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
, then group elements are assumed to act as
homeomorphismIn the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between two topological spaces that has a continuous inverse function...
s on
X. The structure of a
G-space is a group homomorphism ρ :
G → Homeo(
X) into the
homeomorphism groupIn 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 the theory of topological spaces and in general...
of
X.
Similarly, if
X is a
differentiable manifoldA differentiable manifold is a type of manifold that is locally similar enough to Euclidean space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...
, then the group elements are
diffeomorphismIn mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...
s. The structure of a
G-space is a group homomorphism ρ :
G → Diffeo(
X) into the diffeomorphism group of
X.
Geometry
From the point of view of the Erlangen programme, one may understand that "all points are the same", in the
geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
of
X. This was true of essentially all geometries proposed before
Riemannian geometryRiemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives in particular local notions of angle, length...
, in the middle of the nineteenth century.
Thus, for example,
Euclidean spaceIn mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions higher dimensions...
,
affine spaceIn mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. It can be thought of informally as a vector space where one has forgotten which point is the origin. In an affine space, one can subtract points to get vectors, or add a vector to a...
and
projective spaceIn mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R
2 or V=R
3 are the projective line and the projective plane, respectively.The idea of a projective space relates to perspective,...
are all in natural ways homogeneous spaces for their respective
symmetry groupThe symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...
s. The same is true of the models found of
non-Euclidean geometryA non-Euclidean geometry is characterized by a non-vanishing Riemann curvature tensor. Examples of non-Euclidean geometries include the hyperbolic and elliptic geometry, which are contrasted with a Euclidean geometry. The essential difference between Euclidean and non-Euclidean geometry is the...
, of constant
curvatureIn mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...
, such as
hyperbolic spaceIn mathematics, hyperbolic n-space, denoted Hn, is the maximally symmetric, simply connected, n-dimensional Riemannian manifold with constant sectional curvature −1. Hyperbolic space is the principal example of a space exhibiting hyperbolic geometry...
.
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 spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
). 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
minorsIn linear algebra, a minor of a matrix A is the determinant of some smaller square matrix, cut down from A by removing one or more of its rows or columns...
of the 2×4 matrix with columns two basis vectors for the subspace. The geometry of the resulting homogeneous space is the line geometry of
Julius PlückerJulius Plücker was a German mathematician and physicist. He made fundamental contributions to the field of analytical geometry and was a pioneer in the investigations of cathode rays that led eventually to the discovery of the electron. He also vastly extended the study of Lamé curves.- Early...
.
Homogeneous spaces as coset spaces
In general, if
X is a homogeneous space, and
Ho is the stabilizer of some marked point
o in
X (a choice of
originIn mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In a Cartesian coordinate system, the origin is the point where the axes of the system intersect...
), the points of
X correspond to the left
cosetIn mathematics, if G is a group, H is a subgroup of G, and g is an element of G, thenA coset is a left or right coset of some subgroup in G...
s
G/
Ho.
In general, a different choice of origin
o will lead to a quotient of
G by a different subgroup
Ho′ which is related to
Ho by an
inner automorphismIn abstract algebra, an inner automorphism of a group G is a functiondefined bywhere a is a given fixed element of G.The operation axa−1 is called conjugation . Informally, in a conjugation a certain operation is applied, then another one is carried out, and then the initial operation...
of
G. Specifically,
(1)
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 continuous, then
H is a closed subgroup of
G. In particular, if
G is a
Lie groupIn mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
, then
H is a closed
Lie subgroupIn mathematics, a subgroup H of a Lie group G is a Lie subgroup if the inclusion map from H to G is smooth. In particular, this implies that the inclusion map from H to G is an immersion...
by
Cartan's theoremIn mathematics, there are three results in Lie group theory that go by the name Cartan's theorem. They are both named for Élie Cartan.See also Cartan's theorems A and B, results of Henri Cartan....
. Hence
G/
H is a smooth manifold and so
X carries a unique smooth structure compatible with the group action.
If
H is the identity subgroup {
e}, then
X is a
principal homogeneous spaceIn mathematics, a principal homogeneous space, or torsor, for a group G is a homogeneous space X for G such that the stabilizer subgroup of any point is trivial...
.
Example
For example in the line geometry case, we can identify H as a 12-dimensional subgroup of the 16-dimensional
general linear groupIn mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is...
- GL4,
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 that X has dimension 4.
Since the
homogeneous coordinatesIn mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, make calculations possible in projective space just as Cartesian coordinates do in Euclidean space...
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
GrassmannianIn mathematics, a Grassmannian is a space which parameterizes all linear subspaces of a vector space V of a given dimension. For example, the Grassmannian Gr1 is the space of lines through the origin in V, so it is the same as the projective space PV...
, 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 spaceIn mathematics, a prehomogeneous vector space is a finite-dimensional vector space V together with a subgroup G of GL such that G has an open dense orbit in V. Prehomogeneous vector spaces were introduced by Mikio Sato in 1970 and have many applications in geometry, number theory and analysis, as...
was introduced by
Mikio SatoMikio Sato is a Japanese mathematician, who started the field of algebraic analysis. He studied at the University of Tokyo, and then did graduate study in physics as a student of Shin'ichiro Tomonaga...
.
It is a finite-dimensional
vector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
V with a
group actionIn algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
of an
algebraic groupIn algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...
G, such that there is an orbit of
G that is open for the
Zariski topologyIn mathematics, namely algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition. It is due to Oscar Zariski and took a place of particular importance in the field around 1950...
(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
CosmologyCosmology is the study of the Universe in its totality, and by extension, humanity's place in it...
using the general theory of relativity makes use of the
Bianchi classificationIn mathematics, the Bianchi classification, named for Luigi Bianchi, is a classification of the 3-dimensional real Lie algebras into 11 classes, 9 of which are single groups and two of which have a continuum of isomorphism classes...
system. Homogeneous spaces in relativity represent the space part of background
metricsIn mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...
for some
cosmological modelPhysical cosmology, as a branch of astronomy, is the study of the largest-scale structures and dynamics of our universe and is concerned with fundamental questions about its formation and evolution. Cosmology involves itself with studying the motions of the celestial bodies and the first cause....
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 universeThe Mixmaster Universe is a solution to Einstein's field equations of general relativity studied by Charles Misner in an effort to better understand the dynamics of the early universe...
represents an
anisotropicIsotropy is uniformity in all directions. Precise definitions depend on the subject area. The word is made up from Greek iso and tropos . Exceptions, or inequalities, are frequently indicated by the prefix an, hence anisotropy. Anisotropy is also used to describe situations where properties vary...
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 constant", is a constant
rank-three tensorTensors are geometrical entities introduced into mathematics and physics to extend the notion of scalars, vectors, and matrices. Many physical quantities are naturally regarded, not as vectors themselves, but as correspondences between one set of vectors and another...
antisymmetricIn mathematics and theoretical physics, a tensor is antisymmetric on two indices i and j if it flips sign when the two indices are interchanged:An antisymmetric tensor is a tensor for which there are two indices on which it is antisymmetric...
in its lower two indices (on the left-hand side, the brackets denote antisymmetrisation and ";" represents the
covariant differential operatorIn mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach...
). 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 symbolThe Levi-Civita symbol, also called the permutation symbol, antisymmetric symbol, or alternating symbol, is a mathematical symbol used in particular in tensor calculus. It is named after the Italian mathematician and physicist Tullio Levi-Civita.-Definition:In three dimensions, the Levi-Civita...
.