Homogeneous space

Homogeneous space

Overview
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, particularly in the theories of Lie group
Lie group
In 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 group
Algebraic group
In 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 group
Topological group
In 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. A topological group is a mathematical object with both an algebraic structure and a...

s, a homogeneous space for a group
Group (mathematics)
In 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 manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

or topological space
Topological space
Topological 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 continuously
Continuous function
In 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".Continuity of...

by symmetry in a transitive
Transitivity (mathematics)
-In grammar:* Intransitive verb* Transitive verb, when a verb takes an object* Transitivity -In logic and mathematics:* Arc-transitive graph* Edge-transitive graph* Ergodic theory, a group action that is metrically transitive* Vertex-transitive graph...

way. A special case of this is when the topological group, G, in question is the homeomorphism group
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 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 effective (i.e. faithful), although the present article does not. Thus there is a group action
Group action
In 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.
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Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, particularly in the theories of Lie group
Lie group
In 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 group
Algebraic group
In 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 group
Topological group
In 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. A topological group is a mathematical object with both an algebraic structure and a...

s, a homogeneous space for a group
Group (mathematics)
In 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 manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

or topological space
Topological space
Topological 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 continuously
Continuous function
In 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".Continuity of...

by symmetry in a transitive
Transitivity (mathematics)
-In grammar:* Intransitive verb* Transitive verb, when a verb takes an object* Transitivity -In logic and mathematics:* Arc-transitive graph* Edge-transitive graph* Ergodic theory, a group action that is metrically transitive* Vertex-transitive graph...

way. A special case of this is when the topological group, G, in question is the homeomorphism group
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 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 effective (i.e. faithful), although the present article does not. Thus there is a group action
Group action
In 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 category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...

, then the elements of G are assumed to act as automorphism
Automorphism
In 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 homomorphism
Homomorphism
In 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 automorphism
Automorphism
In 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 space
Topological space
Topological 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 homeomorphism
Homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...

s on X. The structure of a G-space is a group homomorphism ρ : G → Homeo(X) into the homeomorphism group
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 the theory of topological spaces and in general...

of X.

Similarly, if X is a differentiable manifold
Differentiable manifold
A differentiable manifold is a type of manifold that is locally similar enough to a linear 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 diffeomorphism
Diffeomorphism
In 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 geometry
Geometry
Geometry 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 geometry
Riemannian geometry
Riemannian 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 space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

, affine space
Affine space
In mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point...

and projective space
Projective space
In 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=R2 or V=R3 are the projective line and the projective plane, respectively....

are all in natural ways homogeneous spaces for their respective symmetry group
Symmetry group
The 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 geometry
Non-Euclidean geometry
Non-Euclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much...

of constant curvature
Curvature
In 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 space
Hyperbolic space
In mathematics, hyperbolic space is a type of non-Euclidean geometry. Whereas spherical geometry has a constant positive curvature, hyperbolic geometry has a negative curvature: every point in hyperbolic space is a saddle point...

.

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
Vector space
A 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 GL4 acts transitively on those. We can parameterize them by line co-ordinates: these are the 2×2 minors
Minor (linear algebra)
In 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 4x2 matrix with columns two basis vectors for the subspace. The geometry of the resulting homogeneous space is the line geometry of Julius Plücker
Julius Plücker
Julius 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 origin
Origin (mathematics)
In 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 coset
Coset
In mathematics, if G is a group, and 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 automorphism
Inner automorphism
In abstract algebra an inner automorphism is a functionwhich, informally, involves a certain operation being applied, then another one performed, and then the initial operation being reversed...

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 group
Lie group
In 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 subgroup
Lie subgroup
In mathematics, a Lie subgroup H of a Lie group G is a Lie group that is a subset of G and such that the inclusion map from H to G is an injective immersion and group homomorphism. According to Cartan's theorem, a closed subgroup of G admits a unique smooth structure which makes it an embedded Lie...

by Cartan's theorem
Cartan's theorem
In mathematics, three results in Lie group theory are called Cartan's theorem, named after Élie Cartan:See also Cartan's theorems A and B, results of Henri Cartan, and Cartan's lemma for various other results attributed to Élie and Henri Cartan....

. Hence G/H is a smooth manifold and so X carries a unique smooth structure
Smooth structure
In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows one to perform mathematical analysis on the manifold....

compatible with the group action.

If H is the identity subgroup {e}, then X is a principal homogeneous space
Principal homogeneous space
In 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...

.

One can go further to double coset
Double coset
In mathematics, an double coset in G, where G is a group and H and K are subgroups of G, is an equivalence class for the equivalence relation defined on G by...

spaces, notably Clifford–Klein forms Γ\G/H, where Γ is a discrete subgroup (of G) acting properly discontinuously.

Example

For example in the line geometry case you can identify the the homozygeous we can identify H as a 12-dimensional subgroup of the 16-dimensional general linear group
General linear group
In 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 invertible...

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 coordinates
Homogeneous coordinates
In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry much as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points,...

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
Grassmannian
In mathematics, a Grassmannian is a space which parameterizes all linear subspaces of a vector space V of a given dimension. For example, the Grassmannian Gr is the space of lines through the origin in V, so it is the same as the projective space P. The Grassmanians are compact, topological...

, 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
Prehomogeneous vector space
In 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 Sato
Mikio 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 space
Vector space
A 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 action
Group action
In 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 group
Algebraic group
In 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 topology
Zariski topology
In 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 GL1 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

Cosmology
Cosmology
Cosmology is the discipline that deals with the nature of the Universe as a whole. Cosmologists seek to understand the origin, evolution, structure, and ultimate fate of the Universe at large, as well as the natural laws that keep it in order...

using the general theory of relativity makes use of the Bianchi classification
Bianchi classification
In 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 metrics
Metric (mathematics)
In 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 model
Physical cosmology
Physical cosmology, as a branch of astronomy, is the study of the largest-scale structures and dynamics of the universe and is concerned with fundamental questions about its formation and evolution. For most of human history, it was a branch of metaphysics and religion...

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
Mixmaster universe
The 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 anisotropic
Isotropy
Isotropy is uniformity in all orientations; it is derived from the Greek iso and tropos . Precise definitions depend on the subject area. 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 constants", form a constant order-three tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...

antisymmetric in its lower two indices (on the left-hand side, the brackets denote antisymmetrisation and ";" represents the covariant differential operator
Covariant derivative
In 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 given...

). 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
Levi-Civita symbol
The Levi-Civita symbol, also called the permutation symbol, antisymmetric symbol, or alternating symbol, is a mathematical symbol used in particular in tensor calculus...

.