In 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 *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 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.

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*.^{ [1] } 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:

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*.

For example, if *X* is a topological space, then group elements are assumed to act as homeomorphisms on *X*. The structure of a *G*-space is a group homomorphism *ρ* : *G* → Homeo(*X*) into the homeomorphism group 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):

- 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, point stabilizer orthogonal group, corresponding to hyperboloid model):
**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 (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)

From the point of view of the Erlangen program, one may understand that "all points are the same", in the geometry 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, 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). 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 of Julius Plücker.

In general, if *X* is a homogeneous space, and *H*_{o} is the stabilizer of some marked point *o* in *X* (a choice of origin), 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.

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

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 of *G*. In particular, if *G* is a Lie group, then *H* is a Lie subgroup by Cartan's theorem. 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 space.

One can go further to *double* coset spaces, notably Clifford–Klein forms *Γ*\*G*/*H*, where *Γ* is a discrete subgroup (of *G*) acting properly discontinuously.

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 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.

The idea of a prehomogeneous vector space was introduced by Mikio Sato.

It is a finite-dimensional vector space *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".

Physical cosmology using the general theory of relativity makes use of the Bianchi classification 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.^{ [2] }

A homogeneous space of *N* dimensions admits a set of Killing vectors.^{ [3] } 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.

- ↑ We assume that the action is on the
*left*. The distinction is only important in the description of*X*as a coset space. - ↑ Lev Landau and Evgeny Lifshitz (1980),
*Course of Theoretical Physics vol. 2: The Classical Theory of Fields*, Butterworth-Heinemann, ISBN 978-0-7506-2768-9 - ↑ Steven Weinberg (1972),
*Gravitation and Cosmology*, John Wiley and Sons

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 group. It is, loosely speaking, the symmetry group of the object.

In mathematics, a **diffeomorphism** is an isomorphism 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.

In the mathematical field of representation theory, **group representations** describe abstract groups in terms of bijective linear transformations of vector spaces; in particular, they can be used to represent group elements as invertible matrices so that the group operation can be represented by matrix multiplication. Representations of groups are important because they allow many group-theoretic problems to be reduced to problems in linear algebra, which is well understood. They are also important in physics because, for example, they describe how the symmetry group of a physical system affects the solutions of equations describing that system.

In mathematics, a **group action** on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism group of the structure. It is said that the group *acts* on the space or structure. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in it. In particular, it acts on the set of all triangles. Similarly, the group of symmetries of a polyhedron acts on the vertices, the edges, and the faces of the polyhedron.

In group theory, the **symmetry group** of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient space which takes the object to itself, and which preserves all the relevant structure of the object. A frequent notation for the symmetry group of an object *X* is *G* = Sym(*X*).

In mathematics, the **orthogonal group** in dimension *n*, denoted O(*n*), is the group of distance-preserving transformations of a Euclidean space of dimension *n* that preserve a fixed point, where the group operation is given by composing transformations. The orthogonal group is sometimes called the **general orthogonal group**, by analogy with the general linear group. Equivalently, it is the group of *n*×*n* orthogonal matrices, where the group operation is given by matrix multiplication; an orthogonal matrix is a real matrix whose inverse equals its transpose. The orthogonal group is an algebraic group and a Lie group. It is compact.

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 physicist Hendrik Lorentz.

In mathematics, the **affine group** or **general affine group** of any affine space over a field K is the group of all invertible affine transformations from the space into itself.

In mathematics, a **principal bundle** is a mathematical object that formalizes some of the essential features of the Cartesian product *X* × *G* of a space *X* with a group *G*. In the same way as with the Cartesian product, a principal bundle *P* is equipped with

- An action of
*G*on*P*, analogous to (*x*,*g*)*h*= for a product space. - A projection onto
*X*. For a product space, this is just the projection onto the first factor, (*x*,*g*) ↦*x*.

In mathematics, a **frame bundle** is a principal fiber bundle F(*E*) associated to any vector bundle *E*. The fiber of F(*E* ) over a point *x* is the set of all ordered bases, or *frames*, for *E*_{x}. The general linear group acts naturally on F(*E *) via a change of basis, giving the frame bundle the structure of a principal GL(*k*, **R**)-bundle.

In mathematics, a **generalized flag variety** is a homogeneous space whose points are flags in a finite-dimensional vector space *V* over a field **F**. When **F** is the real or complex numbers, a generalized flag variety is a smooth or complex manifold, called a **real** or **complex****flag manifold**. Flag varieties are naturally projective varieties.

In the mathematical field of differential geometry, a **Cartan connection** is a flexible generalization of the notion of an affine connection. It may also be regarded as a specialization of the general concept of a principal connection, in which the geometry of the principal bundle is tied to the geometry of the base manifold using a solder form. Cartan connections describe the geometry of manifolds modelled on homogeneous spaces.

In differential geometry, a ** G-structure** on an

In mathematics, the **Stiefel manifold** is the set of all orthonormal *k*-frames in That is, it is the set of ordered orthonormal *k*-tuples of vectors in It is named after Swiss mathematician Eduard Stiefel. Likewise one can define the complex Stiefel manifold of orthonormal *k*-frames in and the quaternionic Stiefel manifold of orthonormal *k*-frames in . More generally, the construction applies to any real, complex, or quaternionic inner product space.

In mathematics, a **symmetric space** is a pseudo-Riemannian manifold whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, leading to consequences in the theory of holonomy; or algebraically through Lie theory, which allowed Cartan to give a complete classification. Symmetric spaces commonly occur in differential geometry, representation theory and harmonic analysis.

In mathematics, **D _{3}** (sometimes alternatively denoted by

In mathematics, a **Gelfand pair** is a pair *(G,K)* consisting of a group *G* and a subgroup *K* that satisfies a certain property on restricted representations. The theory of Gelfand pairs is closely related to the topic of spherical functions in the classical theory of special functions, and to the theory of Riemannian symmetric spaces in differential geometry. Broadly speaking, the theory exists to abstract from these theories their content in terms of harmonic analysis and representation theory.

In mathematics, a **Hermitian symmetric space** is a Hermitian manifold which at every point has an inversion symmetry preserving the Hermitian structure. First studied by Élie Cartan, they form a natural generalization of the notion of Riemannian symmetric space from real manifolds to complex manifolds.

In mathematics, the **classical groups** are defined as the special linear groups over the reals **R**, the complex numbers **C** and the quaternions **H** together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, complex and quaternionic finite-dimensional vector spaces. Of these, the **complex classical Lie groups** are four infinite families of Lie groups that together with the exceptional groups exhaust the classification of simple Lie groups. The **compact classical groups** are compact real forms of the complex classical groups. The finite analogues of the classical groups are the **classical groups of Lie type**. The term "classical group" was coined by Hermann Weyl, it being the title of his 1939 monograph *The Classical Groups*.

In mathematics, and more precisely in topology, the **mapping class group** of a surface, sometimes called the **modular group** or **Teichmüller modular group**, is the group of homeomorphisms of the surface viewed up to continuous deformation. It is of fundamental importance for the study of 3-manifolds via their embedded surfaces and is also studied in algebraic geometry in relation to moduli problems for curves.

- John Milnor & James D. Stasheff (1974)
*Characteristic Classes*, Princeton University Press ISBN 0-691-08122-0 - Takashi Koda An Introduction to the Geometry of Homogeneous Spaces from Kyungpook National University
- Menelaos Zikidis Homogeneous Spaces from Heidelberg University
- Shoshichi Kobayashi, Katsumi Nomizu (1969)
*Foundations of Differential Geometry*, volume 2, chapter X, (Wiley Classics Library)

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