In mathematics, a homogeneous space is, very informally, a space that looks the same everywhere as one moves through it, with movement given by the action of a group. Homogeneous spaces occur in the theories of Lie groups, algebraic groups and topological groups. More precisely, 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 that 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. If X is an object of the category C, then the structure of a G-space is a homomorphism:
ρ : G → A u t C ( X ) {\displaystyle \rho :G\to \mathrm {Aut} _{\mathbf {C} }(X)}
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, 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): Sn−1 ≅ O(n) / O(n−1). This is true because of the following observations: First, Sn−1 is the set of vectors in Rn with 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 of Rn, then the complement is an (n − 1)-dimensional vector space that is invariant under an orthogonal transformation from O(n − 1). This shows us why we can construct Sn−1 as a homogeneous space. Oriented sphere (special orthogonal group): Sn−1 ≅ SO(n) / SO(n − 1) Projective space (projective orthogonal group): Pn−1 ≅ PO(n) / PO(n − 1) Flat (zero curvature): Euclidean space (Euclidean group, point stabilizer is orthogonal group): En ≅ E(n) / O(n) Negative curvature: Hyperbolic space (orthochronous Lorentz group, point stabilizer orthogonal group, corresponding to hyperboloid model): Hn ≅ O+(1, n) / O(n) Oriented hyperbolic space: SO+(1, n) / SO(n) Anti-de Sitter space: AdSn+1 = O(2, n) / O(1, n) Others Affine space over field K (for affine group, point stabilizer general 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 includes Minkowski space Mn ≅ ISO(n-1,1) / SO(n,1) or Galilean and Carrollian spaces.
Geometry 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 GL4 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.
Homogeneous spaces as coset spaces In general, if X is a homogeneous space of G, and Ho is the stabilizer of some marked point o in X (a choice of origin), the points of X correspond to the left cosets G/Ho, 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 {e}, then X is the G-torsor, which explains why G-torsors are often described intuitively as "G with forgotten identity". In general, a different choice of origin o will lead to a quotient of G by a different subgroup Ho′ that is related to Ho by an inner automorphism of G. Specifically,
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