Homogeneous space
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.
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).
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.
The same is true of the models found of non-Euclidean geometry of constant curvature, such as hyperbolic space.
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, where g is any element of G for which go = o′.
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 by looking for the stabilizer of the subspace spanned by the first two standard basis vectors.
There are also homogeneous spaces of relevance in physics that are non-lorentzian, for example Galilean, Carrollian or Aristotelian spacetimes.
[5] 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 ξ(a)i, where the object Cabc, 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).
