a homogeneous space

Some spaces look the same from every point — there is no special 'center'. A sphere is like this: standing at any point, the view of the whole sphere is indistinguishable from the view at any other point, because a rotation carries one to the other. A homogeneous space is a manifold with exactly this property, made precise: a group of symmetries acts transitively, so the symmetries can move any point to any other.

Concretely, if a Lie group G acts smoothly and transitively on a manifold M, then fixing a basepoint p and letting H = G_p be its stabilizer gives a diffeomorphism M ~ G/H, the set of left cosets gH with the quotient smooth structure. Conversely, for any closed subgroup H of a Lie group G, the coset space G/H is canonically a smooth manifold of dimension dim G - dim H, and G acts on it transitively by left translation — so every homogeneous space arises as a G/H. The recipe to identify one: find a transitive symmetry group, pick a convenient point, and compute its stabilizer. Sphere: S^n = SO(n+1)/SO(n). Real projective space: RP^n = O(n+1)/(O(n) x O(1)). The Grassmannian of k-planes in R^n: Gr(k, n) = O(n)/(O(k) x O(n-k)).

Homogeneous spaces are the most symmetric manifolds and serve as the model spaces of geometry — flat space, the sphere, and hyperbolic space are the three constant-curvature homogeneous spaces (the space forms). They also give invariant geometric structures for free: any tensor at the basepoint invariant under H spreads by the action to an invariant tensor on all of G/H. The honesty caveat: a homogeneous space need not be a group itself (S^2 is homogeneous but admits no Lie group structure), and writing M = G/H is not unique — the same manifold can be presented by different pairs (G, H).