Definition:Homogeneous Riemannian Manifold

Definition
Let $\struct {M, g}$ be a Riemannian manifold.

Let $\map {\text {Iso}} {M, g}$ be the set of all isometries from $M$ to itself.

Suppose $\map {\text {Iso}} {M, g}$ acts on $\struct {M, g}$ transitively:


 * $\forall p, q \in M : \exists \phi \in \map {\text {Iso}} {M, g} : \map \phi p = q$

Then $\struct {M, g}$ is called the homogeneous Riemannian manifold.