\begin{definition}[Definition:Homogeneous Riemannian Manifold]
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'''.
{{NamedforDef|Bernhard Riemann}}
\end{definition}