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.

Source of Name

This entry was named for Bernhard Riemann.

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 3$: Model Riemannian Manifolds. Symmetries of Riemannian Manifolds