# Definition:Frame-Homogeneous Riemannian Manifold

## Definition

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

Let $\map O M$ be the set of the disjoint union of all orthonormal frames for all tangent spaces of $M$:

- $\ds \map O M = \bigsqcup_{p \mathop \in M} \set {\text{all orthonormal bases for } T_p M}$

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

Let $\phi$ be an isometry.

Let the induced action of $\map {\operatorname {Iso}} {M, g}$ on $\map O M$ be defined by using the differential of $\phi$ to push an orthonormal basis at $p \in M$ forward to an orthonormal basis at $\map \phi p$:

- $\phi \circ \tuple {b_1, \dots, b_n} = \tuple {\map {d \phi_p} {b_1}, \ldots, \map {d \phi_p} {b_2} }$

Suppose the induced action is transitive on $\map O M$.

That is, suppose for all $p, q \in M$ and all choices of orthonormal bases at $p$ and $q$, there is an isometry taking $p$ to $q$ and the chosen basis at $p$ to the one at $q$.

Then $\struct {M, g}$ is said to be **frame-homogeneous**.

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## Sources

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