Reductive Klein geometry

See also reductive Cartan geometry.

Definition

The same as reductive Cartan geometry.

Characterization of reductive Klein geometries

Question for MO of myself here. As a conclusion:
"So a reductive homogeneous space, in the sense of Sharpe's book, is precisely a homogeneous space with invariant connection on the bundle $G\to G/H$, i.e. precisely a homogeneous space with an $H$-module decomposition $\mathfrak{g}=\mathfrak{h}\oplus \mathfrak{p}$."
"The choice of $\mathfrak{p}$ is not given a priori: rather, every $G$-invariant connection on $M=G/H$ determines one such $\mathfrak{p}$ and vice versa, so the choices of $\mathfrak{p}$ are arbitrary $\text{Ad}H$-invariant complements to $\mathfrak{h}\subseteq \mathfrak{g}$.""

Riemannian homogeneous spaces

A Riemannian manifold $(M,g)$ is a homogeneous Riemannian manifold if its isometry group $I(M)$ acts transitively on $M$. It is known that all Riemannian homogeneous spaces are reductive (said in the introduction of this).