Reductive Cartan geometry

(see also reductive Klein geometry)
A Cartan geometry $P$ of type $(G,H)$ on $M$ is called reductive if the Lie algebra $\mathfrak{g}$ seen as a $Ad(H)$-module (see this) can be decomposed

with $\mathfrak{g}$ and $\mathfrak{h}$ being the corresponding Lie algebras.

Interpretation

Let's denote $M=P/H$. In a reductive Cartan geometry we can identify $T_{\pi (p)}M$ with a subspace of $\mathfrak{g}$ complementary to $\mathfrak{h}$ and invariant under the adjoint action of $H$. So we have a kind of "horizontal" direction. Let's see.

You can fix a choice of $\mathfrak{p}$ and you get a projection $f:\mathfrak{g}\to \mathfrak{h}$. This way, the Cartan connection (or the MC form, if we are in a Klein geometry) can be split

The 1-form $\omega =A_{\mathfrak{h}}$ is the 1-form of a principal connection on a principal bundle, and $\ker (A_{\mathfrak{h}})$ describe the horizontal subspaces. The map $d{\pi }_p:\ker (A_{\mathfrak{h}})\to T_{\pi (p)}M$ is a isomorphism, since it is surjective, and dimensions agree. Observe that if $V\in V_{p}P$ then $A_p(V)={\omega }_p(V)$.

Personal belief
Conversely, given a Cartan geometry and a principal connection on its principal bundle such that $A_p(V)={\omega }_p(V)$ we can define a projection $f:\mathfrak{g}\to \mathfrak{h}$ given by

and then $\mathfrak{g}=\text{im}(f)\oplus \ker f=\mathfrak{h}\oplus \mathfrak{p}$, since the short exact sequence

split. I think it can be proved that $\mathfrak{p}$ is a $Ad(H)$-module and that $f$ is well-defined, i.e., independent of $p\in P$. See my own question in MO.
$◼$

In any case, if we have a reductive Cartan geometry, we have a isomorphism between $T_{\pi (p)}M$ and $\mathfrak{p}$:

This means that reductive Cartan geometries are geometries in which a little displacement on the base manifold from $x:=\pi (p)\in M$ to a nearby point $x^{\prime }\in M$ corresponds to a canonical element $v\in \mathfrak{p}\subset \mathfrak{g}$. A kind of "canonical translation" or transvection (see "Symmetric Space Cartan Connections and Gravity in Three and Four Dimensions" by Derek Wise page 3). That is, we obtain a one-parameter subgroup of the group of $G$ of privileged transformations of the geometry, associated to this little displacement. This is in total analogy with what happens in the case of affine space and usual translations.

Conversely, given a Klein geometry in which there is an isomorphism from $T_{\pi (p)}M$ to a subspace $\mathfrak{p}$ of $\mathfrak{g}$ then the short exact sequence

splits, and $\mathfrak{g}=\mathfrak{h}\oplus \mathfrak{p}$. Of course, several properties have to be checked...

Continue with Jacob Erickson...