What is the space parametrising the curved sub-Cartan geometries of a flat Cartan geometry?

Question

I'm basically wondering how to make "curved" the first column of the diagram $\require{AMScd}$ \begin{CD} P_1 @>\textrm{inclusion} >> G\\ @V \omega_0 V P_1\cap P_2 V @V\omega V P_2 V\\ P_1/(P_1\cap P_2) @= G/P_2 \end{CD} where $M:=G/P_2$ is a compact homogeneous space, such that the smaller group $P_1\subset G$ still acts transitively on $M$. Above, I regard $M$ as a flat Cartan geometry of type $(G,P_2)$, with Maurer-Cartan form $\omega$.

Simultaneously, I want to regard $M$ as a (non necessarily flat) Cartan geometry of type $(P_1,P_1\cap P_2)$, which is compatible with the flat $(G,P_2)$-type one, in the sense that the first column of $\require{AMScd}$ \begin{CD} \mathcal{G}_\sigma @>\textrm{inclusion} >> G\\ @V \omega_\sigma V P_1\cap P_2 V @V\omega V P_2 V\\ M @= M \end{CD} is no longer flat, i.e., $\mathcal{G}_\sigma$ is a principal sub-bundle of $G$ (and not necessarily a subgroup).

Then I see a nonempty family (it contains at least the flat case) $$ \{(\mathcal{G}_\sigma\to M, \omega_\sigma)\}_{\sigma\in \boldsymbol{\Sigma}} $$ of compatible Cartan geometries of type $(P_1,P_1\cap P_2)$.

MAIN QUESTION: what is the space $\boldsymbol{\Sigma}$ parametrising these compatible "sub-Cartan geometries"?

A rather brutal reasoning led me to believe that $$ \boldsymbol{\Sigma}=\Gamma(\pi), $$ where $\pi:G/N_{P_2}(P_1\cap P_2)\longrightarrow G/P_2=M$. Indeed, the generic fibre of $\pi$ is $P_2/N_{P_2}(P_1\cap P_2)$, which tells how many subgroups exist in $P_2$, isomorphic to $P_1\cap P_2$: hence, a section $\sigma$ of $\pi$ allows me to construct a principal $(P_1\cap P_2)$-subbundle $\mathcal{G}_\sigma$ of $G$.

Finally, suppose that $\boldsymbol{\Sigma}$ admits a geometric description (similar, at least in spirit, to my brute attempt above) in terms of natural structures associated to $M$.

SIDE QUESTION: what is the correct way to "pull-back" the Maurer-Cartan form $\omega$ from $G$ to $\mathcal{G}_\sigma$ thus obtaining $\omega_\sigma$? (In the sense that it should be a pull-back which introduce somehow the curvature.)

If my question is well-posed, then I cannot believe that this matter has not been given attention before: any reference will be appreciated!

Answer 1

I don't think that this is a real answer to your question, but there is a general concept of extension functors for Cartan geometries. Such a functor extending Cartan geometries of type $(G,P)$ to Cartan geometries of type $(K,L)$ is determined by a homomorphism $i:P\to L$ and a linear map $\alpha:\mathfrak g\to\mathfrak k$ which is equivariant for the actions of $P$ and $L$ over $i$, restricts to $i'$ on $\mathfrak p$ and induces a linear isomorphism $\mathfrak g/\mathfrak p\to \mathfrak k/\mathfrak l$. Given a Cartan geometry $(\mathcal G\to M,\omega)$ you extend the bundle as $\mathcal G\times_P L$ and construct $\omega_{\alpha}$ as an equivariant extension of $\alpha\circ\omega$. The curvature of the resulting geometry is a combination of two terms, one coming from the curvature of the initial geometry, the other from the obstruction against $\alpha$ being a homomorphism of Lie algebras. I am aware of intersting examples of extension functors producing a non-flat geometry out of a flat one, for example the chains of a spherical CR structure define a non-flat path geometry. However, I don't know of an example of obtaining a flat extension from a non-flat geomtry. Indeed, I think that this would need a rather peculiar setup.

These extension functors are discussed in sections 1.5.15 and 1.5.16 of the book on parabolic geometries by J. Slovak and myself ( http://bookstore.ams.org/surv-154/ ).

Another related concept is holonomy reductions of Cartan geometries, but again this only produces locally flat Cartan geometries when starting from a locally flat one.