Let
$\require{AMScd}$
\begin{CD}
 G\_2/(P\_1\cap P\_2) @= G\_2/(P\_1\cap P\_2)=:\mathbb{I}\\
 @V \lambda V V @VV \pi V\\
 \mathbb{Q}\_5:=G\_2/P\_1 && G\_2/P\_2=:\mathbb{N}\_5
\end{CD}
be the Tits fibration of the exceptional Lie group $G\_2$ (see [Landsberg and Manivel](https://arxiv.org/pdf/math/9810140.pdf), § 4.1, for the general definition, and [Bryant](https://services.math.duke.edu/~bryant/Cartan.pdf), pag. 12, for the particular example of $G\_2$).


I recall that $\mathbb{I}$ is a $\mathbb{P}^1$-bundle over the 5-dimensional contact manifold $\mathbb{N}\_5$, such that, for any $p\in \mathbb{N}\_5$, the fibre $\pi^{-1}(p)$ is a twisted cubic in $\mathbb{P}\mathcal{C}\_p\cong\mathbb{P}^3$ - the projectivised contact space at $p$. In particular, we can think of $\mathbb{I}$ as a sub-bundle of $\mathbb{P}\mathcal{C}$, which in turn is a sub-bundle of $\mathbb{P}T\mathbb{N}\_5$.


Let now $\gamma\subset\mathbb{N}\_5$ be a curve, and denote by $T\gamma\subset T\mathbb{N}\_5|\_\gamma$ its (rank-one) tangent bundle. Since each fibre of $T\gamma$ is a tangent line to $\mathbb{N}\_5$, it makes sense to regard $T\gamma$ as a curve in $\mathbb{P}T\mathbb{N}\_5$.


Then one can prove that $$\mathbb{Q}\_5=\{\gamma\mid T\gamma\subset\mathbb{I} \}\, .\quad\quad (^\*)$$ 


Now $\mathbb{N}\_5$ is just the homogeneous model of Cartan geometries of type $(G\_2,P\_2)$, and I'm wodering **to what extent the Tits fibration can be recast in a non-flat setting**.


More pecisely, let $M$ be a Cartan geometry of type $(G\_2,P\_2)$. In particular, there exists a $\mathbb{P}^1$-bundle $\mathbb{I}\to M$, such that $\mathbb{I}\subset\mathbb{P}TM$. This means that definition $(^\*)$ can be repeated verbatim, viz.
$$
Q\_M:=\{\gamma\textrm{ curve in }M\mid T\gamma\subset\mathbb{I}\}\, .
$$



> 
> **QUESTION**. It is obvious that $Q\_M$ is a Cartan geometry of type $(G\_2,P\_1)$? If yes, what is a good reference for that?
> 
> 
> 


*This is related to [a previous question](https://mathoverflow.net/q/250590/22606), which led to interesting discussions, but it is still unanswered (see also [this one](https://mathoverflow.net/q/229825/22606)).*



What is the curved version of the Tits fibration for $G_2$?
This is an instance of the general construction of correspondence spaces and twistor spaces as described in my article [MR2139714](http://www.ams.org/mathscinet/search/publdoc.html?pg1=INDI&s1=294326&vfpref=html&r=31&mx-pid=2139714 "MR2139714") and in Chapter 4 of the book of Jan Slovak and myself on parabolic geometries. A Cartan geometry of of type $(G\_2,P\_2)$ can be described as a 5-dimensional contact manifold $(M,H)$ together with an auxilliary rank-two bundle $E\to M$ such that $H\cong S^3E$. (This is an equivalent way to phrase the twisted cubic you mention.) Now the correspondence space $N:=\mathcal CM$ is the total space of the projectivization of $E$ and thus a circle bundle over $M$. Denoting by $\mathcal G\to M$ the Cartan bundle, it is easy to see that $N\cong \mathcal G\times\_{P\_2}(P\_1\cap P\_2)\cong\mathcal G/(P\_1\cap P\_2)$. This shows that there is a canonical projection $\pi:\mathcal G\to N$, which is a principal fiber bundle with structure group $P\_1\cap P\_2$. The canonical Cartan connection $\omega$ associated to the original geometry by definition also is a Cartan connection on this bundle, thus defining a parabolic geometry of type $(G\_2,P\_1\cap P\_2)$ on $N$. 


This shows that $TN\cong\mathcal G\times\_{P\_1\cap P\_2}(\mathfrak g/(\mathfrak p\_1\cap\mathfrak p\_2))$ and hence $\mathfrak p\_1/(\mathfrak p\_1\cap\mathfrak p\_2)$ defines a line subbundle in $TN$. For this you can form a local leaf space $Z$, which is called a twistor space, so locally (on $N$) you have a double fibration picture. But you won't get a Cartan geometry on $Z$ unless the initial geometry is locally flat. I am not entirely sure how things look like in the example you are considering but I think that you can identify the pre-image of $Z$ in $\mathcal G$ with an open subset in a principal $P\_1$-bundle over $Z$. But carrying $\omega$ over to this bundle, the result will be $P\_1$-equivariant (thus defining a Cartan connection) if and only if the initial geometry is locally flat. So there is no way to directly get a Cartan geometry in non-flat cases. 


Of course, one could try to directly construct a rank two-distribution on a local leaf space $Z$ from the given data (and it would be easy to find candidates), maybe imposing some restrictions on the curvature of the initial geometry. I have not tried that, but I wouldn't be very optimistic. 


