Up to topology, the 5D homogeneous space
$$
G\_2/P
$$
of the (real form of the) 14D exceptional Lie group $G\_2$ is the 5D jet space
$$
M:=J^1(2,1)=\{(x,y,u,p,q)\}
$$
of scalar functions in two independent variables.



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> WHY? Because $G\_2/P=S^2\times S^3\approx\mathbb{P}T^\*S^3\approx\mathbb{P}T^\*\mathbb{R}^3=\mathbb{R}^2\times(\mathbb{R}\times\mathbb{R}^{2\ast})=J^2(2,1)$, where $\mathbb{R}^2$ is the plane of independent variables, $\mathbb{R}$ the line of the dependent one, and $\mathbb{R}^{2\ast}$ the plane of the derivatives of the latter w.r.t. the former.
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Now $M$ comes equipped with a flag of distributions, viz
$$
(M,V,C)\, ,
$$
where $V=\langle\partial\_p,\partial\_q\rangle$ is the 2D vertical and $C=V\oplus\langle \partial\_x+p\partial\_u, \partial\_y+q\partial\_u\rangle$ the 4D contact one.


Now there is a result (I've learned it from P. Nurowski) saying that



> 
> (THEOREM) There is a one-to-one correspondence between $(G\_2,P)$-type Cartan geometries and generic 2D distributions in dimension 5.
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Since - as textbooks recite - a $(G\_2,P)$-type Cartan geometry is obtained by rolling without slipping the homogeneous space $G\_2/P\approx M$, I guess that a generic 2D distribution in dimension five is obtained by "rolling without slipping" the vertical distribution $V$ of the homogeneous model $M$. This should be the essence of the theorem.


**My point** is simple: if I can "roll" the vertical distribution $V$, why I cannot "roll" the contact distribution $C$ as well?



> 
> **QUESTION:** Is it true that a $(G\_2,P)$-type Cartan geometry in dimension five is the same as a manifold equipped with a $(2,4)$-type flag of distributions $(\widetilde{V},\widetilde{C})$? Under which circumstances $\widetilde{C}$ is contact? (This makes sense, being so at least for the flat case $M$.)
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Maybe the question is ill-posed, due to my still poor grasp on Cartan geometries, but I hope that its essence will be clear: the "flat model", i.e., $M$, is a contact manifold equipped with a distinguished 2D contact-subdistribution (no doubts about that), and I'm just wondering how much of this structure descends to the "curved cases". 


*"I'm just a simple man trying to make my way in the universe of Cartan geometries"* - J. Fett.



Contact distributions on $(G_2,P)$-type Cartan geometries in dimension 5
As Ben wrote, the question appears to conflate two different parabolic geometries of type $\newcommand{bfD}{{\bf D}}\newcommand{bfE}{{\bf E}}\newcommand{bfH}{{\bf H}}G\_2$:


Let $\Bbb V$ be the standard (i.e., $7$-dimensional irreducible) representation of $\mathfrak{g}\_2$ (either the split real or the complex form); recall that $G\_2$ is the stabilizer of cross product map $\times : \Bbb V \times \Bbb V \to \Bbb V$. The inclusion $G\_2 \hookrightarrow SO(3, 4)$ ($G\_2 \hookrightarrow SO(7, \Bbb C)$) determines an indefinite, nondegenerate, symmetric bilinear form $H$ on $\Bbb V$.


The "first" parabolic subgroup $P\_1$ (corresponding to a cross on the first node of the Dynkin diagram of $G\_2$ in the usual Bourbaki ordering) is the stabilizer of an isotropic $1$-dimensional subspace of $\Bbb V$. The cone $\mathcal C$ of nonzero isotropic vectors inherits an invariant filtration of tangent distributions, whose fibers at $Y \in \mathcal C$ are $$\ker (Z \mapsto Y \times Z) \subset \operatorname{im} (Z \mapsto Y \times Z) \subset T\_Y \mathcal C$$ of dimensions $3, 4, 6$ (to apply the cross product, we implicitly use here the identifications determined by the canonical isomorphism $T\_Y \Bbb V \leftrightarrow \Bbb V$). By linearity, this descends to a filtration $$\bfD \subset \bfD' \subset T\Bbb Q\_5,$$ of dimensions $2, 3, 5$, on the null quadric $\Bbb{Q}\_5 := \Bbb P(\mathcal C) \subset \Bbb P(\Bbb V)$, which is diffeomorphic to $(\Bbb S^2 \times \Bbb S^3) / \Bbb Z\_2$, where $\Bbb Z\_2$ acts by the antipodal map on both factors. Since the ingredients are $G\_2$-invariant, so is $\bfD$ under the induced action on $T\Bbb Q\_5$, and as one expects, it turns out that $[\bfD, \bfD] = \bfD'$ and $[\bfD', \bfD] = T\Bbb Q\_5$. (NB there are no $G\_2$-invariant linear or hyperplane distributions on $T\Bbb Q\_5$.)


On the other hand, consider differential equations of the form $z' = F(x, y, y', y'', z)$. Any function $F(x, y, p, q, z)$ determines a total derivative $D\_x := \partial\_x + p \partial\_y + q \partial\_p + F \partial\_z$ on the corresponding partial jet space $J^{2, 0}(\Bbb R, \Bbb R) \cong \Bbb R^5\_{xypqz}$. Suitably regarded, the vertical fibers of the jet truncation map $J^{2, 0}(\Bbb R, \Bbb R) \to J^{1, 0}(\Bbb R, \Bbb R)$ are spanned by $\partial\_q$. Computing directly shows that the distribution $$\bfD\_F := \langle D\_x, \partial\_q \rangle$$ is generic iff $F\_{qq}$ vanishes nowhere; conversely, a theorem of (I believe) Monge states than any generic $2$-plane distribution on a $5$-manifold is locally equivalent to $\bfD\_F$ for some function $F$. If $F(x, y, p, q, z) := q^2$, then the resulting distribution has infinitesimal symmetry algebra isomorphic to $\mathfrak{g}\_2$, so by a general fact about parabolic geometries the distribution $(J^{2, 0}(\Bbb R, \Bbb R), \bfD\_F)$ corresponding to the differential equation $z' = (y'')^2$ is everywhere locally diffeomorphic to the homogeneous model distribution $(\Bbb Q\_5, \bfD)$ above. In particular, it follows from this that *neither of the distributions $V$ and $\widetilde{C}$ are invariant under the action of the infinitesimal symmetry algebra $\mathfrak{g}\_2$ of $(J^{2, 0}(\Bbb R, \Bbb R), \bfD\_F)$*.


The correct statement of the theorem you mention is that there is an equivalence of categories between generic $2$-plane distributions on $5$-manifolds and *normal, regular* parabolic geometries of type $(G\_2, P\_1)$. (See the end of Subsubsection 4.3.2 in Cap & Slovak's book, *Parabolic Geometries*.)


On the other hand, we can consider the action of $G\_2$ on the space of isotropic $2$-planes in $\Bbb V$. This action has two orbits, according to whether the cross product $\times$ restricts to the zero map on each $2$-plane. (Bryant calls the $2$-planes on which the restriction is zero *special* in his highly enjoyable lecture notes [*Elie Cartan and Geometric Duality*](https://services.math.duke.edu/%7Ebryant/Cartan.pdf) [pdf], which treats the correspondence space construction for $G\_2 / P\_1 \leftarrow G\_2 / (P\_1 \cap P\_2) \to G\_2 / P\_2$, as well as the analogous construction for $A\_2$ and $B\_2 \cong C\_2$. NB that this article seems contains a few typos, replacing $\Bbb N\_5$, introduced in a moment, with $\Bbb Q\_5$, which is the essential apparent confusion in the question here.) The isotropy subgroup of a point in the $5$-dimensional space $\Bbb N\_5$ of special $2$-planes is the "second" parabolic $P\_2 \subset G\_2$. Analogously to the situation for the first parabolic, we can view $\Bbb N\_5$ as a subset of $\Bbb P^{13} = \Bbb P(\mathfrak{g}\_2)$, but its geometry is apparently much more complicated than that of $\Bbb Q\_5$: In the complex case, $\Bbb N\_5$ is a variety of degree $18$, and its complete intersection with three hyperplanes in a general configuration is a K3 surface of genus $10$, but NB other geometric descriptions of this space (which look less daunting to non-algebraic geometers like myself) are available, too. Apparently this is worked out in the paper of Borcea cited below, but I can't find an ungated copy. See also the accessible historical survey paper of Agricola, also cited below.


Now, $\Bbb N\_5$ inherits an invariant contact distribution $\bfH$. (Surely this can be written down with some much effort in terms of the cross product on $\Bbb V$, but to my knowledge this hasn't been done anywhere.) Moreover, the representation $P\_2$ induces on each fiber of $\bfH$ turns out to be a trivial extension of a representation of $GL(2, \Bbb F) \subset P\_2$, and this representation is isomorphic to $S^3 \Bbb F^2$ (this representation is conformally symplectic and so determines equivalently a nondegenerate cone in each fiber of $\bfH$). All of the $G\_2$-invariant structure on $\Bbb N\_5$ can be recovered from these objects, corresponding to the fact that a (normal, regular) parabolic geometry of type $(G\_2, P\_2)$ is a $5$-manifold $M$ equipped with a "$G\_2$ contact structure", which is a contact structure $\bfH \subset TM$ together with an auxiliary rank-$2$ vector bundle $\bfE \to M$ and a vector bundle isomorphism $S^3 \bfE \stackrel{\cong}{\to} \bfH$ such that the Levi bracket $\bfH \times \bfH \to TM / \bfH$ (the tensorial map induced by the Lie bracket) is invariant under the induced action of $\mathfrak{sl}(\bfE)$. See Subsubsection 4.2.8 of Cap & Slovak's book. I don't know of any sensible analog of the Monge (quasi-)normal form $z' = F(\cdots)$ for $G\_2$ contact structures, but I would be pleased to hear about one.


There are connections between these two types of parabolic geometry beyond the mentioned correspondence space construction. See this [preprint of Leistner, Nurowski, & Sagerschnig](http://arxiv.org/pdf/1601.03979v2.pdf).



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> I. Agricola, [Old and New on the Exceptional Group $G\_2$](http://www.ams.org/notices/200808/tx080800922p.pdf) [pdf], *Notices Amer. Math. Soc.* **55**(8) (2008), 922-929.
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> C. Borcea, Smooth global complete intersections in certain compact homogeneous complex manifolds, *J. Reine Angew. Math.* **344** (1983), 65–70.
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> A. Cap, J. Slovak, *Parabolic geometries I: Background and general theory*. Math. Surveys Monogr. **154**, Amer. Math. Soc., Providence, RI, 628pp.
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> T. Leistner, P. Nurowski, K. Sagerschnig, *New relations between $G\_2$-geometries in dimensions $5$ and $7$*, Internat. J. Math. **28**(13) (2017). [arXiv:1601.03979](http://arxiv.org/abs/1601.03979)
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