 Nurowski's conformal
structures associated to a generic 2-plane field on a 5-manifold.  Let 
$\mathcal{D}$ be a distribution of rank 2 on a manifold $M$ of dimension 5.  We
assume that $M$ is connected and, for convenience, oriented (which is
equivalent to the condition that $\mathcal{D}$ is oriented).  (Our arguments and
results all 
carry over to the nonorientable case upon replacing below $G_2$ by 
$\{\pm I\} G_2\subset O(3,4)$ and $P$ by 
$\{A\in \{\pm I\} G_2:Ae_0=\lambda e_0, \lambda>0\}$.)    
Let $X$, $Y$ be a local basis for the sections of $\mathcal{D}$.  $\mathcal{D}$ is
said to be generic if    
$X$, $Y$, $[X,Y]$, $[X,[X,Y]]$, $[Y,[X,Y]]$ are linearly independent at
each point.  In this case we denote by
$\mathcal{D}^1=\operatorname{span}\{X,Y,[X,Y]\}$ the derived rank 3 distribution.     
In \cite{C}, Cartan solved the equivalence problem for such
distributions by canonically associating to $\mathcal{D}$ a principal bundle 
$\mathcal{B}\rightarrow M$ and 
Cartan connection $\omega$ taking values in the Lie algebra $\mathfrak{g}_2$ of
$G_2$, the split real form of the exceptional group.  The structure   
group of $\mathcal{B}$ is the parabolic subgroup $P\subset G_2$ fixing a null ray.
In the appendix we fix our conventions,  
review the Cartan connection and derive two properties which will be 
needed in the sequel.  

In \cite{N1}, Nurowski showed that there is a conformal structure on $M$ of
signature $(2,3)$ 
naturally determined by $\mathcal{D}$.  This follows from the existence 
of Cartan's canonical bundle and connection and the relation between the
relevant groups.  One way to see this is as follows.  It is a general   
fact that the tangent bundle $TM$ of a manifold $M$ with a Cartan geometry
of type $(\mathfrak{g},P)$ is associated to the adjoint representation of $P$ on
$\mathfrak{g}/\mathfrak{p}$.  We claim that the adjoint
action of our $P$ on $\mathfrak{g}_2/\mathfrak{p}\cong \mathbb R^5$ preserves up to scale a
quadratic form of signature $(2,3)$.  This implies the existence
of a canonical conformal structure by the associated bundle construction.
Now $G_2$ is a subgroup of $SO(3,4)$.  Let $P_c$ be the subgroup of  
$SO(3,4)$ fixing the same null ray, with Lie algebra $\mathfrak{p}_c$.  Then
$P\subset P_c$ and $\mathfrak{s}\mathfrak{o}(3,4)/\mathfrak{p}_c\cong \mathfrak{g}_2/\mathfrak{p}$.  The adjoint
action of $P_c$ on $\mathfrak{s}\mathfrak{o}(3,4)/\mathfrak{p}_c$ preserves up to scale a
quadratic form of signature $(2,3)$; this is the reason a Cartan geometry
of type $(\mathfrak{s}\mathfrak{o}(3,4),P_c)$ induces a conformal structure.  Since the  
adjoint action of $P$ on $\mathfrak{g}_2/\mathfrak{p}$ can be regarded as the
restriction of the adjoint action of $P_c$, it preserves the same quadratic
form up to scale.  

As discussed in the introduction, any generic $\mathcal{D}$ can be written locally
in the form \eqref{Dform} for some smooth function $F$ such that
$F_{qq}$ is nonvanishing, and $\mathcal{D}$ defined by \eqref{Dform} is
generic for any such $F$.  The choice $F=q^2$ gives the homogeneous model
distribution.  For $\mathcal{D}$ in this form, Nurowski gave a  
formula in \cite{N1}, \cite{N2} for a representative metric $g_F$
of the conformal class 
such that the components of $g_F$ and $g_F^{-1}$ are polynomials in $F$,
the derivatives of $F$ of orders $\leq 4$, and $F_{qq}^{-1}$, with 
coefficients which are universal functions of the local coordinates.  The
metric $g_F$ is flat for $F=q^2$.  
In \cite{N2}, he considered the case $F=q^2+\sum_{k=0}^6a_kp^k + bz$ with
$a_k$, $b\in \mathbb R$, 
and gave an explicit formula for the ambient metric $\widetilde{g}_F$ in normal form
relative to $g_F$.  For these $g_F$, $g_\rho$ in \eqref{normalform} is a
polynomial in $\rho$ of degree $\leq 2$.  In \cite{LN}, Leistner-Nurowski
showed that the holonomy of $\widetilde{g}_F$ is contained in $G_2$ for all values of
the $a_k$'s and $b$, and is equal to $G_2$ if one of $a_3$, $a_4$, $a_5$ 
or $a_6$ is nonzero.  That the holonomy is contained in $G_2$ is
proved by exhibiting explicitly a suitable parallel 3-form (or equivalently
a nonisotropic parallel spinor) for $\widetilde{g}_F$.  

In \cite{HS}, Hammerl-Sagerschnig characterized those conformal structures
of signature $(2,3)$ which arise from a generic distribution $\mathcal{D}$ by the
existence of a parallel tractor 3-form $\chi$ compatible with 
the tractor metric $h$ in the sense that
$$
(U\mathbin{\intprod} \chi)\wedge (V\mathbin{\intprod} \chi)\wedge \chi = \lambda\, h(U,V)\, dv,\qquad  
U,V\in \mathcal{T}_x,
$$
where $dv$ denotes the tractor volume form and $\lambda$ is some positive
constant.  As explained in \cite{HS}, the existence of such a  
$\chi$, which is all that we need here,  
follows easily from the realization of $\Lambda^3\mathcal{T}^*$ and its connection as
associated to the $(\mathfrak{g}_2,P)$ Cartan connection.  Namely, since the 3-form
$\varphi\in\Lambda^3\mathbb R^7{}^*$ defining $G_2$ is fixed by $P$, the constant
function $\varphi$ on $\mathcal{B}$ is $P$-equivariant, so determines a
section $\chi$ of the associated bundle $\Lambda^3\mathcal{T}^*$.  Since 
$\varphi$ is fixed by all of $G_2$ and the associated covariant derivative
is given in terms of the action of $\mathfrak{g}_2$ and differentiation by vector
fields on $\mathcal{B}$, it follows that $\chi$ is parallel.  The compatibility
condition follows from the fact that $h$ has a similar realization 
as associated to the quadratic form determined by $\varphi$.    

Let $\mathcal{D}$ be a generic 2-plane field on $M$, with $\mathcal{D}$ and $M$
real-analytic.  In the following, $\widetilde{g}$ will denote a  
real-analytic ambient metric for Nurowski's conformal structure associated
to $\mathcal{D}$, with domain 
a sufficiently small dilation-invariant neighborhood $\cGt$ of $\mathcal{G}$
diffeomorphic to $\mathbb R_+\times  M\times \mathbb R$.         


\bigskip
\noindent
{\it Proof of Theorem~\ref{holonomycontained}.}
By the result of Hammerl-Sagerschnig, there is a nonzero
parallel section $\chi\in\Gamma(\Lambda^3\mathcal{T}^*)$ compatible with the
tractor metric.  By Theorem~\ref{main}, $\chi$ has an extension as a
real-analytic 3-form 
$\widetilde{\chi}$ on some dilation-invariant neighborhood $\cGt$ of $\mathcal{G}$ which is
parallel with respect to $\widetilde{g}$.  $\widetilde{\chi}$ is compatible with $\widetilde{g}$ by analytic
continuation, so it follows that $\operatorname{Hol}(\cGt,\widetilde{g})\subset G_2$.
\hfill$\square$\medskip 

Next we discuss Cartan's basic curvature invariant of generic 2-plane
fields.  In Cartan's derivation this invariant arises directly from 
the structure equations.  We describe how it can be realized as a piece of
the Weyl tensor of the conformal structure.

Let $\mathcal{D}$