If $M$ is a real smooth manifold of dimension $n+1$, by $D\in\mathbb{P}T^*M$ I mean a tangent hyperplane at some point of $M$. I denote by $b$ the canonical projection of the $(2n+1)$-dimensional manifold $\mathbb{P}T^*M$ onto $M$.

The hyperplane distribution $$ C_D:=b_\ast^{-1}(D)\subset T_D(\mathbb{P}T^*M) $$ fits into the exact sequence $$ 0\longrightarrow D^*\otimes N\longrightarrow C\longrightarrow D\longrightarrow 0\quad\quad (\bullet) $$ of vector bundles over $\mathbb{P}T^*M$. Symbol $D$ denotes the tautological bundle, whereas $N$ is the normal line bundle $$ N_D:=\frac{T_{b(D)}M}{D}\, . $$ By construction, sequence $(\bullet)$ is $GL(D)$-equivariant. By decomposing $\Lambda^2C^*$ into irreducible constituents, one finds that the unique one-dimensional component corresponds precisely to the conformal class $[\omega]$ of the symplectic form induced by the canonical contact structure $C_D$ on $\mathbb{P}T^*M$. In other words, sequence $(\bullet)$ defines a $CSp(2n)$-structure on $\mathbb{P}T^*M$. (By $CSp(2n)$ I mean the group preserving the standard symplectic structure on $\mathbb{R}^{2n}$, up to proportionality.)

PRELIMINARY QUESTION: what distinguishes the $CSp(2n)$-structure on $\mathbb{P}T^*M$ I just obtained, among all possible others? In particular, what reflects the fact that it comes from a contact structure?

**EDIT (after Bryant comment): the preliminary question is a little silly, since the $CSp(2n)$-structure is not defined on the whole of $\mathbb{P}T^*M$, but only on its contact distribution; the main question below, however, still makes sense.**

Suppose now that $M$ comes equipped with a $G$-structure (not necessarily integrable), e.g., a Riemannian metric.

MAIN QUESTION: is there a canonical $\widehat{G}$-reduction of the above $CSp(2n)$-structure on $\mathbb{P}T^*M$, induced by the $G$-structure on $M$? (Symbol $\widehat{G}$ explained below)

**Why do I expect it to be so?**

In the presence of the additional structure $G$ on $M$, the sequence $(\bullet)$ is also $G_D$-equivariant, and, as before, it can be used to decompose the tensor algebra of $C$ into irreducible pieces. Suppose that a one-dimensional component will be found: than an its generator $T$ defines a conformal class $[T]$ of a tensor on $C$. If $\widehat{G}$ denotes the symmetry group of $[T]$, then we have a $\widehat{G}$-reduction of the $CSp(2n)$-structure on $\mathbb{P}T^*M$, i.e., a "contact $\widehat{G}$-structure".

In the case of a Riemannian metric $M$, I have seen some guy constructing out of it a split signature metric on $\mathbb{P}T^*M$, but without relying so much on representation theory. So, I was wondering if a general procedure exists to pass from $G$-structures on $M$ to contact structures on $\mathbb{P}T^*M$, and if using $(\bullet)$ is indeed a nice trick. Then it would be nice to know how the integrability of the $G$-structure reflects on the integrability of the $\widehat{G}$-structure.

Any reference on this concern will be appreciated!