Does a $G$-structure on $M$ automatically descend to a contact $G$-structure on $\mathbb{P}T^*M$?

Question

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!

Answer 1

The $CSp(2n)$-structure you describe makes sense and is an equivalent encoding of the contact structure on the projectivized cotangent bundle. Let us start with a general contact manifold $N$, $H\subset TM$, then consider the line bundle $Q:=TN/H$. Then one can view $H\subset TN$ as a filtration of the tangent bundle and consider the associated graded vector bundle $gr(TN)=H\oplus Q$. The Lie bracket of vector fields induces a skew-symmetric bundle map $H\times H\to Q$ which makes the fibers of $gr(TN)$ into a nilpotent graded Lie algebra isomorphic to a Heisenberg algebra. This shows that you get a natural frame bundle for $gr(TN)$ with structure group the automorphism group of the Heisenberg group. This automorphism group is isomorphic to $CSp(2n)$ since any automorphism is determined by its restriction to $H$. So you can equivalently view it as a reduction of the linear frame bundle of $H$ to $CSp(2n)\subset GL(2n,\mathbb R)$. This can be vastly generalized, possible keywords are "filtered manifolds", "symbol algebras", and "nilpotent geometry".

In particular, you can do all that for the canonical contact structure on $\mathbb PT^*M$. Locally, there is nothing special about the resulting structure (since all contact structures are locally isomorphic), globally you have facts like the existence of a global contact form and of an involutive Legendrean distribution, which are not available in general.

Concerning reductions to some $\hat G$ coming from $G$-structures on $M$, there certainly is the issue of non-transitivity mentioned in the comments by @RobertBryant . However there are some interesting cases in which there are open orbits (say for Riemannian or conformal structures). Some intersting examples have been worked out in the context of parabolic geometries, like a classical projective structure on $M$ giving rise to a Lagrangean contact structure on $\mathbb PT^*M$.