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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!

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    $\begingroup$ You don't get a true $\mathrm{CSp}(2n)$-structure on $X(M) = \mathbb{P}(T^*M)$ since you haven't taken into account the rest of the tangent bundle of $X(M)$, only the codimension-$1$ subbundle $C$ (which does have a structure reduction to $\mathrm{CSp}(2n)$, of course). The minimal $\mathrm{Diff}(M)$-invariant $G$-structure on $X(M)$ takes into account the $\mathrm{Diff}(M)$-invariant filtration $D^*\otimes N\subset C\subset TX(M)$ of bundles as well as the 'contact' section of $(T/C)\otimes\Lambda^2(C^*)$. As for the prolongation of a $G$-structure to $X(M)$, yes, in some cases, it exists. $\endgroup$ – Robert Bryant Feb 1 '16 at 9:54
  • $\begingroup$ @RobertBryant: thanks for spotting my rough mistake! The sense of the main question, however, does not suffer from it. Can you please point out a good departing point, in the literature, to start learning about this prolongation process? (You said by yourself that, "in some cases, it exists"). $\endgroup$ – Giovanni Moreno Feb 1 '16 at 11:53
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    $\begingroup$ I'm afraid that I don't know a good reference, probably because there aren't that many cases in which the unmodified process works. The obvious case when it does is when the group $G\subset\mathrm{GL}(n{+}1,\mathbb{R})$ acts transitively on the space of codimension-$1$ subspaces of $\mathbb{R}^{n+1}$ (which includes, for example, the Riemannian, Hermitian, almost complex, and almost symplectic structures; these are probably the most important cases, though there are a few important exceptional cases as well). When this transitivity fails, one generally runs into problems. $\endgroup$ – Robert Bryant Feb 1 '16 at 12:48
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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 $\mathbb PT^*M$.

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  • $\begingroup$ Very illuminating explanation: thanks! However, I still detect a certain conflict between @Robert Bryant's first comment above and the first sentence of your answer. You claim that the $CSp(2n)$-structure is 'equivalent' to the contact structure $H$: I can easily understand that given the latter, you obtain the former, but what about the other way around? As Bryant said, we don't have a genuine $CSp(2n)$-structure, but just a $CSp(2n)$-reduction of the frames of $H$: in particular, we already have $H$! In this sense, I fail to appreciate the 'equivalence'. $\endgroup$ – Giovanni Moreno Feb 1 '16 at 17:02
  • $\begingroup$ Concerning the $\hat{G}$-reduction, do you have any reference? It's enough one title: I'll track back the citations by myself! $\endgroup$ – Giovanni Moreno Feb 1 '16 at 17:05
  • $\begingroup$ Sorry for not coming back for a long time. I agree that in some sense $H$ has to be there before the reduction. But any corank one subbundle in dimension $2n+1$ gives you a redcution of the associated graded to $GL(2n)\times GL(1)$ and the contact distributions are eactly those, for which there is a further reduction to $CSp(2n)$. With a reference for the reductions it is a bit complicated since these results are quite roundabout. Some of them are treated in Sections 4.4. and 4.5 of the book bookstore.ams.org/surv-154 . $\endgroup$ – Andreas Cap Apr 9 '16 at 9:57

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