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I was asking myself if there exists a sort of canonical relation between the standard contact structure on $J^1 N$ and $J^1 M$, for an arbitrary submanifold $N$ of $M$.

My starting point is that, constructing the spaces of $k$-jets and their structures of smooth manifolds, it is remarked that, for any smooth manifold $P$:
while, mapping $\phi\in C^\infty(M,N)$ to $J^k(P,\phi):J^k(P,M)\to J^k(P,N),\ j^k_x f\mapsto j_x^k (\phi\circ f)$, it is possible to define the covariant endofunctor $J^k(P,\cdot)$ of the category of smooth manifolds and maps,
instead, it is possible only to define a contravariant functor $J^k(P,\cdot)$ from the category of diffeomorfisms to the category of smooth maps, by mapping $\phi\in\mathrm{Diffeo}(M,N)$ to $J^k(P,\phi):J^k(P,N)\to J^k(P,M),\ j^k_x f\mapsto j^k_{\phi^{-1}(x)} (f\circ\phi)$.

So it seems to me that this is not the way to relate the $1$-jets space of a manifold $M$ and that of one of its submanifold $N$.
But I imagine that the entire information on $J^1 N$ is already contained in $J^1 M$, even if I do not see (up to now) how to extract it with some invariantly defined procedure

My question is:

Given a submanifold $N$ of $M$, under what conditions is there a sort of canonical relation between the standard contact structures of $J^1 N$ and $J^1 M$? and in what terms is it expressed?

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Not sure what you want but jets are just infinitesimal data about functions. So what you want to ask is: what is the canonical relation between functions on $M$ and functions on $N$? That then tells you what the corresponding canonical relation between jet bundles are. – Deane Yang Jul 18 '11 at 14:16
up vote 5 down vote accepted

Well, the short answer is that, when $N$ is a submanifold of $M$, the contact manifold $J^1(N)$ is a subquotient of $J^1(M)$. The point is that restriction of germs of functions to submanifolds implies that, if $S\subset J^1(M)$ is the submanifold consisting of those $1$-jets with source a point of $N$, then the canonical submersion $S\to J^1(N)$ induced by restriction of germs has the property that it pulls back the canonical contact form on $J^1(N)$ to be the pullback to $S$ of the canonical contact form on $J^1(M)$. I don't know that there's much more to say than that.

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Dear Robert Bryant, thanks for the attention. In such a way, if I am correct, the surjective summersion $S\to J^1 N$, defined by $j^1_x f\mapsto j^1_x (f|_N)$, identifies $J^1 N$ with the space of the leaves for the integral foliation of $Z$ on $S$. Here I mean by $Z$ the characteristic distribution of the corank 1 distribution $C\cap TS$, the pull-back to $S$ of the standard contact distribution $C$ on $J^1 M$. Thank you again. – Giuseppe Tortorella Jul 18 '11 at 20:17

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