Legendrian Tubular Neighborhood Theorem Given a Lagrangian submanifold $L\subset(M,\omega)$ of a symplectic manifold, we have Alan Weinstein's celebrated Lagrangian tubular neighborhood theorem. I now look for the analog on Legendrian submanifolds $K\subset(Y,\lambda)$ of contact manifolds. As the Lagrangian neighborhood construction essentially relies on the Moser method, I don't think it'd be too hard to build a Legendrian neighborhood by using the analogous Gray stability.
However, I am specifically questioning whether I can get a Legendrian neighborhood theorem directly from the Lagrangian neighborhood theorem: By passing to the symplectization $(\mathbb{R}\times Y,d(e^t\lambda)$), a Legendrian submanifold $K$ becomes a Lagrangian submanifold $\mathbb{R}\times K$. I would love to project some Lagrangian tubular neighborhood down into $Y$ to get a desired neighborhood of $K$, but can this actually be done?
(I spoke with Alan, and he said this might be achieved somehow by viewing all of our constructions equivariantly using the translation $\mathbb{R}$-action on our bundle $\mathbb{R}\times Y\to Y$.)
 A: Basically, Weinstein's theorem says that you can embed $T^*L$ into $M$ like that:
$$
T^*L\cong NL\cong \mathcal{T}_L\subseteq M,
$$
where $NL$ is the normal bundle and $\mathcal{T}_L$ is a tubular neighborhood, in such a way that the canonical symplectic form on $T^*L$ is the pull-back of $\omega$. So, the "contact counterpart" of above chain of identifications should read
$$
J^1K\cong NK\cong \mathcal{T}_K\subseteq Y
$$
where now $J^1K$ is the first-order jet bundle of (smooth) functions on $K$. Notice that all manifolds appearing above are $(2n+1)$-dimensional, and all bundles are over $K$ with $(n+1)$-dimensional fibers. I'm sure this observation can be found in the literature about jet spaces and/or contact manifolds.
My guess is that the above embedding, at least locally, can be always found. Concerning the real question, i.e., whether the canonical contact form on $J^1K$ is the pull-back of $\lambda$, I'm not able to answer, but I feel it is not true!
A: The closest analog I am aware of to the Weinstein model for Lagrangian submanifolds in the contact setting is the so-called "Lychagin chart:" Every Legendrian submanifold $L$ has a neighborhood contactomorphic to a neighborhood of the zero section of the space of one-jets on $L$. See Banyaga, "The Structure of Classical Diffeomorphism Groups," Section 6.2, also Lychagin, "Local classification of nonlinear first-order partial differential equations," Russ. Math. Surv. 39 (1975), 105-175.
