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 $\mathbb{R}\times Y\to Y$.)

toLegendrian neighborhood (whereas the answers thus far are only showing me how to get a Legendrian neighborhood). $\endgroup$ – Chris Gerig Sep 10 '13 at 20:10