**Context**

According to Arnol'd, a contact structure on a smooth manifold $M$ is given by a corank 1 tangent distribution $C$ which is maximally non-integrable; this means that, for any local $1$-form $\alpha$ on $M$, $$\ker\alpha=C\Longrightarrow d\alpha\textrm{ is non-degenerate on }C.$$
A contact structure is said to be co-orientable if there exists a global $1$-form $\alpha$ on $M$ such that $\ker\alpha=C$.

Then a contact manifold $(M,C)$ is constituted by a smooth manifold $M$ equipped with a contact structure $C$. Necessarily we have that

- $\dim M=2n+1$, and
- if $N\subset M$ is an integral manifold of $C$, i.e. $TN\subset C$, then $\dim M\le n$.

In particular $n$-dimensional integral manifolds are called Legendrian submanifolds of $(M,C)$.

A diffeomorphism $\phi:M_1\to M_2$ is said to a contactomorphism of $(M_1,C_1)$ onto $(M_2,C_2)$ if it satisfies $$(T\phi)C_1=C_2.$$

**Questions**

Let us remark that the current context is slightly different than the one adopted in the previous question, where only co-orientable contact manifolds are examined.

So I am wondering myself if, in our case, it is possible to get an analogous Legendrian Tubular Neighbourhood Theorem.

Let $N$ be a legendrian submanifold of $(M,C)$. Is again possible to find open neighborhoods $U$ and $V$ of $N$ respectively in $M$ and $J^1(N,\mathbb R)$, such that there exists a contactomorphism $$\phi:(U,C|_U)\to(V,\mathscr C|_V),\textrm{ with }\phi|_N=\operatorname{id}_N?$$ Above $\mathscr C$ is the Cartan distribution on $J^1(N,\mathbb R)$,and $N$ is canonically identified with $j^1 0$.

Actually, this question should admit some equivalent reformulations like as:

- is the line bundle $(TM)/C$ trivial over $N$?
- is there a local contact form for $(M,C)$ which is defined on a whole neighborhood of $N$ in $M$?

Probably, in the current context, there does not exist something like a Legendrian Tubular Neighborhood Theorem, but until now I have not been able to point out a counter-example. In such a negative case, I would also ask

there exists a classification of legendrian embeddings?

As usual any feedback is welcome.