Is there a Legendrian Neighbourhood Theorem also for non-cooriented contact manifolds?

Question

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.

Answer 1

This is really a comment, but I don't have enough reputation to leave one (or to vote the question up, which I like indeed!). As pointed out by the comments of Petya and G_infinity, you should not expect a local normal form theorem of the kind that is asked for in the question, as the line bundle restricted to N need not be trivial.

Let me, therefore, suggest a different take to the question, by suggesting that the model contact structure proposed is not the correct one. Let $p: L := TM/C|_N \to N$ be the restriction of the line bundle to the Legendrian submanifold $N$. Denote by $J^1 L$ the space of 1-jets of sections of $L \to N$, that is $$ J^1 L : = \{ j^1_x \sigma \mid x \in N \text{ and } \sigma \in \Gamma_{\mathrm{loc}}(L) \}. $$

There are natural projections $\mathrm{pr} : J^1 L \to L$ and $p \circ \mathrm{pr}: J^1 L \to N$ given by $\mathrm{pr}(j^1_x \sigma) = \sigma(x)$, and $p \circ \mathrm{pr}(j^1_x \sigma) = x$ respectively. The smooth manifold $J^1 L$ carries a canonical contact structure $H$, which is defined as the kernel of a 1-form $\theta_{\mathrm{can}} \in \Omega^1(J^1 L ; \mathrm{pr}^* p^* L)$, which is known in the literature on exterior differential systems and the geometric theory of PDEs as the Cartan form. Before defining $\theta_{\mathrm{can}}$, observe that the line bundle $p^* L \to L$ can be identified as the vertical tangent bundle $\ker Dp \subset TL \to L$, by identifying the tangent space to each fibre $L_x$ with the fibre itself (recall that the fibre is a vector space). Therefore, given a point $q = j^1_x \sigma$ and a vector $X_q \in T_q J^1 L$, set $$ \theta_{\mathrm{can}}(X_q) := D_q\mathrm{pr}(X_q) - D_x \sigma \circ D_{\sigma(x)}p \circ D_q \mathrm{pr} (X_q); $$ this is easily seen to take value in the appropriate line bundle, and its kernel $H$ is a contact distribution. The Cartan form $\theta_{\mathrm{can}}$ generalises the canonical contact 1-form on $J^1(N;\mathbb{R})$; in particular, a (locally defined) section $s \in \Gamma_{\mathrm{loc}}(J^1L)$ satisfies $s^*\theta_{\mathrm{can}} = 0$ if and only if $s = j^1 \sigma$, for some $\sigma \in \Gamma_{\mathrm{loc}}(L)$.

Thus it makes sense to modify the question slightly to allow an open neighbourhood of the zero section of $p \circ \mathrm{pr} : J^1 L \to N$ as a local model for an open neighbourhood of the Legendrian submanifold $N \subset (M,C)$, with contact structure $H$ defined above. That is, to look for open neighbourhoods $U$ of $N \subset M$ and $V$ of the zero section of $J^1L \to N$, and a contactomorphism $\phi: (U, C|_U) \to (V,H|_V)$ with $\phi|_N = id_N$. I don't know the answer to this different question, but I will think about it and see whether I can get anywhere with it.