Realizing closed manifolds as Legendrian submanifolds of the standard contact vector space

Question

I started learning some basic contact geometry, in particular its flexible side, and I got stuck with the following naive question. Given a closed manifold of dimension $n$, we can always embed it into $\mathbb R^{2n+1}$, say by Whitney embedding theorem. I wonder whether this manifold can always be "approximated" by some loose Legendrian submanifolds there. In other words, I wonder whether every closed $n$-manifold admits a Legendrian embedding into $\mathbb R^{2n+1}$, maybe even with some controlled classical invariants like Maslov class (say, 0) and Thurston-Bennequin number.

Answer 1

There are obstructions: For example, if $N^n\subset\mathbb{R}^{2n+1}$ is a Legendrian submanifold, then $\nu$, the normal bundle of $N^n$, is isomorphic to $\tau\oplus TN$, where $\tau$ is the trivial bundle. Thus, $\tau\oplus TN\oplus TN$ is isomorphic to $\nu\oplus TN$, which is trivial. In particular, it follows that the total Stiefel-Whitney and Pontrjagin classes of $TN$ must square to zero.

Since this doesn't happen for, say, $N=\mathbb{CP}^{2}$, it follows that $\mathbb{CP}^2$ cannot be immersed as a Legendrian submanifold of $\mathbb{R}^9$.

Answer 2

The obstruction here is in admitting a formal Legendrian embedding, i.e., there must be an embedding $f : N \to \Bbb R^{2n+1}$ which is covered by a 1-parameter family $F_t : TN \to T\Bbb R^{2n+1}$ of bundle monomorphisms such that $F_0 = df$ and $F_1(T_p N) \subset T_{f(p)}\Bbb R^{2n+1}$ is a Legendrian subspace. In particular, one must have the conditions imposed by Robert Bryant's answer.

If this is true, then one can run the proof of Murphy's $h$-principle: every formal Legendrian embedding is homotopic to a $\varepsilon$-Legendrian embedding, and therefore admits a front projection over which it is graphical. We may project to the front, wrinkle by Eliashberg-Mishachev's wrinkled embedding theorem, and then lift it back up. This gives a topological embedding of $N$ in $\Bbb R^{2n+1}$ which is Legendrian everywhere except embedded disjoint $S^{n-2}$'s which correspond to unfurled swallowtail singularities.

If one can now introduce a loose chart, then this may be used to cancel the swallowtail singularities by markings. This can be done by taking a connect sum with a loose topological unknot $S^n \to \Bbb R^{2n+1}$, obtained by spinning a zigzag. This doesn't change the topology of $N$, so we are done.