Let $(M, \xi = \text{ker}\,\alpha)$ be a compact contact manifold with non-empty boundary. Vaguely asked, is there any natural geometric structure on the boundary $\partial M$ induced from the contact structure on $M$ which is inner to $\partial M$?

Let me give an analogous example in the symplectic case: Let $(B, \omega)$ be a symplectic manifold with non-empty boundary and suppose that the boundary $\partial B$ is of *contact type*, that is

- $\omega = d\lambda$ near $\partial B$ and
- the Liouville vector field $X_{\lambda}$, defined by $\iota_{X_{\lambda}} \omega = \lambda$, is positively transverse to the boundary $\partial B$.

Then the pair $(M, \xi) := (\partial B, \text{ker}\,\lambda|_{\partial B})$ is a contact manifold. Note that the same contact manifold $(M, \xi)$ can arise from (many) different symplectic manifolds $(B, \omega)$ under the above construction. In other words, you can start with a contact manifold $(M, \xi)$ and ask if there is a symplectic manifold $(B, \omega)$ with boundary $\partial B \cong M$ which will induce the given contact structure $\xi$ on $M$.

In this sense I say that $\xi$ is a geometric structure inner to $M$ (and not fully dependent on the/an ambient symplectic manifold $(B, \omega)$ which it came from).

**Question:** $(M, \xi = \text{ker}\,\alpha)$ be a compact contact manifold with non-empty boundary. Is there any nice/reasonable/natural condition on the boundary $\partial M$ which would ensure that $\partial M$ inherits a nice/reasonable/natural geometric structure from $\xi$, inner to $\partial M$?

I suspect that many people thought about this, but I haven't encountered any satisfactory answer so far. Any ideas are very welcome!