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 $\partial M$?




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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


1. $\omega = d\lambda$ near $\partial B$ and
2. 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).




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**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!



Boundary geometry of a contact manifold
Maybe what you're looking for are convex hypersurfaces (due to Emmanuel Giroux). They are mostly used in dimension 3, but can be used in every dimension (but things are more complex ... and: being convex in high dimension is not a *generic* property that you could automatically obtain by small perturbations).


A hypersurface $\Sigma$ in a contact manifold is called **convex** if there exists a contact vector field $X$ on a neighborhood of $\Sigma$ being transverse to it.
Some of the properties induced by $X$ on $\Sigma$ are the following:
The dividing set $\Gamma\_X$ of $\Sigma$ is the hypersurface along which $X$ is tangent to the contact structure $\xi$. One can easily decompose $\Sigma$ into different pieces by plugging $X$ into a contact form $\alpha$ and tracing the domains $\Sigma\_+$ and $\Sigma\_-$, where $f:= \alpha(X)$ is either positive or negative with $\Gamma\_X = \Sigma\_+ \cap \Sigma\_-$.


One can check that $\Gamma\_X$ is a contact submanifold with induced distribution $\xi\_\Gamma := T\Gamma\_X \cap \xi$. The two remaining domains $\Sigma\_+$ and $\Sigma\_-$ are Liouville domains, where the Liouville form is given by $\lambda\_\pm := 1/f \alpha$. The dividing set is a convex boundary for both Liouville domains.


Changing the auxiliary contact vector field $X$ changes all structures, but up to isotopy they are preserved so that one can really think of the structures as something inherent to the hypersurface and not depending on any particular $X$.


Maybe you were having something like this in mind?


