Boundary geometry of a contact manifold

Question

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

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).

---

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!

Answer 1

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?

Answer 2

I think that there's not really much about local geometry to say beyond this: There are two kinds of points on the boundary $B$, those $b\in B$ for which $T_bB = \xi_b$ (these points form a closed set) and those $b\in B$ for which $T_bB\not=\xi_b$ (these form an open set).

One could say that the boundary $B$ (of dimension, say, $2n$) is contact nondegenerate if $T_bB\not=\xi_b$ for all $b\in B$. (Obviously this is stable under small perturbations of the boundary.) In this case, there is a well-defined line field $L$ on $B$ that is the kernel of the nonvanishing $(2n{-}1)$-form $A = \alpha\wedge(\mathrm{d}\alpha)^{n-1}$, which is well-defined up to a multiple on $B$. There is a little bit more structure in the sense that, on an open subset $U\subset B$ on which the line field is simple (i.e., there exists a surjective submersion $\pi:U\to N^{2n-1}$ whose fibers are the leaves of the line field $L$ in $U$) there is an induced contact structure on the local quotient $N$.

Generically, the line field $L$ will not be globally simple, in particular, the space $X$ of integral curves of $L$ in $B$ will not support the structure of a smooth $(2n{-}1)$-manifold for which the natural mapping $\pi:B\to X$ is a submersion, so it won't necessarily make sense to speak of an 'induced contact structure on $X$. However, when the line field is globally simple, $X$ will inherit the structure of a contact manifold.

Locally, there is no more structure to be found. The local structure is simply that of a submersion of a $2n$-manifold $B$ onto a contact $(2n{-}1)$-manifold $X$.

N.B.: You might find it interesting that all this is just a geometric formulation of the classical method of Cauchy characteristics for a first-order, scalar PDE. The theory of so-called 'singular solutions' developed in the 19th century is really just an attempt to understand the geometry of the situation when $B$ has points of tangency with the contact plane field.