4
$\begingroup$

Is there any work on manifolds (perhaps contact) with symplectic boundary (not asking about the boundary of a symplectic manifold)?

$\endgroup$
4
  • $\begingroup$ Perhaps it would be good to make your question more precise : are you asking about manifolds whose boudary can be endowed with a symplectic structure ? $\endgroup$ May 8, 2012 at 14:54
  • 1
    $\begingroup$ One way to start thinking about this would be to understand what type of structure $M\times\mathbb{R}$ can have if $M$ is equipped with a symplectic form. If $\omega=-d\theta$ is exact then $dt+\theta$ is a contact form on $M\times\mathbb{R}$, however this exclude compact $M$'s. On the other side, the good notion of boundary of contact manifolds seems to be that of convex hypersurfaces, which outside a dividing set are exact symplectic. $\endgroup$
    – Noz
    May 8, 2012 at 18:14
  • $\begingroup$ For closed symplectic manifold, here springerlink.com/content/m080126681712458 some symplectic manifold arises as boundary of quasi-symplectic manifold which are manifold whith a closed $2$-form with one dimensionnal kernel (in the case of $M\times\mathbb{R}$ it is simply $\omega$. I don't know whether or not those type of manifold have been extensively studied. $\endgroup$
    – Noz
    May 8, 2012 at 18:15
  • $\begingroup$ Maybe one could make a definition like this. A filling of symplectic manifold $(M,\omega)$ is a $2n+1$ manifold with boundary $M$ and with stable Hamiltonian structure $(\Omega,\lambda)$ such that $\Omega\vert_M=\omega$. $\endgroup$
    – Noz
    May 8, 2012 at 18:27

2 Answers 2

3
$\begingroup$

For contact manifolds there is the notion of a convex hypersurface due to Giroux, ie, a hypersurface that is transverse to a contact vector field. A convex hypersurface is composed by two exact symplectic domains (not necessarily connected) glued together along the dividing set (the dividing set is a hypersurface in the hypersurface) that carries a natural contact structure (see also this question).

I don't know if this helps, since the boundary is not a closed symplectic manifold, but it is a very natural object to consider in contact topology.

$\endgroup$
2
$\begingroup$

There is a book by Guillemin, Gizburg and Karshon: Moment Maps, Cobordisms, and Hamiltonian Group Actions.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.