My question is motivated by the following question.


[How transitive are the actions of symplectomorphism groups ?](https://mathoverflow.net/questions/61994/how-transitive-are-the-actions-of-symplectomorphism-groups)


A subset $X$ of a symplectic manifold $M^{2n}$ is called $\it displaceable$ if there is a Hamiltonian (or, symplectic if you want weaker condition) diffeomorphism $f\colon M \rightarrow M$ such that $f(X) \cap X = \phi$. Now assume $X$ is a $k$-dimensional closed submanifold of $M$.


When $k=0$, $X$ is displaceable as explained in the above question. When $k=n$, some $X$ is not displaceable since we know many nondisplaceable Lagrangians (such as great circle in $S^2$). Then when $k<n$, can we expect that $X$ is displaceable?


If so, can we expect more? Suppose two diffeomorphic submanifolds $X$ and $Y$ of dimension $k<n$ represent the same homology class. Is there a symplectomorphism sending $X$ to $Y$? Are there any topological obstructions for such symplectomorphism?



Displaceability of submanifolds
There is a result of Basak Gurel which says that a nowhere coisotropic submanifold of a symplectic manifold is displaceable (by a Hamiltonian diffeomorphism) as soon as it's infinitesimally displaceable (meaning its normal bundle admits a nowhere vanishing section). A submanifold $X$ of a symplectic manifold $(M,\omega)$ is nowhere coisotropic if for every $x \in X$ we have $(T\_xX)^\omega \not\subset T\_xX$. This trivially holds for submanifolds of dimension $<n$. Therefore a $k$-dimensional submanifold $X$ with $k < n$ will be displaceable as soon as its normal bundle admits a nonvanishing section (I can't immediately see whether this always holds for such a submanifold). Link: <http://arxiv.org/abs/math/0702091>


