On the strict Arnold chord property and coisotropic submanifolds of complex projective space (Q2796764)

Let \(M\) be a manifold and \(\alpha\) a contact form on \(M\). We say that \((M,\alpha)\) has the strict chord property if for every nonempty closed Legendrian submanifold \(L\subset M\), there exists a characteristic for the form \(\alpha\) that intersect \(L\) at least twice. The paper is concerned with the following strict chord problem: Find conditions on \((M,\alpha)\) under which it has the strict chord property. 30 years ago, V. I. Arnold has conjectured that for \(n\geq 2\) any contact form on \(\mathbb S^{2n-1}\) inducing the standard structure has the strict chord property. The main result is that this property holds for every contact form on a manifold if and only if all its characteristics are closed and of same period, and the contact form embeds into the product of \(\mathbb R^{2n}\) and an exact symplectic manifold.NEWLINENEWLINENEWLINEIn particular, this result confirms V. I. Arnold's conjecture for the standard form on \(\mathbb S^{2n-1}\).