Local rigidity, symplectic homeomorphisms, and coisotropic submanifolds

TL;DR

This paper introduces a new notion of local rigidity for points on subsets of symplectic manifolds, proves its invariance under symplectic homeomorphisms, and characterizes coisotropic submanifolds as those with all points locally rigid, simplifying existing rigidity proofs.

Contribution

It defines local rigidity for points on subsets, shows its invariance under symplectic homeomorphisms, and characterizes coisotropic submanifolds via this property, providing a simplified proof of known $C^0$-rigidity results.

Findings

Coisotropic submanifolds have all points locally rigid.

Local rigidity is invariant under symplectic homeomorphisms.

Simplified proof of $C^0$-rigidity of coisotropic submanifolds.

Abstract

We introduce the notion of a point on a locally closed subset of a symplectic manifold being "locally rigid" with respect to that subset, prove that this notion is invariant under symplectic homeomorphisms, and show that coisotropic submanifolds are distinguished among all smooth submanifolds by the property that all of their points are locally rigid. This yields a simplified proof of the Humili\`ere-Leclercq-Seyfaddini theorem on the $C^0$-rigidity of coisotropic submanifolds. Connections are also made to the "rigid locus" that has previously been used in the study of Chekanov-Hofer pseudometrics on orbits of closed subsets under the Hamiltonian diffeomorphism group.