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arXiv · 1808.08459

Chekanov's dichotomy in contact topology

Daniel Rosen·Jun Zhang
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Abstract

In this paper we study submanifolds of contact manifolds. The main submanifolds we are interested in are contact coisotropic submanifolds. Based on a correspondence between symplectic and contact coisotropic submanifolds, we can show contact coisotropic submanifolds admit a $C^0$-rigidity, similar to Humili\`ere-Leclercq-Seyfaddini's coisotropic rigidity on symplectic manifolds. Moreover, based on Shelukhin's norm defined on the contactomorphism group, we define a Chekanov type pseudo-metric on the orbit space of a fixed submanifold of a contact manifold. Moreover, we can show a dichotomy of (non-)degeneracy of this pseudo-metric when the dimension of this fixed submanifold is equal to the one for a Legendrian submanifold. This can be viewed as a contact topology analogue to Chekanov's dichotomy of (non-)degeneracy of Chekanov-Hofer's metric on the orbit space of a Lagrangian submanifold.

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Daniel Rosen, Jun Zhang. 2018-08-25. Chekanov's dichotomy in contact topology. https://doi.org/10.4310/mrl.2020.v27.n4.a9

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