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

Chekanov's dichotomy in contact topology