Geometric structures on contactomorphism groups and contact rigidity in jet spaces

TL;DR

This paper introduces geometric structures like biinvariant partial orders and metrics on subgroups of contactomorphisms in jet spaces, revealing contact rigidity phenomena and restrictions on contactomorphisms.

Contribution

It establishes the existence of biinvariant structures on contactomorphism groups of jet spaces and proves contact rigidity results using generating functions.

Findings

Existence of biinvariant partial orders on contactomorphism subgroups

Existence of biinvariant integer-valued metrics on certain groups

Restrictions on contactomorphisms displacing subsets in jet spaces

Abstract

For a closed connected manifold N, we establish the existence of geometric structures on various subgroups of the contactomorphism group of the standard contact jet space J^1N, as well as on the group of contactomorphisms of the standard contact T*N \times S^1 generated by compactly supported contact vector fields. The geometric structures are biinvariant partial orders (for J^1N and T*N \times S^1) and biinvariant integer-valued metrics (T*N\times S^1 only). Also we prove some forms of contact rigidity in T*N \times S^1, namely that certain (possibly singular) subsets of the form X \times S^1 cannot be disjoined from the zero section by a contact isotopy, and in addition that there are restrictions on the kind of contactomorphisms of T*N\times S^1 which are products of pairwise commuting contactomorphisms generated by vector fields supported in sets of the form U \times S^1 with U \subset T*N Hamiltonian displaceable. The method is that of generating functions for Legendrians in jet spaces.