Geometric structures on contactomorphism groups and contact rigidity in jet spaces

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 TN \times S^1 generated by compactly supported contact vector fields. The geometric structures are biinvariant partial orders (for J^1N and TN \times S^1) and biinvariant integer-valued metrics (TN\times S^1 only). Also we prove some forms of contact rigidity in TN \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 TN\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 TN Hamiltonian displaceable. The method is that of generating functions for Legendrians in jet spaces.