Geometry of Contactomorphism Groups, Contact Rigidity, and Contact Dynamics 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 J¹N, as well as on the group of contactomorphisms of the standard contact T*N×S¹ generated by compactly supported contact vector fields. The geometric structures are biinvariant partial orders (for J¹N and T*N×S¹) and biinvariant integer-valued metrics (T*N×S¹ only). We prove some forms of contact rigidity in T*N×S¹, namely that certain (possibly singular) subsets of the form X×S¹ cannot be disjoined from the zero section by a contact isotopy, and in addition that there are restrictions on commuting products of contactomorphisms. Finally, we prove multiplicity results for orbits of certain contact flows in T*N×S¹ with Legendrian boundary conditions, which in particular apply to Reeb chords. The method is that of generating functions for Legendrians in jet spaces.