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. [ABSTRACT FROM PUBLISHER]