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For a closed connected manifold N, we establish the existence of geometricstructures on various subgroups of the contactomorphism group of the standardcontact jet space J^1N, as well as on the group of contactomorphisms of thestandard contact T*N \times S^1 generated by compactly supported contact vectorfields. The geometric structures are biinvariant partial orders (for J^1N andT*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 thatcertain (possibly singular) subsets of the form X \times S^1 cannot bedisjoined from the zero section by a contact isotopy, and in addition thatthere are restrictions on the kind of contactomorphisms of T*N\times S^1 whichare products of pairwise commuting contactomorphisms generated by vector fieldssupported in sets of the form U \times S^1 with U \subset T*N Hamiltoniandisplaceable. The method is that of generating functions for Legendrians in jetspaces.

Geometric structures on contactomorphism groups and contact rigidity in jet spaces