The discriminant and oscillation lengths for contact and Legendrian isotopies

Vincent Colin (1) , Sheila Sandon (2, 1)

1 LMJL - Laboratoire de Mathématiques Jean Leray
2 IRMA - Institut de Recherche Mathématique Avancée

We define an integer-valued non-degenerate bi-invariant metric (the discriminant metric) on the universal cover of the identity component of the contactomorphism group of any contact manifold. This metric has a very simple geometric definition, based on the notion of discriminant points of contactomorphisms. Using generating functions we prove that the discriminant metric is unbounded for the standard contact structures on R^2n x S^1 and RP^2n+1. On the other hand we also show by elementary arguments that the discriminant metric is bounded for the standard contact structures on R^2n+1 and S^2n+1. As an application of these results we deduce that the contact fragmentation norm is unbounded for R^2n x S^1 and RP^2n+1. By elaborating on the construction of the discriminant metric we then define an integer-valued bi-invariant pseudo-metric, which we call the oscillation pseudo-metric, that is non-degenerate if and only if the contact manifold is orderable in the sense of Eliashberg and Polterovich and, in this case, it is compatible with the partial order. Finally we define the discriminant and oscillation lengths of a Legendrian isotopy, and prove that they are unbounded for T*B x S 1 for any closed manifold B, for RP^2n+1 and for some 3-dimensional circle bundles.

Mots clés: Bi-invariant metrics; contactomorphism group; discriminant and translated points of contactomorphisms; Legendrian isotopies; orderability of contact manifolds; generating functions

Domaines: Mathématiques [math]

Vincent Colin, Sheila Sandon. The discriminant and oscillation lengths for contact and Legendrian isotopies. Journal of the European Mathematical Society, 2015, 17, pp.1657 - 1685. ⟨10.4171/JEMS/542⟩. ⟨hal-01465260⟩