The discriminant and oscillation lengths for contact and Legendrian isotopies

Vincent Colin ; Sheila Sandon

Journal of the European Mathematical Society, Tome 17 (2015) no. 7, pp. 1657-1685

Résumé

We define an integer-valued non-degenerate bi-invariant metric (the \emph{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 R2n×S1 and RP2n+1. On the other hand we also show by elementary arguments that the discriminant metric is bounded for the standard contact structures on R2n+1 and S2n+1. As an application of these results we get that the contact fragmentation norm is unbounded for R2n×S1 and RP2n+1. By elaborating on the construction of the discriminant metric we then define an integer-valued bi-invariant pseudo-metric, that we call the \emph{oscillation pseudo-metric}, which 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×S1 for any closed manifold B, for RP2n+1 and for some 3-dimensional circle bundles.

Détail

DOI : 10.4171/jems/542

Classification : 53-XX, 00-XX
Keywords: Bi-invariant metrics, contactomorphism group, discriminant and translated points of contactomorphisms, Legendrian isotopies, orderability of contact manifolds, generating functions