JournalsjemsVol. 17, No. 7pp. 1657–1685

The discriminant and oscillation lengths for contact and Legendrian isotopies

  • Vincent Colin

    Université de Nantes, France
  • Sheila Sandon

    Université de Strasbourg, France
The discriminant and oscillation lengths for contact and Legendrian isotopies cover

Abstract

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\mathbb{R}^{2n}\times S^1 and RP2n+1\mathbb{R}P^{2n+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\mathbb{R}^{2n+1} and S2n+1S^{2n+1}. As an application of these results we get that the contact fragmentation norm is unbounded for R2n×S1\mathbb{R}^{2n}\times S^1 and RP2n+1\mathbb{R}P^{2n+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 TB×S1T^{\ast}B\times S^1 for any closed manifold BB, for RP2n+1\mathbb{R}P^{2n+1} and for some 33-dimensional circle bundles.

Cite this article

Vincent Colin, Sheila Sandon, The discriminant and oscillation lengths for contact and Legendrian isotopies. J. Eur. Math. Soc. 17 (2015), no. 7, pp. 1657–1685

DOI 10.4171/JEMS/542