The discriminant and oscillation lengths for contact and Legendrian\n isotopies

Abstract

We define an integer-valued non-degenerate bi-invariant metric (the\ndiscriminant metric) on the universal cover of the identity component of the\ncontactomorphism group of any contact manifold. This metric has a very simple\ngeometric definition, based on the notion of discriminant points of\ncontactomorphisms. Using generating functions we prove that the discriminant\nmetric is unbounded for the standard contact structures on R^{2n} x S^1 and\nRP^{2n+1}. On the other hand we also show by elementary arguments that the\ndiscriminant metric is bounded for the standard contact structures on R^{2n+1}\nand S^{2n+1}. As an application of these results we get that the contact\nfragmentation norm is unbounded for R^{2n} x S^1 and RP^{2n+1}. By elaborating\non the construction of the discriminant metric we then define a second\ninteger-valued bi-invariant metric, that we call the discriminant oscillation\nmetric. This second metric is non-degenerate if and only if the contact\nmanifold is orderable in the sense of Eliashberg and Polterovich and, in this\ncase, it is compatible with the partial order. Finally we define the\ndiscriminant and oscillation lengths of a Legendrian isotopy, and prove that\nthey are unbounded for T^{\\ast}B x S^1 for any closed manifold B, for RP^{2n+1}\nand for some 3-dimensional circle bundles.\n