The discriminant and oscillation lengths for contact and Legendrian isotopies

We define an integer-valued non-degenerate bi-invariant metric (the metric) on the universal cover of the identity component of the group of any contact manifold. This metric has a very simple definition, based on the notion of discriminant points of. Using generating functions we prove that the discriminant is unbounded for the standard contact structures on R²ⁿ x S^1 and²ⁿ⁺¹. On the other hand we also show by elementary arguments that the metric is bounded for the standard contact structures on R²ⁿ⁺¹ S²ⁿ⁺¹. As an application of these results we get that the contact norm is unbounded for R²ⁿ x S^1 and RP²ⁿ⁺¹. By elaborating the construction of the discriminant metric we then define a second-valued bi-invariant metric, that we call the discriminant oscillation. This second metric is non-degenerate if and only if the contact is orderable in the sense of Eliashberg and Polterovich and, in this, it is compatible with the partial order. Finally we define the and oscillation lengths of a Legendrian isotopy, and prove that are unbounded for T\B x S^1 for any closed manifold B, for RP²ⁿ⁺¹ for some 3-dimensional circle bundles.