接触同位素和勒让德同位素的判别长度和振荡长度
The discriminant and oscillation lengths for contact and Legendrian isotopies
摘要：
在任意切触流形的切触同胚群的单位连通区的泛覆盖上，定义了一个整数值非退化双不变度量（判别度量）。这个度量有一个非常简单的几何定义，基于接触同构的判别点的概念。利用生成函数证明了R^{2n} x S^1和RP^{2n+1}上标准切触结构的判别度量是无界的.另一方面，我们还通过初等论证证明了R^{2n+1}和S^{2n+1}上标准切触结构的判别度量是有界的.作为这些结果的一个应用，我们得到了对于R^{2n} x S^1和RP^{2n+1}，切触分裂范数是无界的.通过详细说明判别度量的构造，我们定义了第二个整数值双不变度量，我们称之为判别振荡度量。第二个度量是非退化的当且仅当接触流形在Eliashberg和Polterovich意义下是可序的，并且在这种情况下，它与偏序相容。最后定义了勒让德合痕的判别长度和振荡长度，并证明了它们对任意闭流形B，RP^{2n+1}和某些三维圆丛对T^{ \ast }B x S^1是无界的.
英文摘要：
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 get 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 a second integer-valued bi-invariant metric, that we call the discriminant oscillation metric. This second metric 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^{\ast}B x S^1 for any closed manifold B, for RP^{2n+1} and for some 3-dimensional circle bundles.
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