# The discriminant and oscillation lengths for contact and Legendrian isotopies

Journal of the European Mathematical Society (2015)

- Volume: 017, Issue: 7, page 1657-1685
- ISSN: 1435-9855

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topColin, Vincent, and Sandon, Sheila. "The discriminant and oscillation lengths for contact and Legendrian isotopies." Journal of the European Mathematical Society 017.7 (2015): 1657-1685. <http://eudml.org/doc/277258>.

@article{Colin2015,

abstract = {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 $\mathbb \{R\}^\{2n\}\times S^1$ and $\mathbb \{R\}P^\{2n+1\}$. On the other hand we also show by elementary arguments that the discriminant metric is bounded for the standard contact structures on $\mathbb \{R\}^\{2n+1\}$ and $S^\{2n+1\}$. As an application of these results we get that the contact fragmentation norm is unbounded for $\mathbb \{R\}^\{2n\}\times S^1$ and $\mathbb \{R\}P^\{2n+1\}$. By elaborating on the construction of the discriminant metric we then define an integer-valued bi-invariant pseudo-metric, that we call the 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 $T^\{\ast \}B\times S^1$ for any closed manifold $B$, for $\mathbb \{R\}P^\{2n+1\}$ and for some $3$-dimensional circle bundles.},

author = {Colin, Vincent, Sandon, Sheila},

journal = {Journal of the European Mathematical Society},

keywords = {Bi-invariant metrics; contactomorphism group; discriminant and translated points of contactomorphisms; Legendrian isotopies; orderability of contact manifolds; generating functions; bi-invariant metrics; discriminant metrics; contactomorphism group; generating functions; oscillation pseudo-metric; Legendrian isotopies; zizag norm},

language = {eng},

number = {7},

pages = {1657-1685},

publisher = {European Mathematical Society Publishing House},

title = {The discriminant and oscillation lengths for contact and Legendrian isotopies},

url = {http://eudml.org/doc/277258},

volume = {017},

year = {2015},

}

TY - JOUR

AU - Colin, Vincent

AU - Sandon, Sheila

TI - The discriminant and oscillation lengths for contact and Legendrian isotopies

JO - Journal of the European Mathematical Society

PY - 2015

PB - European Mathematical Society Publishing House

VL - 017

IS - 7

SP - 1657

EP - 1685

AB - 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 $\mathbb {R}^{2n}\times S^1$ and $\mathbb {R}P^{2n+1}$. On the other hand we also show by elementary arguments that the discriminant metric is bounded for the standard contact structures on $\mathbb {R}^{2n+1}$ and $S^{2n+1}$. As an application of these results we get that the contact fragmentation norm is unbounded for $\mathbb {R}^{2n}\times S^1$ and $\mathbb {R}P^{2n+1}$. By elaborating on the construction of the discriminant metric we then define an integer-valued bi-invariant pseudo-metric, that we call the 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 $T^{\ast }B\times S^1$ for any closed manifold $B$, for $\mathbb {R}P^{2n+1}$ and for some $3$-dimensional circle bundles.

LA - eng

KW - Bi-invariant metrics; contactomorphism group; discriminant and translated points of contactomorphisms; Legendrian isotopies; orderability of contact manifolds; generating functions; bi-invariant metrics; discriminant metrics; contactomorphism group; generating functions; oscillation pseudo-metric; Legendrian isotopies; zizag norm

UR - http://eudml.org/doc/277258

ER -

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