Invariants of contact structures and transversally oriented foliations
type: Item
instance of: scholarly article
title: Invariants of contact structures and transversally oriented foliations
zbMATH Open document ID: 0902.53024
DOI: 10.1007/BF00129900
author: Augustin Banyaga
published in: Annals of Global Analysis and Geometry
publication date: 1998-12-07T00:00:00Z
review text: Let \(S\) be a contact structure on a manifold \(M\) defined by a global contact 1-form \(\omega\). Denote by \(\text{Diff}_S(M)\) the group of automorphisms of \((M, S): \phi\in \text{Diff}_S(M)\) iff \(\phi\in \text{Diff}(M)\) and \(\phi^\ast\omega =u^\omega_\phi\cdot\omega\) for a positive scalar function \(u^\omega_\phi\) on \(M\). Also, denote by \(h_S(M)\) the Lie algebra of the group \(\text{Diff}_S(M)\): a field \(X\) on \(M\) lies in \(h_S(M)\) iff \(\mathcal L_X\omega= v^\omega_\phi\cdot\omega\) for a scalar function \(v^\omega_\phi\). In the paper under review, the author shows that the maps \(D_\omega: \text{Diff}_S(M)\ni \phi\mapsto \log u^\omega_{\phi^{-1}}\) and \(\Delta_\omega: h_S(M)\ni X\mapsto v^\omega_{\phi^{-1}}\) are suitable 1-cocycles and define cohomology classes in \(H^1(\text{Diff}_S(M), C^\infty (M))\) and \(H^1(h_S(M), C^\infty (M))\), respectively. These classes are shown to be nontrivial and independent of the choice of the form \(\omega\) defining \(S\), and thus become invariants of the structure \(S\). Similar constructions are performed for flows (i.e., one-dimensional transversely oriented foliations) and transverse symplectic structures on contact manifolds.
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53C12
zbMATH DE Number: 955343
zbMATH Keywords: contact structure
zbMATH Keywords: transverse symplectic structure
zbMATH Keywords: flow
zbMATH Keywords: cohomology
reviewed by: Paweł Walczak
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00129900
OpenAlex ID: W2088092553
description: scientific article; zbMATH DE number 955343