A bilinear form associated to contact sub-conformal manifolds
type: Item
instance of: scholarly article
title: A bilinear form associated to contact sub-conformal manifolds
zbMATH Open document ID: 1142.53022
author: Elisha Falbel
author: Jose Miguel Martins Veloso
published in: Differential Geometry and its Applications
publication date: 2007-06-13T00:00:00Z
review text: Let \(M\) be a \((2n + 1)\)-dimensional manifold endowed with a sub-Riemannian structure \((M, D, g)\) where \(D\) is a contact distribution on \(M\) and \(g\) is a metric on \(D\). Denote by \((M D, \tilde{g})\) a sub-conformal structure on \(M\) such that \(\tilde{g}\) is a conformal class of sub-Riemannian metrics on \(D\). Consider the \(SO(2n)\) bundle \(E\) of coframes \((\theta, \theta^i)\) on \(M\) such that the \(\theta^i\) restricted to \(D\) are orthonormal, \(\theta\) is the positive contact form defined by \[ d\theta = h_{ij} \theta^i \wedge \theta^j,\quad \text{det}h_{ij} = 1. \] Denote by \(\wedge\) the canonical form of \(H = (h_{ij})\) and \(E_1\) the subset of points of \(E\) such that \(H = \wedge\). The authors have proved the following three results {\parindent=4mm \begin{itemize}\item[{\(\bullet\)}] Suppose \(H\) is constant on \(E_1\). Then there exists a torus bundle \(T\) over \(M\) and a conformal invariant bilinear form \(b\) on \(T\). \item[{\(\bullet\)}] If \(H^2 = -Id\) on \(E_1\), then the torus bundle \(T\) is a \(U(1)\) bundle and \(b\) is a conformal Lorentz metric. \item[{\(\bullet\)}] The bilinear form \(b\) descends to a Lorentz metric on the \(U(1)\) bundle \(N = T/ \exp g\) defined by another conformal invariant bilinear form. \end{itemize}}
Mathematics Subject Classification ID: 53C17
Mathematics Subject Classification ID: 53C15
zbMATH DE Number: 5163861
zbMATH Keywords: contact manifolds
zbMATH Keywords: Fefferman metric
zbMATH Keywords: sub-Riemannian manifolds
Wikidata QID: Q115357831
reviewed by: Krishan Lal Duggal
MaRDI profile type: Publication
full work available at URL: j.difgeo.2006.05.007
OpenAlex ID: W2070814612
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DOI: 10.1016/J.DIFGEO.2006.05.007
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description: scientific article; zbMATH DE number 5163861