A class of conformally flat contact metric 3-manifolds
type: Item
instance of: scholarly article
title: A class of conformally flat contact metric 3-manifolds
zbMATH Open document ID: 1052.53054
DOI: 10.1007/BF03322727
author: Florence Gouli-Andreou
author: Ramesh Sharma
published in: Results in Mathematics
publication date: 2003-11-10T00:00:00Z
review text: A contact metric manifold \(M(\eta,\xi,\phi,q)\) is a \((2n+1)\)-dimensional manifold equipped with a global contact form \(\eta\), a \((1,1)\)-tensor field \(\phi\) and a Riemannian metric \(q\) satisfying \[ \begin{gathered} \phi^2=-I+ \eta\otimes\xi,\quad \eta(X)= q(\xi,X),\\ d\eta(X,Y)= q(X,\phi Y),\end{gathered} \] for all vector fields \(X\), \(Y\) on \(M\), where \(\xi\) is the characteristic (Reeb) vector field. In this paper, the authors show in the 3-dimensional case that if the Riemannian metric is conformally flat and its Ricci curvature vanishes along \(\xi\) then the scalar curvature is a divergence and is nonpositive. One consequence is that if \(M\) is also compact then the metric is flat.
Mathematics Subject Classification ID: 53D10
Mathematics Subject Classification ID: 53C21
zbMATH DE Number: 2002346
zbMATH Keywords: conformally flat
zbMATH Keywords: contact metric 3-manifold
zbMATH Keywords: scalar curvature
reviewed by: R. Hind
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2002346