A class of conformally flat contact metric 3-manifolds (Q1412976)

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.