A conformally flat contact Riemannian \((\kappa,\mu)\)-space (Q5939132)

Let \(M\) be an odd dimensional manifold with a contact form \(\eta\). Let \(\xi\) be the characteristic vector field of \(\eta\) determined by \(\eta (\xi)=1\) and \(d\eta (\xi,X)=0\) for all vector fields \(X\). Then there exists a Riemannian metric \(g\) and a \((1,1)\) tensor \(\varphi\) such that \[ \eta(X)= g(X, \xi),\;d\eta(X,Y)= g\bigl(X, \varphi(Y) \bigr),\quad \varphi^2(X)= -X+ \eta (X) \xi \] for all \(X,Y\). Then \(M=(M,\eta, \varphi,g)\) is called a contact Riemannian manifold. Due to Blair, Koufogiorgos and Papantoniou a contact Riemannian manifold \(M\) is called a \((\kappa,\mu)\)-space, if the following equation is satisfied \[ R(X,Y)\xi= \kappa\bigl( \eta(Y)X-\eta(X)Y \bigl)+ \mu \bigl(\eta (Y) h(X)- \eta (X)h (Y)\bigr), \] where \(\kappa,\mu\) are constants and \(h(X)\) is 1/2 of the Lie differential of \(X\) along \(\xi\). The main results in the paper are the following: Theorem A. Let \(M\) be a conformally flat \((\kappa, \mu)\)-space. Then \(M\) is a 3-dimensional flat manifold or a Sasakian manifold of constant curvature 1. Theorem B. Let \(M\) be a \((\kappa,\mu)\)-space with vanishing C-Bochner curvature tensor. Then \(M\) is a Sasakian manifold.