On a class of contact Riemannian manifolds (Q1586737)

\textit{D. E. Blair, T. Koufogiorgos} and \textit{B. Papantoniou} [Isr. J. Math. 91, No. 1-3, 189-214 (1995; Zbl 0837.53038)] have introduced a class of contact Riemannian manifolds \((M,\varphi,\xi,\eta,g)\) (\(M\) in short), whose curvature satisfies the condition \((\ast)\quad R(X,Y)\xi=\kappa(\eta(Y)X-\eta(X)Y)+\mu(\eta(Y)hX-\eta(X)hY),\) where \(\kappa\), \(\mu\) are constants and \(h=(1/2)\mathcal{L}_{\xi}\varphi\) (the Lie derivative of \(\varphi\) in the direction \(\xi\)). The author proves here the following results. Let \(M\) be a contact Riemannian manifold satisfying \((\ast)\) and \(\dim M=2n+1\). (1) If \(M\) is additionally locally symmetric, then (a) \(M\) is locally the product of a flat \((n+1)\)-dimensional manifold and an \(n\)-dimensional manifold of positive constant curvature 4, or (b) \(M\) is a space of constant curvature 1 (in this case the contact structure \((\varphi,\xi,\eta,g)\) is Sasakian). (2) If \(M\) is Ricci-parallel, then (a) \(M\) is locally the product of a flat \((n+1)\)-dimensional manifold and an \(n\)-dimensional manifold of positive constant curvature 4, or (b) \(M\) is an Einstein-Sasakian manifold.