Einstein-like and conformally flat contact metric three-manifolds (Q2703560)

Let \((M,\eta, g,\xi, \phi)\) be a contact metric three-manifold. In this paper the author gives the following interesting classification results. NEWLINENEWLINENEWLINEa) The Ricci tensor of \(M\) is cyclic-parallel if and only if \(M\) is locally isometric to a naturally reductive homogeneous space. NEWLINENEWLINENEWLINEb) If the Ricci tensor of \(M\) is a Codazzi tensor and the torsion \(\tau = L_{\xi}g\) satisfies the condition \(\nabla _{\xi}\tau =2a\tau\), where \(a\) is a function constant along the geodesic foliation generated by the characteristic vector field \(\xi\), then \(M\) has constant sectional curvature \(0\) or \(+1\). NEWLINENEWLINENEWLINEc) If \(M\) is conformally flat and the torsion \(\tau\) satisfies the condition \(\nabla _{\xi}\tau =2a\tau\), \(a=const\), then \(M\) has constant sectional curvature \(0\) or \(+1\). NEWLINENEWLINENEWLINENote that the additional condition on \(\tau\) used in b) and c) is a natural condition for a homogeneous contact metric three-manifold [see the reviewer, Ill. J. Math. 42, 243-256 (1998; Zbl 0906.53031)].