On 3-dimensional Kenmotsu manifolds (Q1431057)

By simple calculations the standard metric of constant sectional curvature \(K=-1\) is characterized as the only 3-dimensional Kenmotsu manifold satisfying certain curvature conditions. One of them is the Bianchi type equality \(\nabla_{[i}R_{jk]}=0\) (cyclic sum in \(i,j,k)\) for the Ricci tensor. Another one is the assumption that the Ricci tensor is \(\eta\)-parallel. This is based on the fact that the notion of a Kenmotsu manifold is not scale invariant: By definition the sectional curvature in any \((X,\xi)\)-plane is \(-1\) where \(\xi\) is the structure vector field and \(X\) is orthogonal to \(\xi\). Therefore the metric is of constant sectional curvature if and only if its scalar curvature is equal to \(-6\).