Covariant derivative of the curvature tensor of Kenmotsu manifolds (Q2119255)

Kenmotsu manifolds are almost contact Riemannian manifolds \((M, \varphi, \xi, \eta, g)\) that satisfy the equations \begin{align*} (\nabla_{X}\varphi)Y & = -\eta(Y)\varphi(X) - g(X, \varphi(Y))\xi \\ \nabla_{X}\xi &= X-\eta(X)\xi \end{align*} for all vector fields \(X\) and \(Y\). Here \(\nabla\) is the Levi-Civita connection for \(g\). The author introduces a new \((1,3)\)-tensor \(T\) on Kenmotsu manifolds that detects whether a Kenmotsu manifold is \(\eta\)-Einstein. Moreover, he shows that every \(3\)-dimensional Kenmotsu manifold is a generalized pseudo-symmetric manifold.