Nearly trans-Sasakian manifolds (Q2785897)

\textit{K. Kenmotsu} [Tôhoku Math. J. 24, 93-103 (1972; Zbl 0245.53040)] studied the class of almost contact metric manifolds \(M(\varphi,\xi,\eta,g)\) such that the \((0,3)\)-tensor field \(T\), defined below, vanishes identically on \(M\): NEWLINE\[NEWLINE T(X,Y,Z):=((\nabla_X\varphi)Y,Z)-\eta(Y)g(\varphi Z,X)-\eta(Z)g(\varphi X,Y). NEWLINE\]NEWLINE In the geometrical literature, such manifolds are usually said to be Kenmotsu manifolds [cf., e.g., \textit{D. Janssens} and \textit{L. Vanhecke}, Kodai Math. J. 4, 1-27 (1981; Zbl 0472.53043)]. However, the author calls them trans-Sasakian manifolds. By a nearly trans-Sasakian manifold he means an almost contact metric manifold for which the tensor \(T\) is fully symmetric.NEWLINENEWLINENEWLINEIn the paper under review, certain analytic formulas concerning the covariant derivatives of the structure tensors \(\varphi\), \(\xi\), \(\eta\) of such manifolds are obtained, and it is proved that such manifolds are trans-Sasakian if they are additionally normal.