On some important characterizations of Lorentz para-Kenmotsu manifolds on some special curvature tensors (Q6650983)

The $W_8$-curvature and $W_9$-curvature tensors on a semi-Riemannian manifold $M^n$ are defined as\N\[\NW_8(Z_1, Z_2) Z_3 = R(Z_1, Z_2) Z_3 - \frac{1}{n-1} [S(Z_2, Z_3) Z_1- S(Z_1, Z_2) Z_3]\N\]\Nand \N\[\NW_9(Z_1, Z_2) Z_3 = R(Z_1, Z_2) Z_3 - \frac{1}{n-1} [S(Z_1, Z_2) Z_3- g(Z_2, Z_3) QZ_1], \N\]\Nrespectively. Here $R, S$ and $Q$ are the Riemannian curvature tensor, the Ricci tensor and the Ricci operator of the manifold $M^n$, respectively.\N\NIn the present work, Lorentzian para-Kenmotsu manifolds are studied and described in terms of the $W_8$-curvature and $W_9$-curvature tensors. The following two results are very important: \N\NTheorem 1. If $M^n$ is a $W_8$-flat Lorentzian para-Kenmotsu manifold, then $M^n$ is an Einstein manifold. \N\NTheorem 2. If $M^n$ is a $W_9$-flat Lorentzian para-Kenmotsu manifold, then $M^n$ is an $\eta$-Einstein manifold.\N\NSome results on Lorentzian para-Kenmotsu manifolds admitting almost $\eta$-Ricci solitons are obtained. In particular, it is proved that an $\eta$-Ricci soliton on a $W_8$-Ricci semisymmetric Lorentzian para-Kenmotsu manifold is always shrinking.\N\NSome characterizations of invariant submanifolds of Lorentzian para-Kenmotsu manifolds in terms of $W_8$-curvature and $W_9$-curvature tensors are also presented.