Investigations of warped product manifolds admitting $W^{\ast}$-curvature tensor with applications

One significant geometric invariant with relativistic consequences is the $W^{\ast}$-curvature tensor. This article examines the $W^{\star}$-curvature tensor on warped product manifolds $N=\bar{N}\times_{F}\tilde{N}$. In a $W^{\star}$-curvature flat warped product manifold, it is shown that $\tilde{N}$ is Einstein and the $W^{\star}$-curvature tensor vanishes on $\tilde{N}$. Also, necessary and sufficient conditions are given to guarantee that $\bar{N}$ is of constant curvature. The form of the Riemann and Ricci tensors of $\bar{N}$, $\tilde{N}$ are stated. The Ricci tensor of the fiber manifold is of Codazzi type if the $W^{\star}$-curvature tensor is divergence-free, whereas the base manifold's Ricci tensor is of Codazzi type if the Hessian of the warping function is of Codazzi type. In a $W^{\star}$-symmetric warped product manifold, many characterizations of Cartan-symmetry of the factor manifolds are obtained. Finally, many relativistic applications are discussed.