  First, we show that a warped product of a line and a fiber manifold is weakly
conformally flat and quasi Einstein if and only if the fiber is Einstein. Next,
we characterize and classify contact (in particular, $K$-contact) Riemannian
manifold satisfying weakly (and doubly weakly) conformally flat and
quasi-Einstein ($\eta$-Einstein) conditions. Finally, we provide local
classification and characterization of a semi-Riemannian (including the
4-dimensional spacetime) with harmonic Weyl tensor and a non-homothetic
conformal (including closed) vector field, in terms of Petrov types and Bach
tensor.
