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First, we show that a warped product of a line and a fiber manifold is weaklyconformally flat and quasi Einstein if and only if the fiber is Einstein. Next,we characterize and classify contact (in particular, $K$-contact) Riemannianmanifold satisfying weakly (and doubly weakly) conformally flat andquasi-Einstein ($\eta$-Einstein) conditions. Finally, we provide localclassification and characterization of a semi-Riemannian (including the4-dimensional spacetime) with harmonic Weyl tensor and a non-homotheticconformal (including closed) vector field, in terms of Petrov types and Bachtensor.

Characterizations Of Weakly Conformally Flat And Quasi Einstein Manifolds