The main aim of this study is to investigate the effects of the
P
−
curvature flatness,
P
−
divergence-free characteristic, and
P
−
symmetry of a warped product manifold on its base and fiber (factor) manifolds. It is proved that the base and the fiber manifolds of the
P
−
curvature flat warped manifold are Einstein manifold. Besides that, the forms of the
P
−
curvature tensor on the base and the fiber manifolds are obtained. The warped product manifold with
P
−
divergence-free characteristic is investigated, and amongst many results, it is proved that the factor manifolds are of constant scalar curvature. Finally,
P
−
symmetric warped product manifold is considered.