Research Article A Study of Generalized Projective P - Curvature Tensor on Warped Product Manifolds Uday Chand De, 1 Abdallah Abdelhameed Syied , 2 Nasser Bin Turki, 3 and Suliman Alsaeed 4 1 Department of Pure Mathematics, University of Calcutta, 35, Ballygaunge Circular Road, Kolkata 700019, West Bengal, India 2 Department of Mathematics, Faculty of Science, Zagazig University, P.O. Box 44519, Zagazig, Egypt 3 Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia 4 Applied Science College, Department of Mathematical Sciences, Umm Al-Qura University, P.O. Box 715, Makkah 21955, Saudi Arabia Correspondence should be addressed to Abdallah Abdelhameed Syied; a.a_syied@yahoo.com Received 18 October 2021; Accepted 14 December 2021; Published 27 December 2021 Academic Editor: Antonio Masiello Copyright © 2021 Uday Chand De et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e 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. e 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. 1. Introduction Curvature tensors play a significant role in mathematics and physics. is is why many researchers have introduced and studied many curvature tensors in various ways, as well as they have shown the importance of these curvature tensors. For instance, the deviation of a space from constant cur- vature is measured by the concircular curvature tensor (for more details, see [1]). e Weyl curvature tensor describes the distorting but volume-preserving tidal effects of gravi- tation on a material body. e P−curvature tensor was first coined by De et al. in 2021 [2]. is curvature tensor is a good generalization of projective [3], conharmonic [4], M−projective [5], and the set of W i −curvature tensors which was introduced by Pokhariyal and Mishra [6–10]. is curvature tensor is given by P ijkl � a 0 R ijkl + a 1 g ij R kl + a 2 g ik R jl + a 3 g il R jk + a 4 g jk R il + a 5 g jl R ik + a 6 g kl R ij , (1) where a i are constants, R ijkl is the Riemann tensor, and R ij is the Ricci tensor [2]. e authors studied this curvature tensor on pseudo-Riemannian manifolds and space times of general relativity. It is proved that pseudo-Riemannian manifolds M will be Einstein manifold if M admits a traceless P−curvature tensor and will be of constant scalar curvature if M is of P−curvature flat. Pseudo-Riemannian manifolds with P−divergence-free characteristic were investigated in Gray’s seven subspaces. As a final point, they studied perfect fluid space times when the P−curvature tensor is flat, and in this case, many interesting results are obtained. Geometers have considered all well-known curvature tensors on the warped product manifolds. For instance, Hindawi Journal of Mathematics Volume 2021, Article ID 7882356, 10 pages https://doi.org/10.1155/2021/7882356