On the existence of various generalizations of semisymmetric and pseudosymmetric type manifolds

Abstract: The objective of the paper is to investigate a sequential study of different generalizations of semisymmetric and pseudosymmetric manifolds with their proper existence by several spacetimes. In the literature of differential geometry, there are many generalizations of such notions in various directions by involving different curvature tensors. In this paper, we have systematically and consecutively reviewed various generalized notions of semisymmetry, such as, Ricci semisymmetry, conformal semisymmetry, pseudosymmetry, Ricci pseudosymmetry, Ricci generalized pseudosymmetry, conformal pseudosymmetry, Ricci generalized Weyl pseudosymmetry and also many other semisymmetry type conditions. Most importantly, we have exhibited a plenty of suitable examples to examine the proper existence of such geometric structures, and they are physically significant as several spacetimes admit such geometric structures. By considering an immersion of Schwarzschild black hole metric, we have also provided a new example of Ricci pseudosymmetric manifolds. Finally, the interesting character of Weyl projective curvature tensors of type $(1,3)$ and $(0,4)$ are exhibited with their interesting examples.