Pseudosymmetry, Ricci soliton and Curvature Inheritance symmetries of Friedmann Lema\^itre Robertson Walker spacetime
The Friedmann--Lema\^{i}tre--Robertson--Walker (FLRW) spacetime, which was first proposed by Friedmann (1922--1924) and Lema\^{i}tre (1927) and subsequently developed by Robertson and Walker (1935), is an isotropic and homogeneous cosmological model of the universe. This paper addresses a significant gap in the differential geometry literature by providing a comprehensive examination of the curvature properties of the FLRW spacetime. It is demonstrated that the FLRW spacetime satisfies the curvature condition R \cdot R - Q(S, R)=L_C Q(g, C) alongside several pseudosymmetric-type conditions related to the conformal and conharmonic curvature tensors. Furthermore, the Tachibana tensors Q(g,C) and Q(S, C) are found to exhibit a linear dependence on the tensor $(C \cdot R + R \cdot C)$. Additionally, the spacetime is shown to be a 2-quasi-Einstein manifold, generalized Roter type and Ein(3). The Ricci tensor is shown to be neither cyclic parallel nor of Codazzi type, yet it satisfies several compatibility requirements concerning the R, C, P, K and W curvature tensors. A thorough analysis of Ricci solitons and curvature inheritance properties reveals that the spacetime admits almost Ricci soliton and $\eta$-Ricci Yamabe soliton structures with respect to the non-Killing soliton vector fields $\frac{\partial}{\partial t}$ and $\frac{\partial}{\partial r}$. Moreover, the spacetime admits generalized curvature inheritance symmetry properties for the Riemann curvature tensor, as well as for the Weyl conformal, concircular, and conharmonic curvature tensors with respect to the coordinate vector field $\frac{\partial}{\partial t}$ and the gradient of $t$. Later, a comparison of the FLRW and Lema\^{i}tre--Tolman--Bondi (LTB) spacetimes is provided in terms of various curvature-related geometric properties and physical characteristics. Finally, a noteworthy conclusion of the entire study is presented.