In this paper, we consider Egorov and Cahen-Wallach symmetric spaces and study the Riemann solitons on these spaces. We prove that Egorov and Cahen-Wallach symmetric spaces admit the Riemann solitons. Also, we classify the Riemann solitons on these spaces and show that the potential vector fields of the Riemann solitons are Killing, Ricci collineation, and Ricci bi-conformal vector fields.

In the present paper we study Ricci pseudo-symmetry, Z-Ricci pseudo-symmetry and concircularly pseudo-symmetry conditions on a mixed generalized quasi-Einstein spacetime $MG(QE{)}_4$. Also, it is proven that if $d\ne \Lambda$, then $MG(QE{)}_4$ spacetime does not admit heat flux, where $d$ and $\Lambda$ are the function and the cosmological constant, respectively. In the end of this paper we construct a non-trivial example of $MG(QE{)}_4$ to prove its existence.