A Study On Some Geometric Properties and Physical Applications of A Mixed Quasi-Einstein Spacetime.

In the present paper we discuss about a set of geometric properties and physical applications of a mixed quasi-Einstein spacetime$[M(QE)_{4}]$, which is a special type of nearly quasi-Einstein spacetime$[N(QE)_{4}]$. Firstly we consider a nearly quasi-Einstein manifold$[N(QE)_{n}]$ along with a mixed quasi-Einstein manifold$[M(QE)_{n}]$ and study some geometric conditions with respect to different curvature tensors on them. Then we consider a mixed quasi-Einstein spacetime and study $W_{2}$-Ricci pseudosymmetry and projective Ricci pseudosymmetry on it. Then we study the conditions $H \cdot S=0$ and $\tilde C \cdot S=0$ in an $M(QE)_{4}$ spacetime, where $H, \tilde C$ are the conharmonical curvature tensor and semiconformal curvature tensor respectively. Next we consider a semiconformally flat Ricci pseudosymmetric $M(QE)_{4}$ spacetime and find the nature of that spacetime. In the next section we study about some applications of a perfect fluid $M(QE)_{4}$ spacetime in the general theory of relativity. Followed by we discuss about the Ricci soliton structure in a semiconformally flat $M(QE)_{4}$ spacetime satisfying Einstein field equation without cosmological constant. Finally we give a nontrivial example of a $M(QE)_{4}$ spacetime to establish the existence of it.