Dipankar Debnath

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.

In this paper we have introduced the notion of $\ast -$ Ricci flow and shown that $\ast -$ Ricci soliton which was introduced by Kakimakamis and Panagiotid in 2014, is a self similar soliton of the $\ast -$ Ricci flow. We have also find the deformation of geometric curvature tensors under $\ast -$ Ricci flow. In the last two section of the paper, we have found the $\Im$-functional and $\omega -$ functional for $\ast -$ Ricci flow respectively.