On the application of M-projective curvature tensor in general relativity

In this paper the application of the $M$-projective curvature tensor in the general theory of relativity has been studied. Firstly, we have proved that an $M$-projectively flat quasi-Einstein spacetime is of a special class with respect to an associated symmetric tensor field, followed by the theorem that a spacetime with vanishing $M$-projective curvature tensor is a spacetime of quasi-constant curvature. Then we have proved that an $M$-projectively flat quasi-Einstein spacetime is infinitesimally spatially isotropic relative to the unit timelike vector field $\xi$. In the next section we have proved that an $M$-projectively flat Ricci semi-symmetric quasi-Einstein spacetime satisfying a definite condition is an $N(\frac{2l-m}{6})$-quasi Einstein spacetime. In the last section, we have firstly proved that an $M$-projectively flat perfect fluid spacetime with torse-forming vector field $\xi$ satisfying Einstein field equation with cosmological constant represents an inflation, then we have found out the curvature of such spacetime, followed by proving the theorem that the spacetime also becomes semi-symmetric under these conditions. Lastly, we have found out the square of the length of the Ricci tensor in this type of spacetime and also proved that if an $M$-projectively flat perfect fluid spacetime satisfying Einstein field equation with cosmological constant, with torse-forming vector field $\xi$ admits a symmetric $(0,2)$ tensor $\alpha$ parallel to $\nabla$ then either $\lambda = \frac{k}{2}(p-\sigma)$ or $\alpha$ is a constant multiple of $g$.