SOME GEOMETRIC AND PHYSICAL PROPERTIES OF PSEUDO $\mathscr{M}^{\ast}$-PROJECTIVE SYMMETRIC MANIFOLDS

Abstract

In this study we introduce a new tensor in a semi-Riemannian manifold, named the $\mathscr{M}^{\ast}$-projective curvature tensor which generalizes the $m$ projective curvature tensor. We start by deducing some fundamental geometric properties of the $\mathscr{M}^{\ast}$-projective curvature tensor. After that, we study pseudo $\mathscr{M}^{\ast}$-projective symmetric manifolds $\left(PM^{\ast}S\right)_{n}$. A non-trivial example has been used to show the existence of such a manifold. We introduce a series of interesting conclusions. We establish, among other things, that if the scalar curvature $\rho$ is non-zero, the associated $1$-form is closed for a $\left(PM^{\ast}S\right)_{n}$ with $\mathrm{div}M^{\ast}=0$. We also deal with pseudo $\mathscr{M}^{\ast}$-projective symmetric spacetimes, $\mathscr{M}^{\ast}$-projectively flat perfect fluid spacetimes, and $\mathscr{M}^{\ast}$-projectively flat viscous fluid spacetimes. As a result, we establish some significant theorems.