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The object of this paper is to study the properties of flat spacetimes under some conditions regarding the W 2-curvature tensor. In the first section, several results are obtained on the geometrical symmetries of this curvature tensor. It is shown that in a spacetime with W 2curvature tensor filled with a perfect fluid, the energy momentum tensor satisfying the Einstein's equations with a cosmological constant is a quadratic conformal Killing tensor. It is also proved that a necessary and sufficient condition for the energy momentum tensor to be a quadratic Killing tensor is that the scalar curvature of this space must be constant. In a radiative perfect fluid, it is shown that the sectional curvature is constant.

The object of the present paper is to study the nature of LP-Sasakian manifolds admitting the M -projective curvature tensor. It is examined whether this manifold satisfies the condition W (X, Y ).R = 0. Moreover, it is proved that, in the M -projectively flat LP-Sasakian manifolds, the conditions R(X, Y ).R = 0 and R(X, Y ).S = 0 are satisfied. In the last part of our paper, M -projectively flat space-time is introduced and some properties of this space are obtained.Вивчається природа многовидiв Сасакяна, що допускають M -проективний тензор кривизни. Перевiрено, чи задовольняє цей многовид умову W (X, Y ).R = 0. Бiльш того, доведено, що умови R(X, Y ).R = 0 та R(X, Y ).S = 0 виконуються для M -проективно плоских LP-многовидiв Сасакяна. В останнiй частинi роботи введено M -проективно плоский простiр-час та встановлено деякi властивостi цього простору.

As it is known, Einstein manifolds play an important role in geometry as well as in general relativity. Einstein manifolds form a natural subclass of the class of quasi-Einstein manifolds. In this work, we investigate conformal mappings of quasi-Einstein manifolds. Considering this mapping, we examine some properties of these manifolds. After that, we also study some special vector fields under this mapping of these manifolds and some theorems about them are proved.