Narrow Results

The object of this paper is to study spacetimes admitting W₂-curvature tensor. At first we prove that a W₂-flat spacetime is conformally flat and hence it is of Petrov type O. Next, we prove that if the perfect fluid spacetime with vanishing W₂-curvature tensor obeys Einstein's field equation without cosmological constant, then the spacetime has vanishing acceleration vector and expansion scalar and the perfect fluid always behaves as a cosmological constant. It is also shown that in a perfect fluid spacetime of constant scalar curvature with divergence-free W₂-curvature tensor, the energy-momentum tensor is of Codazzi type and the possible local cosmological structure of such a spacetime is of type I, D or O.

The goal of this paper is to study SdS space-time and its gravitational field with consideration of the canonical form and invariants of the curvature tensor. The characteristic of $\lambda$-tensor identifies the type of gravitational field. Gaussian curvature quantities enunciated in terms of curvature invariants.

The aim of this paper is to study special multiply Einstein warped products having an affine connection. Let $M=I{\times }_{f_1}{F}_1\times \cdots {\times }_{f_m}{F}_m$ be a multiply warped product such that $I\subset ℝ$ is an open interval, $dimI=1,$$f_i:I\to (0,\infty ),$$f_i\in {{𝒞}}^{\infty }(I),$$dim{F}_i=k_i\ge 1$ for every $i\in \{1,\dots ,m\},$$m\ge 1,$$dimM=\overline{n}=1+{\sum }_{i=1}^m{k}_i$ and $\overline{\nabla }$ an affine connection on $M.$ We compute the warping functions that make $M$ an Einstein space in the following cases:

• ^(a)$\overline{\nabla }$ is a semi-symmetric metric/non-metric connection and all the fibers are Ricci flat.
• ^(b)$\overline{\nabla }$ is a quarter-symmetric metric/non-metric connection and all the fibers are Ricci flat.

In this paper we review basic properties of manifolds endowed with paraquaternionic structures and mixed 3-structures. Also, we investigate the existence of mixed 3-structures on normal semi-invariant submanifolds of paraquaternionic Kähler manifolds.