Narrow Results

Electrodynamics in curved space-time can be studied in the Eastwood–Singer gauge, which has the advantage of respecting the invariance under conformal rescalings of the Maxwell equations. Such a construction is here studied in Einstein spaces, for which the Ricci tensor is proportional to the metric. The classical field equations for the potential are then equivalent to first solving a scalar wave equation with cosmological constant, and then solving a vector wave equation where the inhomogeneous term is obtained from the gradient of the solution of the scalar wave equation. The Eastwood–Singer condition leads to a field equation on the potential which is preserved under gauge transformations provided that the scalar function therein obeys a fourth-order equation where the highest-order term is the wave operator composed with itself. The second-order scalar equation is here solved in de Sitter space-time, and also the fourth-order equation in a particular case, and these solutions are found to admit an exponential decay at large time provided that square-integrability for positive time is required. Last, the vector wave equation in the Eastwood–Singer gauge is solved explicitly when the potential is taken to depend only on the time variable.

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 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.

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.

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.

The existence of conformal Killing-Yano tensors on higher dimensional spacetimes endowed with positive and negative mixed 3-Sasakian structures is investigated.