ON pseudo $\mathcal{H}$-symmetric Lorentzian manifolds with applications to relativity

Abstract

In this paper, we introduce a new type of curvature tensor named $\mathcal{H}$-curvature tensor of type $(1,3)$ which is a linear combination of conformal and projective curvature tensors. First we deduce some basic geometric properties of $\mathcal{H}$-curvature tensor. It is shown that a $\mathcal{H}$-flat Lorentzian manifold is an almost product manifold. Then we study pseudo $\mathcal{H}$-symmetric manifolds $(P\mathcal{H}S)_{n}(n > 3)$ which recovers some known structures on Lorentzian manifolds. Also, we provide several interesting results. Among others, we prove that if an Einstein $(P\mathcal{H}S)_{n}$ is a pseudosymmetric $(PS)_{n}$, then the scalar curvature of the manifold vanishes, provided $2a+(n-2)b\not=0$ and conversely. Moreover, we deal with pseudo $\mathcal{H}$-symmetric perfect fluid spacetimes and obtain several interesting results. Also, we present some results of the spacetime satisfying divergence free $\mathcal{H}$-curvature tensor. Finally, we construct a non-trivial Lorentzian metric of $(P\mathcal{H}S)_4$.