A note on generalized weakly ℋ-symmetric manifolds and relativistic applications

In this work, generalized weakly Formula: see text-symmetric space-times Formula: see text are investigated, where Formula: see text is any symmetric Formula: see text tensor. It is proved that, in a nontrivial Formula: see text space-time, the tensor Formula: see text has a perfect fluid form. Accordingly, sufficient conditions for a nontrivial generalized weakly Ricci symmetric space-time Formula: see text to be either an Einstein space-time or a perfect fluid space-time are obtained. Also, conditions for space-times admitting either a generalized weakly symmetric energy-momentum tensor or a generalized weakly symmetric Formula: see text tensor to be Einstein or perfect fluid space-times are provided.