DOI: 10.36890/iejg.1788384 ISSN: 1307-5624

Hyper Generalized Pseudo-Ricci Symmetric Manifolds

Rana Pratap Singh Yadav, Bhagwat Prasad, Sudhakar Kumar Chaubey
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Symmetric spacetimes such as Ricci symmetric manifolds $\mathcal{(RSM}s)$, pseudo-Ricci manifolds $\mathcal{(PRSM}s)$, Ricci recurrent manifolds $\mathcal{(RRM}s)$, semi-Ricci symmetric manifolds $\mathcal{(SRSM}s)$, semi-pseudo-Ricci symmetric manifolds $\mathcal{(SPRSM}s)$, almost pseudo-Ricci symmetric manifolds $\mathcal{(APRSM}s)$, and extended pseudo-Ricci symmetric manifolds $\mathcal{(EPRSM}s)$ have played a significant role in the development of differential geometry and mathematical physics, particularly in finding exact solutions to Einstein's field equations. Motivated by these structures, we introduce a new class of Riemannian manifolds, termed hyper generalized pseudo-Ricci symmetric manifolds $\mathcal{H(GPRS)}_n$, characterized by a specific condition on the covariant derivative of the Ricci tensor involving three associated $1$-forms. This class encompasses several well-known symmetric spaces as special cases, thereby providing a unified framework for their study. We establish the existence of such manifolds with an illustration demonstrating the existence of this new class of manifolds and explore their geometric properties, including scalar curvature behavior, eigenvalue structure under pseudo-cyclic Ricci tensor $\mathcal{(PCRT})$, and conditions under which they exhibit quasi-constant curvature. Further, we investigate conformally flat $\mathcal{H(GPRS)}_n$, study their structure in the context of $\mathbb{GRW}$ spacetimes, and analyze their decomposition under Gray's classification.

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