On mixed super quasi-Einstein manifolds with Ricci-Bourguignon solitons

TL;DR

The paper addresses the geometry and physics of mixed super quasi-Einstein manifolds equipped with Ricci-Bourguignon solitons. It develops curvature identities under conformal Ricci pseudosymmetry, analyzes soliton structures with a torse-forming generator, and connects these to the Einstein field equations via the space-matter tensor, including a 4D explicit example. Key findings show that covariantly constant data reduces the Einstein equation to a linear combination of geometric tensors, that vanishing space-matter tensor yields a mixed quasi-constant curvature structure, and that RBC solitons enforce a pseudo generalized quasi-Einstein geometry with eigenvalue relations for the governing tensor, along with energy-density constraints. This framework links generalized quasi-Einstein geometry to physically meaningful spacetimes and provides a tangible example demonstrating the existence of mixed SQE spacetimes.

Abstract

This paper delves into the study of mixed super quasi-Einstein manifolds of dimension $n$ (for short, ${\rm M^{n}_{SQE}}$), focusing on their geometric and physical attributes. Initially, we explore several properties of ${\rm M^{n}_{SQE}}$, including conformal Ricci pseudosymmetry, Einstein's field equation, and the space-matter tensor. Subsequently, we characterize mixed super quasi-Einstein manifolds that admit Ricci-Bourguignon solitons. We establish that if the generator vector field is torse-forming, the manifold reduces to a pseudo generalized quasi-Einstein manifold, and we provide a detailed characterization of the eigenvalue problem associated with the symmetric tensor. To substantiate our findings, we construct an example to illustrate the existence of mixed super quasi-Einstein spacetime.

Theorems & Definitions (35)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Definition 2.3