quasi-Einstein Manifolds Research Articles

Notes on conformal and non-conformal transformation metrics of generalized quasi-Einstein manifolds

Notes on conformal and non-conformal transformation metrics of generalized quasi-Einstein manifolds

Characterization of Bach and Cotton Tensors on a Class of Lorentzian Manifolds

In this work, we aim to investigate the characteristics of the Bach and Cotton tensors on Lorentzian manifolds, particularly those admitting a semi-symmetric metric ω-connection. First, we prove that a Lorentzian manifold admitting a semi-symmetric metric ω-connection with a parallel Cotton tensor is quasi-Einstein and Bach flat. Next, we show that any quasi-Einstein Lorentzian manifold admitting a semi-symmetric metric ω-connection is Bach flat.

A note on closed quasi-Einstein manifolds

A note on closed quasi-Einstein manifolds

PARA-SASAKIAN MANIFOLD ADMITTING RICCI-YAMABE SOLITONS WITH QUARTER SYMMETRIC METRIC CONNECTION

In the year 2019, Guler and Crasmareanu [6] conducted an investigation into another geometric flow known as the Ricci-Yamabe map. This map is nothing but a scalar combination of the Ricci and the Yamabe flow [7]. The primary objective of the current paper is to provide a characterization of a Ricci Yamabe soliton on a para-Sasakian manifold [17]. To commence, we prove that a para-Sasakian manifold admits a nearly quasi-Einstein manifold. Moreover, we discuss whether such a manifold is shrinking, expanding, or steady. Subsequently, we generalize these findings to Ricci-Yamabe solitons on para-Sasakian manifolds equipped with a quarter symmetric metric connection. Lastly, we furnish an illustration of a three-dimensional para-Sasakian manifold admitting a Ricci-Yamabe soliton which satisfies our results.

A study of mixed super quasi-Einstein manifolds with applications to general relativity

In this paper, we explore a set of geometric properties of Mixed Super Quasi-Einstein (MSQE) manifolds and provide examples of both Riemannian and Lorentzian MSQE manifolds to demonstrate their existence. Furthermore, we examine MSQE spacetimes in the context of the space-matter tensor, discussing several related properties. Finally, we establish the existence of an MSQE spacetime through a nontrivial example.

Some characterizations of Quasi-Einstein and doubly product manifold

This research paper explores the properties of quasi-Einstein manifolds with a unit concircular generator vector field within doubly warped product structures. The paper begins by investigating the characteristics of quasi-Einstein manifolds that possess a unit concircular generator vector field. Subsequently, it analyzes the behavior of the Hessian, Riemann, and Ricci vector fields in the context of a doubly warped product. Moreover, the impact of the quasi-Einstein condition on both the doubly warped product and its factor manifolds is studied. Finally, conditions are derived to determine whether a doubly warped product satisfying the quasi-Einstein condition can be classified as a warped or a direct product. By presenting these findings, this study contributes to the comprehension of quasi-Einstein manifolds and their interactions within doubly warped product structures.

Pseudo generalized quasi-Einstein manifolds with applications to general relativity

Quasi-Einstein manifold and generalized quasi-Einstein manifold are the generalization of Einstein manifold. In the present paper we discuss about a set of some geometric properties of pseudo generalized quasi-Einstein manifold and we give three and four examples (both Riemannian and Lorentzian) of pseudo generalized quasi-Einstein manifold to show the existence of such manifold. We also discuss PG(QE)4 spacetime with space-matter tensor and some properties related to it. Lastly we prove the existence of a pseudo generalized quasi-Einstein spacetime by constructing a non-trivial example.

ON RICCI SEMI-SYMMETRIC SUPER QUASI-EINSTEIN HERMITIAN MANIFOLD

The object of the present paper is to study the Bochner Ricci semi-symmetric super quasi-Einstein Hermitian manifold and a holomorphically projective Ricci-semi symmetric super quasi Einstein Hermitian manifold.

Quasi-Einstein manifolds admitting a closed conformal vector field

In this article, we investigate quasi-Einstein manifolds admitting a closed conformal vector field. Initially, we present a rigidity result for quasi-Einstein manifolds with constant scalar curvature and carrying a non-parallel closed conformal vector field. Moreover, we prove that quasi-Einstein manifolds admitting a closed conformal vector field can be conformally changed to constant scalar curvature almost everywhere. Finally, we obtain a characterization for quasi-Einstein manifolds endowed with a non-parallel gradient conformal vector field.

Characterization of Ricci Almost Soliton on Lorentzian Manifolds

Ricci solitons (RS) have an extensive background in modern physics and are extensively used in cosmology and general relativity. The focus of this work is to investigate Ricci almost solitons (RAS) on Lorentzian manifolds with a special metric connection called a semi-symmetric metric u-connection (SSM-connection). First, we show that any quasi-Einstein Lorentzian manifold having a SSM-connection, whose metric is RS, is Einstein manifold. A similar conclusion also holds for a Lorentzian manifold with SSM-connection admitting RS whose soliton vector Z is parallel to the vector u. Finally, we examine the gradient Ricci almost soliton (GRAS) on Lorentzian manifold admitting SSM-connection.

Compact gradient Einstein-type manifolds with boundary satisfying the weakly Einstein condition

In this article, we study a family of compact gradient Einstein-type manifolds with boundary, introduced by Freitas and Gomes in [7], satisfying the weakly Einstein condition. We prove the rigidity results for this case when the scalar curvature is constant. As a special case of gradient Einstein-type manifold, we also give the rigidity result of n≥4 dimensional compact weakly Einstein (m,ρ)-quasi-Einstein manifold with boundary.

Rigidity Results On Generalized m-Quasi Einstein Manifolds with Associated Affine Killing Vector Field.

We study a non-trivial generalized $m$-quasi Einstein manifold $M$ with finite $m$ and associated divergence-free affine Killing vector field, and show that $M$ reduces to an $m$-quasi Einstein manifold. In addition, if $M$ is complete, then it splits as the product of a line and an $(n-1)$-dimensional negatively Einstein manifold. Finally, we show that the same result holds for a complete non-trivial $m$-quasi Einstein manifold $M$ with finite $m$ and associated affine Killing vector field.

Several geometric properties on mixed quasi-Einstein manifolds with soliton structures

The object of this paper is to study mixed quasi-Einstein manifolds of dimension [Formula: see text] (for short, [Formula: see text]) and their geometric properties. We discuss some geometric properties of [Formula: see text] such as Ricci semisymmetric, concircular Ricci pseudosymmetric and [Formula: see text] Ricci pseudosymmetric. Furthermore, we discuss the mixed quasi-Einstein manifolds with Ricci soliton and Riemann soliton, respectively. Finally, we construct an example of mixed quasi-Einstein manifold.

A Note on Generalized Solitons

In this paper, we initiate the study of a generalized soliton on a Riemannian manifold, we find a characterization for the Euclidean space, and in the compact case, we find a sufficient condition under which it reduces to a quasi-Einstein manifold. We also find sufficient conditions under which a compact generalized soliton reduces to an Einstein manifold. Note that Ricci solitons being self-similar solutions of the heat flow, this topic is related to the symmetry in the geometry of Riemannian manifolds. Moreover, generalized solitons being generalizations of Ricci solitons are naturally related to symmetry.

Solitonic Aspect of Relativistic Magneto-Fluid Spacetime with Some Specific Vector Fields

The target of the current research article is to investigate the solitonic attributes of relativistic magneto-fluid spacetime (MFST) if its metrics are Ricci–Yamabe soliton (RY-soliton) and gradient Ricci–Yamabe soliton (GRY-soliton). We exhibit that a magneto-fluid spacetime filled with a magneto-fluid density ρ, magnetic field strength H, and magnetic permeability μ obeys the Einstein field equation without the cosmic constant being a generalized quasi-Einstein spacetime manifold (GQE). In such a spacetime, we obtain an EoS with a constant scalar curvature R in terms of the magnetic field strength H and magnetic permeability μ. Next, we achieve some cauterization of the magneto-fluid spacetime in terms of Ricci–Yamabe solitons with a time-like torse-forming vector field ξ and a φ(Ric) vector field. We establish the existence of a black hole in the relativistic magneto-fluid spacetime by demonstrating that it admits a shrinking Ricci–Yamabe soliton and satisfies the time-like energy convergence criteria. In addition, we examine the magneto-fluid spacetime with a gradient Ricci–Yamabe soliton and deduce some conditions for an equation of state (EoS) ω=−15 with a Killing vector field. Furthermore, we demonstrate that the EoS ω=−15 of the magneto-fluid spacetime under some constraints represents a star model and a static, spherically symmetric perfect fluid spacetime. Finally, we prove that a gradient Ricci–Yamabe soliton with the conditions μ=0 or H=2; μ≠0, H>2 and obeying the equation of state ω=−15 is conceded in a magneto-fluid spacetime, and a naked singularity with a Cauchy horizon subsequently emerges, respectively.

Almost co-Kähler manifolds and [formula omitted]-quasi-Einstein solitons

The present paper aims to investigate (m,ρ)-quasi-Einstein metrics on almost co-Kähler manifolds M. It is proven that if a (κ,μ)-almost co-Kähler manifold with κ<0 is (m,ρ)-quasi-Einstein manifold, then M represents a N(κ)-almost co-Kähler manifold and the manifold is locally isomorphic to a solvable non-nilpotent Lie group. Next, we study the three dimensional case and get the above mentioned result along with the manifold M3 becoming an η-Einstein manifold. We also show that there does not exist (m,ρ)-quasi-Einstein structure on a compact (κ,μ)-almost co-Kähler manifold of dimension greater than three with κ<0. Further, we prove that an almost co-Kähler manifold satisfying η-Einstein condition with constant coefficients reduces to a K-almost co-Kähler manifold, provided ma1≠(2n−1)b1 and m≠1. We also characterize perfect fluid spacetime whose Lorentzian metric is equipped with (m,ρ)-quasi Einstein solitons and acquired that the perfect fluid spacetime has vanishing vorticity, or it represents dark energy era under certain restriction on the potential function. Finally, we construct an example of an almost co-Kähler manifold with (m,ρ)-quasi-Einstein solitons.

An analysis of symmetry groups of generalized $m$-quasi-Einstein manifolds

In this paper, emphasis is placed on how the behavior of the solutions of a system of PDEs is affected by the geometry of generalized $m$-quasi-Einstein manifold, and vice versa. Considering an $n$-dimensional generalized $m$-quasi-Einstein manifold which

Characterization of sequential warped product gradient Ricci-Bourguignon soliton

In this paper, westudy characterization of sequential warped product gradient Ricci-Bourguignon soliton. We derive applications of some vector fields like torse-forming vector field, torqued vector field, conformal vector field on Ricci-Bourguignon soliton. We show that for torse-forming vector field, a Ricci- Bourguignon soliton becomes an almost quasi-Einstein manifold. Next, we obtain the inheritance properties of the Einstein-like sequential warped product gradient Ricci-Bourguignon almost soliton of class type P,A,B. We prove that, when the manifold is complete, the potential function depends only on M1 and M3 must be an Einstein manifold. We also present for a gradient Ricci-Bourguignon soliton sequential warped product, the warping functions are constants under some certain conditions.

Quasi-Einstein manifolds admitting conformal vector fields

Quasi-Einstein manifolds admitting conformal vector fields

Static near-horizon geometries and rigidity of quasi-Einstein manifolds

Static vacuum near horizon geometries are solutions $(M,g,X)$ of a certain quasi-Einstein equation on a closed manifold $M$, where $g$ is a Riemannian metric and $X$ is a closed 1-form. It is known that when the cosmological constant vanishes, there is rigidity: $X$ vanishes and consequently $g$ is Ricci flat. We study this form of rigidity for all signs of the cosmological constant. It has been asserted that this rigidity also holds when the cosmological constant is negative, but we exhibit a counter-example. We show that for negative cosmological constant if $X$ does not vanish identically, it must be incompressible, have constant norm, and be nontrivial in cohomology, and $(M,g)$ must have constant scalar curvature and zero Euler characteristic. If the cosmological constant is positive, $X$ must be exact (and vanishing if $\dim M=2$). Our results apply more generally to a broad class of quasi-Einstein equations on closed manifolds. We extend some known results for quasi-Einstein metrics with exact 1-form $X$ to the closed $X$ case. We consider near horizon geometries for which the vacuum condition is relaxed somewhat to allow for the presence of a limited class of matter fields. An appendix contains a generalization of a result of Lucietti on the Yamabe type of quasi-Einstein compact metrics (with arbitrary $X$).