Ricci Yamabe Soliton on Genearlized Sasakian Space Form

Universe

In this article, a Ricci soliton and *-conformal Ricci soliton are examined in the framework of trans-Sasakian three-manifold. In the beginning of the paper, it is shown that a three-dimensional trans-Sasakian manifold of type (α,β) admits a Ricci soliton where the covariant derivative of potential vector field V in the direction of unit vector field ξ is orthogonal to ξ. It is also demonstrated that if the structure functions meet α2=β2, then the covariant derivative of V in the direction of ξ is a constant multiple of ξ. Furthermore, the nature of scalar curvature is evolved when the manifold of type (α,β) satisfies *-conformal Ricci soliton, provided α≠0. Finally, an example is presented to verify the findings.

2018

The aim of the present paper is to study Ricci solitons in Kenmotsu manifolds under $D$-homothetic deformation. We analyzed behaviour of Ricci solitons when potential vector field is orthogonal to Reeb vector field and pointwise collinear with Reeb vector field. Further we prove Ricci solitons in $D$-homothetically transformed Kenmotsu manifolds are shrinking.

Results in Mathematics

Ricci-like solitons on Sasaki-like almost contact B-metric manifolds are the object of study. Cases, where the potential of the Ricci-like soliton is the Reeb vector field or pointwise collinear to it, are considered. In the former case, the properties for a parallel or recurrent Riccitensor are studied. In the latter case, it is shown that the potential of the considered Ricci-like soliton has a constant length and the manifold is η-Einstein. Other curvature conditions are also found, which imply that the main metric is Einstein. After that, some results are obtained for a parallel symmetric second-order covariant tensor on the manifolds under study. Finally, an explicit example of dimension 5 is given and some of the results are illustrated.

Journal of Nonlinear Mathematical Physics

We study conformal $$\eta$$ η -Einstein solitons on the framework of trans-Sasakian manifold in dimension three. Existence of conformal $$\eta$$ η -Einstein solitons on trans-Sasakian manifold is discussed. Then we find some results on trans-Sasakian manifold which are conformal $$\eta$$ η -Einstein solitons where the Ricci tensor is cyclic parallel and Codazzi type. We also consider some curvature conditions with addition to conformal $$\eta$$ η -Einstein solitons on trans-Sasakian manifold. We also use torse-forming vector fields in addition to conformal $$\eta$$ η -Einstein solitons on trans-Sasakian manifold. Finally, an example of conformal $$\eta$$ η -Einstein solitons on trans-Sasakian manifold is constructed.

Annales Mathematicae Silesianae, 2018

In this paper we study the nature of Ricci solitons in D-homo-thetically deformed Kenmotsu manifolds. We prove that η -Einstein Kenmotsu metric as a Ricci soliton remains η -Einstein under D-homothetic deformation and the scalar curvature remains constant.

2019

In this paper, we study Ricci-semisymmetric and Ricci pseudo-symmetric generalized (k,µ)-space forms along with characterization of generalized (k,µ)-space forms satisfying the curvature conditions Q(g,S) = 0 and Q(S,R) = 0. Further, we study Ricci solitons in generalized (k,µ)-space forms and obtained some interesting results.

2021

The object of this paper is to study Ricci solitons and gradient Ricci solitons in Da-homothetically deformed K-contact and N(k)-contact metric manifolds. M.S.C. 2010: 53C20, 53C44.

Axioms

In this paper, we study the properties of ϵ-Kenmotsu manifolds if its metrics are *η-Ricci-Yamabe solitons. It is proven that an ϵ-Kenmotsu manifold endowed with a *η-Ricci-Yamabe soliton is η-Einstein. The necessary conditions for an ϵ-Kenmotsu manifold, whose metric is a *η-Ricci-Yamabe soliton, to be an Einstein manifold are derived. Finally, we model an indefinite Kenmotsu manifold example of dimension 5 to examine the existence *η-Ricci-Yamabe solitons.

Mathematics

In the present paper, we characterize m-dimensional ζ-conformally flat LP-Kenmotsu manifolds (briefly, (LPK)m) equipped with the Ricci–Yamabe solitons (RYS) and gradient Ricci–Yamabe solitons (GRYS). It is proven that the scalar curvature r of an (LPK)m admitting an RYS satisfies the Poisson equation Δr=4(m−1)δ{β(m−1)+ρ}+2(m−3)r−4m(m−1)(m−2), where ρ,δ(≠0)∈R. In this sequel, the condition for which the scalar curvature of an (LPK)m admitting an RYS holds the Laplace equation is established. We also give an affirmative answer for the existence of a GRYS on an (LPK)m. Finally, a non-trivial example of an LP-Kenmotsu manifold (LPK) of dimension four is constructed to verify some of our results.

arXiv (Cornell University), 2024

This article aims to investigate the characteristics of (α, β)-Ricci-Yamabe Soliton (briefly: (α, β) − (RY S) n) and its spacetime. The inclusion of killing vector field and the Lorentzian metrics make the Ricci-Yamabe soliton richer and interesting. We study the cosmological and dust fluid model on (RY S) 4 equipped with Lorentzian para Sasakian (LP S) 4 spacetime. The cases of η-parallel Ricci tensor and the Poisson structure have been studied on (RY S) n equipped with (LP S) n manifold. Gradient (RY S) n equipped with (LP S) n manifold also reveal. Finally, we establish an example of four-dimensional LP-Sasakian manifold (LP S) 4 that satisfy (α, β) − (RY S) 4 and some results.