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Journal ArticleDOI

∗-Ricci Soliton within the frame-work of Sasakian and (κ,μ)-contact manifold

24 May 2018-International Journal of Geometric Methods in Modern Physics (World Scientific Publishing Company)-Vol. 15, Iss: 07, pp 1850120
TL;DR: In this article, it was shown that if a complete Sasakian metric is an almost gradient ∗-Ricci soliton, then it is either positive or null-Sakian.
Abstract: We prove that if a Sasakian metric is a ∗-Ricci Soliton, then it is either positive Sasakian, or null-Sasakian. Next, we prove that if a complete Sasakian metric is an almost gradient ∗-Ricci Solit...
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TL;DR: In this paper, the authors considered the case of *-Ricci soliton in the framework of a Kenmotsu manifold and proved that soliton constant λ is zero.
Abstract: Abstract In this paper, we consider *-Ricci soliton in the frame-work of Kenmotsu manifolds. First, we prove that if (M, g) is a Kenmotsu manifold and g is a *-Ricci soliton, then soliton constant λ is zero. For 3-dimensional case, if M admits a *-Ricci soliton, then we show that M is of constant sectional curvature –1. Next, we show that if M admits a *-Ricci soliton whose potential vector field is collinear with the characteristic vector field ξ, then M is Einstein and soliton vector field is equal to ξ. Finally, we prove that if g is a gradient almost *-Ricci soliton, then either M is Einstein or the potential vector field is collinear with the characteristic vector field on an open set of M. We verify our result by constructing examples for both *-Ricci soliton and gradient almost *-Ricci soliton.

29 citations

Journal ArticleDOI
TL;DR: In this paper, the authors studied ∗-η-Ricci soliton on Sasakian manifolds and obtained some significant curvature properties on the manifold admitting the soliton.
Abstract: In this paper we study ∗-η-Ricci soliton on Sasakian manifolds. Here, we have discussed some curvature properties on Sasakian manifold admitting ∗-η-Ricci soliton. We have obtained some significant...

25 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that if the metric g of M is a *-Ricci soliton, then either M is locally isometric to the product ℍn+1(−4)×ℝn or the potential vector field is strict infinitesimal contact transformation.
Abstract: Abstract Let (M, g) be a non-Kenmotsu (κ, μ)′-almost Kenmotsu manifold of dimension 2n + 1. In this paper, we prove that if the metric g of M is a *-Ricci soliton, then either M is locally isometric to the product ℍn+1(−4)×ℝn or the potential vector field is strict infinitesimal contact transformation. Moreover, two concrete examples of (κ, μ)′-almost Kenmotsu 3-manifolds admitting a Killing vector field and strict infinitesimal contact transformation are given.

24 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that the Ricci soliton of a 3-dimensional Kenmotsu manifold is locally isometric to the hyperbolic 3-space and the potential vector field coincides with the Reeb vector field.
Abstract: Let $(M,\phi,\xi,\eta,g)$ be a three-dimensional Kenmotsu manifold. In this paper, we prove that the triple $(g,V,\lambda)$ on $M$ is a $*$-Ricci soliton if and only if $M$ is locally isometric to the hyperbolic 3-space $\mathbf{H}^3(-1)$ and $\lambda=0$. Moreover, if $g$ is a gradient $*$-Ricci soliton, then the potential vector field coincides with the Reeb vector field. We also show that the metric of a coKahler 3-manifold is a $*$-Ricci soliton if and only if it is a Ricci soliton.

20 citations

References
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Journal ArticleDOI
TL;DR: In this article, a study of contact metric manifolds for which the characteristic vector field of the contact structure satisfies a nullity type condition, condition (*) below, is presented.
Abstract: This paper presents a study of contact metric manifolds for which the characteristic vector field of the contact structure satisfies a nullity type condition, condition (*) below. There are a number of reasons for studying this condition and results concerning it given in the paper: There exist examples in all dimensions; the condition is invariant underD-homothetic deformations; in dimensions>5 the condition determines the curvature completely; and in dimension 3 a complete, classification is given, in particular these include the 3-dimensional unimodular Lie groups with a left invariant metric.

325 citations

Journal ArticleDOI
TL;DR: The Ricci almost soliton as discussed by the authors is a natural extension of the concept of gradient Ricci soliton, and it has been shown to be a generalization of the Ricci Almost Soliton.
Abstract: We introduce a natural extension of the concept of gradient Ricci soliton: the Ricci almost soliton. We provide existence and rigidity results, we deduce a-priori curvature estimates and isolation phenomena, and we investigate some topological properties. A number of differential identities involving the relevant geometric quantities are derived. Some basic tools from the weighted manifold theory such as general weighted volume comparisons and maximum principles at infinity for diffusion operators are discussed. Mathematics Subject Classification (2010): 53C21.

191 citations

Journal ArticleDOI
TL;DR: In this article, the authors studied the problem of finding a (φ, ξ, η, g)-connection in a space with a normal contact structure and proved that the space with such a contact structure is an Einstein space.
Abstract: Introduction. Recently S. Sasaki [3]° defined the notion of (φ, ξ, η, g) structure of a differentiable manifold. Further, S. Sasaki and Y. Hatakeyama [ 4 ] [ 5 ] showed that the structure is closely related to contact structure. By means of this notion, it is shown that a space with a contact structure can be dealt with as we deal with an almost complex space. So, by similar manner, some problems discussed in the latter space may be considered in the former. On the other hand, S. Tachibana [6] [7] proved many interesting theorems in an almost complex space. In this paper, the present author tries to study, in the space with a certain contact structure, the problem corresponding to S. Tachibana's results. We shall devote § 1 to preliminaries and in this section introduce a normal contact structure. In §2, we ennumerate identities which will be useful in the later sections. We shall prove in § 3 that a space with a normal contact structure satisfying VkRjt = 0 be necessarily an Einstein one and that a symmetric space with a normal contact structure reduces to the space of constant curvature respectively. The differential form R is dealt with in § 4, and in this section, we shall show a necessary and sufficient condition that the space be an Einstein space by means of the form R. Finally in § 5, we introduce a certain type of (φ, η,

178 citations