η-Ricci solitons in trans-Sasakian manifolds
01 Aug 2017-Vol. 66, Iss: 2, pp 218-224
TL;DR: In this article, the authors studied the -Ricci solitons in 3-dimensional trans-Sasakian manifolds and showed that they can be computed in 3D.
Abstract: The aim of this paper is to study the -Ricci solitons in 3-dimensionaltrans-Sasakian manifolds
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Journal Article•
TL;DR: In this article, it was shown that if a real hypersurface with constant mean curvature of a complex space form satisfying ▽S = 0 and Sξ = ξ for a smooth function, then the structure vector field ξ is principal.
Abstract: We prove that if a real hypersurface with constant mean curvature of a complex space form satisfying ▽S = 0 and Sξ = ξ for a smooth function , then the structure vector field ξ is principal, where S denotes the Ricci tensor of the hypersurface.
9 citations
Posted Content•
TL;DR: Ricci-like solitons with arbitrary potential are introduced and studied on Sasaki-like almost contact B-metric manifolds in this article, where it is proved that the Ricci tensor of such a soliton is the vertical component of both B-means multiplied by a constant, and explicit examples of Lie groups as manifolds of dimensions 3 and 5 equipped with the structures studied are provided.
Abstract: Ricci-like solitons with arbitrary potential are introduced and studied on Sasaki-like almost contact B-metric manifolds. It is proved that the Ricci tensor of such a soliton is the vertical component of both B-metrics multiplied by a constant. It is established that gradient almost Ricci-like solitons have constant soliton coefficients. Explicit examples of Lie groups as manifolds of dimensions 3 and 5 equipped with the structures studied are provided.
6 citations
Additional excerpts
..., [2], [3], [9], [10], [11], [17], [21], [31], [34])....
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18 May 2018
TL;DR: In this article, it was shown that a symmetric second order covariant tensor in a δ-Lorentzian Trans Sasakian manifold is a constant multiple of metric ten sor.
Abstract: The object of the present research is to study the δ-Lorentzian Trans Sasakian manifolds addmitting the η-Einstein Solitons and gradient Ein stein soliton. It is shown that a symmetric second order covariant tensor in a δ-Lorentzian Trans Sasakian manifold is a constant multiple of metric ten sor. Also an example of η-Einstein soliton in 3-diemsional δ-Lorentzian Trans Sasakian manifold is provided in the region where δ-Lorentzian Trans Sasakian manifold expanding.
3 citations
Cites methods from "η-Ricci solitons in trans-Sasakian ..."
...[30] study the η-Ricci solitns in Trans-Sasakian manifold....
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26 Dec 2018
TL;DR: In this article, the generalized η-Riccisoliton on trans-Sakian manifolds is studied and it is shown that a trans-sakian manifold which also satisfies the generalized gradient η -Ricci soliton equation satisfiessome conditions, is necessarily Einstein manifold.
Abstract: The object of the present research is to study generalized η-Riccisoliton on trans-Sasakian manifolds. It shows that a trans-Sasakian manifoldwhich also satisfies the generalized gradient η-Ricci soliton equation satisfiessome conditions, is necessarily Einstein manifold.
2 citations
03 Dec 2018
TL;DR: In this article, the existence of Ricci solitons on the (LCS)n-manifolds satisfying certain curvature conditions was studied and it was shown that the Ricci-soliton is a quasi-Einstein soliton.
Abstract: In this paper, we consider an η-Ricci soliton on the (LCS)n-manifolds (M,φ, ξ, η, g) satisfying certain curvature conditions likes: R(ξ,X) · S = 0 and W2(ξ,X) · S = 0. We show that on the (LCS)n-manifolds (M,φ, ξ, η, g), the existence of η-Ricci soliton implies that (M, g) is a quasi-Einstein. Further, we discuss the existence of Ricci solitons with the potential vector field ξ. In the end, we construct the non-trivial examples of η-Ricci solitons on the (LCS)n-manifolds.
2 citations
Cites background from "η-Ricci solitons in trans-Sasakian ..."
...The geometrical properties of the Ricci solitons have been studied in ([1]-[5], [7]-[13], [17]-[21], [26], [31], [37], [38], [43]) and by others....
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References
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3,014 citations
"η-Ricci solitons in trans-Sasakian ..." refers background in this paper
...We know [15, 16] that any compact steady or expanding Ricci soliton is Einstein....
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Book•
01 Jan 1976
TL;DR: In this paper, the tangent sphere bundle is shown to be a contact manifold, and the contact condition is interpreted in terms of contact condition and k-contact and sasakian structures.
Abstract: Contact manifolds.- Almost contact manifolds.- Geometric interpretation of the contact condition.- K-contact and sasakian structures.- Sasakian space forms.- Non-existence of flat contact metric structures.- The tangent sphere bundle.
1,259 citations
"η-Ricci solitons in trans-Sasakian ..." refers background in this paper
...In particular, if the metric g is positive definite, then an (ε)-almost contact metric manifold is the usual almost contact metric manifold [5]....
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TL;DR: In this paper, it was shown that sixteen classes of almost Hermitian manifolds can be found in the Euclidean space, and that they are Hermitians in a natural way.
Abstract: It is shown that in a natural way there are precisely sixteen classes of almost Hermitian manifolds.
823 citations
"η-Ricci solitons in trans-Sasakian ..." refers background in this paper
...In [12], Gray-Harvella classification of almost Hermitian manifolds appears as a class W4 of Hermitian manifolds which are closely related to locally conformal Kähler manifolds....
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TL;DR: In this article, Tanno has classified connected almost contact Riemannian manifolds whose automorphism groups have themaximum dimension into three classes: (1) homogeneous normal contact manifolds with constant 0-holomorphic sec-tional curvature if the sectional curvature for 2-planes which contain
Abstract: Recently S. Tanno has classified connected almostcontact Riemannian manifolds whose automorphism groups have themaximum dimension [9]. In his classification table the almost contactRiemannian manifolds are divided into three classes: (1) homogeneousnormal contact Riemannian manifolds with constant 0-holomorphic sec-tional curvature if the sectional curvature for 2-planes which contain
614 citations
TL;DR: In this article, it was shown that there are no compact three-dimensional Ricci solitons other than spaces of constant curvature, which generalizes a result obtained for surfaces by Hamilton.
Abstract: In this short article we show that there are no compact three-dimensional Ricci solitons other than spaces of constant curvature. This generalizes a result obtained for surfaces by Hamilton [4]. The proof involves a careful analysis of the ODE for the curvature which is associated to the Ricci flow.
343 citations
"η-Ricci solitons in trans-Sasakian ..." refers background in this paper
...We know [15, 16] that any compact steady or expanding Ricci soliton is Einstein....
[...]