Non-existence of $$*$$ ∗ -Ricci solitons on $$(\kappa ,\mu )$$ ( κ , μ ) -almost cosymplectic manifolds
TL;DR: In this paper, the authors prove a non-existence result for Ricci solitons on non-cosymplectic manifolds, and prove the same result for almost cosympelous manifolds.
Abstract: In this short note, we prove a non-existence result for $$*$$
-Ricci solitons on non-cosymplectic $$(\kappa ,\mu )$$
-almost cosymplectic manifolds.
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TL;DR: In this article, the Ricci soliton is shown to be Ricci flat and locally isometric with respect to the Euclidean distance of the potential vector field when the manifold satisfies gradient almost.
Abstract: In the present paper, we initiate the study of $$*$$
-
$$\eta $$
-Ricci soliton within the framework of Kenmotsu manifolds as a characterization of Einstein metrics. Here we display that a Kenmotsu metric as a $$*$$
-
$$\eta $$
-Ricci soliton is Einstein metric if the soliton vector field is contact. Further, we have developed the characterization of the Kenmotsu manifold or the nature of the potential vector field when the manifold satisfies gradient almost $$*$$
-
$$\eta $$
-Ricci soliton. Next, we deliberate $$*$$
-
$$\eta $$
-Ricci soliton admitting $$(\kappa ,\mu )^\prime $$
-almost Kenmotsu manifold and proved that the manifold is Ricci flat and is locally isometric to $${\mathbb {H}}^{n+1}(-4)\times {\mathbb {R}}^n$$
. Finally we present some examples to decorate the existence of $$*$$
-
$$\eta $$
-Ricci soliton, gradient almost $$*$$
-
$$\eta $$
-Ricci soliton on Kenmotsu manifold.
8 citations
15 Apr 2021
TL;DR: In this article, it was shown that Bach flat almost coKahler manifold admits Ricci solitons, satisfying the critical point equation (CPE) or Bach flat.
Abstract: In this paper, we study an almost coKahler manifold admitting certain metrics such as $$*$$
-Ricci solitons, satisfying the critical point equation (CPE) or Bach flat. First, we consider a coKahler 3-manifold (M, g) admitting a $$*$$
-Ricci soliton (g, X) and we show in this case that either M is locally flat or X is an infinitesimal contact transformation. Next, we study non-coKahler $$(\kappa ,\mu )$$
-almost coKahler metrics as CPE metrics and prove that such a g cannot be a solution of CPE with non-trivial function f. Finally, we prove that a $$(\kappa , \mu )$$
-almost coKahler manifold (M, g) is coKahler if either M admits a divergence free Cotton tensor or the metric g is Bach flat. In contrast to this, we show by a suitable example that there are Bach flat almost coKahler manifolds which are non-coKahler.
6 citations
TL;DR: In this paper , the authors studied α-cosymplectic manifold and showed that the Ricci tensor tensor is a semisymmetric manifold, which is an extension of the RICCI tensor.
Abstract: In this paper, we study α-cosymplectic manifold
admitting
-Ricci tensor. First, it is shown that a
-Ricci semisymmetric manifold
is
-Ricci flat and a
-conformally flat manifold
is an
-Einstein manifold. Furthermore, the
-Weyl curvature tensor
on
has been considered. Particularly, we show that a manifold
with vanishing
-Weyl curvature tensor is a weak
-Einstein and a manifold
fulfilling the condition
is
-Einstein manifold. Finally, we give a characterization for α-cosymplectic manifold
admitting
-Ricci soliton given as to be nearly quasi-Einstein. Also, some consequences for three-dimensional cosymplectic manifolds admitting
-Ricci soliton and almost
-Ricci soliton are drawn.
4 citations
TL;DR: In this article , a geometric classification of Sasakian manifolds that admit an almost ∗-Ricci soliton (RS) structure (g,ω,X) is presented.
Abstract: This article presents some results of a geometric classification of Sasakian manifolds (SM) that admit an almost ∗-Ricci soliton (RS) structure (g,ω,X). First, we show that a complete SM equipped with an almost ∗-RS with ω≠ const is a unit sphere. Then we prove that if an SM has an almost ∗-RS structure, whose potential vector is a Jacobi vector field on the integral curves of the characteristic vector field, then the manifold is a null or positive SM. Finally, we characterize those SM represented as almost ∗-RS, which are ∗-RS, ∗-Einstein or ∗-Ricci flat.
1 citations
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14 citations
TL;DR: In this paper, the Ricci curvature of the Reeb vector field is invariant to the Riemannian curvature tensor in a 3D almost co-Kahler manifold.
Abstract: Let M3 be a three-dimensional almost coKahler manifold such that the Ricci curvature of the Reeb vector field is invariant along the Reeb vector field. In this paper, we obtain some classification results of M3 for which the Ricci tensor is η-parallel or the Riemannian curvature tensor is harmonic.
13 citations
Posted Content•
26 Jan 2012
TL;DR: In this article, the Riemann curvature tensor tensor of (kappa,\mu,
u)-spaces when they have almost cosymplectic and almost Kenmotsu structures was studied.
Abstract: We study the Riemann curvature tensor of (\kappa,\mu,
u)-spaces when they have almost cosymplectic and almost Kenmotsu structures, giving its writing explicitly This leads to the definition and study of a natural generalisation of the contact metric (\kappa,\mu,
u)-spaces We present examples or obstruction results of these spaces in all possible cases
4 citations