Certain types of metrics on almost coKähler manifolds
15 Apr 2021-pp 1-17
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.
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TL;DR: In this article , it was shown that if the metric of an almost co-K?hler manifold is a Riemann soliton with the soliton vector field, then the manifold is flat.
Abstract: The aim of the present paper is to characterize almost co-K?hler manifolds
whose metrics are the Riemann solitons. At first we provide a necessary and
sufficient condition for the metric of a 3-dimensional manifold to be
Riemann soliton. Next it is proved that if the metric of an almost
co-K?hler manifold is a Riemann soliton with the soliton vector field ?,
then the manifold is flat. It is also shown that if the metric of a (?,
?)-almost co-K?hler manifold with ? < 0 is a Riemann soliton, then the
soliton is expanding and ?, ?, ? satisfies a relation. We also prove that
there does not exist gradient almost Riemann solitons on (?, ?)-almost
co-K?hler manifolds with ? < 0. Finally, the existence of a Riemann soliton
on a three dimensional almost co-K?hler manifold is ensured by a proper
example.
2 citations
TL;DR: In this article, the authors studied generalized Ricci soliton in the framework of paracontact metric manifolds and proved that the scalar curvature r is constant and the squared norm of Ricci operator is constant.
Abstract: In the present paper, we study generalized Ricci soliton in the framework of paracontact metric manifolds. First, we prove that if the metric of a paracontact metric manifold M with $$Q\varphi =\varphi Q$$
is a generalized Ricci soliton (g, X) and if $$X
e 0$$
is pointwise collinear to $$\xi$$
, then M is K-paracontact and $$\eta$$
-Einstein. Next, we consider closed generalized Ricci soliton on K-paracontact manifold and prove that it is Einstein provided $$\beta (\lambda +2n\alpha )
e 1$$
. Next, we study K-paracontact metric as gradient generalized almost Ricci soliton and in this case we prove that (i) the scalar curvature r is constant and is equal to $$-2n(2n+1)$$
; (ii) the squared norm of Ricci operator is constant and is equal to $$4n^2(2n+1)$$
, provided $$\alpha \beta
e -1$$
.
1 citations
TL;DR: In this article, the authors considered CPE on almost f-cosymplectic manifolds and proved that the CPE conjecture is true for almost f cosymetric manifolds.
TL;DR: In this paper , a special class of contact pseudo-Riemannian manifold, called almost * {*} -η-Ricci solitons, is studied and shown to be an Einstein manifold if the potential vector field V is an infinitesimal contact transformation.
Abstract: Abstract In this paper, we aim to study a special type of metric called almost * {*} -η-Ricci soliton on the special class of contact pseudo-Riemannian manifold. First, we prove that a Kenmotsu pseudo-Riemannian metric as an * {*} -η-Ricci soliton is Einstein if either it is η-Einstein or the potential vector field V is an infinitesimal contact transformation. Further, we prove that if a Kenmotsu pseudo-Riemannian manifold admits an almost * {*} -η-Ricci soliton with a Reeb vector field leaving the scalar curvature invariant, then it is an Einstein manifold. Finally, we present an example of * {*} -η-Ricci solitons which illustrate our results.
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TL;DR: In this article, a metric quasi-Einstein metric is defined, where the m -Bakry-Emery Ricci tensor is a constant multiple of the metric tensor.
Abstract: We call a metric quasi-Einstein if the m -Bakry–Emery Ricci tensor is a constant multiple of the metric tensor. This is a generalization of Einstein metrics, it contains gradient Ricci solitons and is also closely related to the construction of the warped product Einstein metrics. We study properties of quasi-Einstein metrics and prove several rigidity results. We also give a splitting theorem for some Kahler quasi-Einstein metrics.
209 citations
170 citations
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TL;DR: In this paper, a metric quasi-Einstein metric is defined, where the Ricci tensor is a constant multiple of the metric tensor, which is a generalization of the Einstein metric.
Abstract: We call a metric quasi-Einstein if the $m$-Bakry-Emery Ricci tensor is a constant multiple of the metric tensor. This is a generalization of Einstein metrics, which contains gradient Ricci solitons and is also closely related to the construction of the warped product Einstein metrics. We study properties of quasi-Einstein metrics and prove several rigidity results. We also give a splitting theorem for some K\"ahler quasi-Einstein metrics.
138 citations
129 citations