# 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

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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

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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.

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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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08 Jan 2002

TL;DR: In this article, the authors describe a complex geometry model of Symplectic Manifolds with principal S1-bundles and Tangent Sphere Bundles, as well as a negative Xi-sectional Curvature.

Abstract: Preface * 1. Symplectic Manifolds * 2. Principal S1-bundles * 3. Contact Manifolds * 4. Associated Metrics * 5. Integral Submanifolds and Contact Transformations * 6. Sasakian and Cosymplectic Manifolds * 7. Curvature of Contact Metric Manifolds * 8. Submanifolds of Kahler and Sasakian Manifolds * 9. Tangent Bundles and Tangent Sphere Bundles * 10. Curvature Functionals and Spaces of Associated Metrics * 11. Negative Xi-sectional Curvature * 12. Complex Contact Manifolds * 13. Additional Topics in Complex Geometry * 14. 3-Sasakian Manifolds * Bibliography * Subject Index * Author Index

1,822 citations

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331 citations