# *-Ricci soliton on (κ, μ)′-almost Kenmotsu manifolds

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01 Jan 1970

294 citations

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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.

3 citations

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TL;DR: In this article, it was shown that if a non-constant solution of the critical point equation of a connected non-compact manifold admits a nonconstant function, then the manifold is locally isometric to the Ricci flat manifold and the function is harmonic.

Abstract: In the present paper, we characterize $$(k,\mu )'$$-almost Kenmotsu manifolds admitting $$*$$-critical point equation. It is shown that if $$(g, \lambda )$$ is a non-constant solution of the $$*$$-critical point equation of a connected non-compact $$(k,\mu )'$$-almost Kenmotsu manifold, then (1) the manifold M is locally isometric to $$\mathbb {H}^{n+1}(-4)$$$$\times $$$$\mathbb {R}^n$$, (2) the manifold M is $$*$$-Ricci flat and (3) the function $$\lambda $$ is harmonic. Finally an illustrative example is presented.

3 citations

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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.

3 citations

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TL;DR: In this article, it was shown that if the metric g represents a Yamabe soliton, then it is locally isometric to the product space and the contact transformation is a strict infinitesimal contact transformation.

Abstract: Let $$(M^{2n+1},\phi ,\xi ,\eta ,g)$$
be a non-Kenmotsu almost Kenmotsu $$(k,\mu )'$$
-manifold. If the metric g represents a Yamabe soliton, then either $$M^{2n+1}$$
is locally isometric to the product space $$\mathbb {H}^{n+1}(-4)\times \mathbb {R}^n$$
or $$\eta $$
is a strict infinitesimal contact transformation. The later case can not occur if a Yamabe soliton is replaced by a gradient Yamabe soliton. Some corollaries of this theorem are given and an example illustrating this theorem is constructed.

2 citations

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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

1,643 citations

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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

539 citations

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01 Jan 1970

294 citations

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