# Critical point equation on a class of almost Kenmotsu manifolds

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.

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28 Apr 2020

TL;DR: In this paper, it was shown that if a $(2n + 1)$-dimensiinal $(k,\mu)'$-almost Kenmotsu manifold admits the Ricci soliton, then the manifold is locally isometric to a Ricci flat manifold.

Abstract: The goal of this paper is to characterize a class of almost Kenmotsu manifolds admitting $\ast$-conformal Ricci soliton. It is shown that if a $(2n + 1)$-dimensiinal $(k,\mu)'$-almost Kenmotsu manifold $M$ admits $\ast$-conformal Ricci soliton, then the manifold $M$ is $\ast$-Ricci flat and locally isometric to $\mathbb{H}^{n+1}(-4) \times \mathbb{R}^n$. The result is also verified by an example.

3 citations

### Cites background from "Critical point equation on a class ..."

...l notions related to the ∗-Ricci tensor. In 2016, the notion of ∗-Ricci soliton ([13]) was introduced. Later in 2019, the notion of ∗-critical point equation [6] was introduced and further studied in [7]). In this paper, we study the notion of ∗-conformal Ricci soliton deﬁned as Deﬁnition 1.1. A Riemannian manifold (M,g) of dimension (2n + 1) ≥ 3 is said to admit ∗-conformal Ricci soliton (g,V,λ) if ...

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TL;DR: In this paper, the authors introduced the notion of $$*$$� -gradient $$\rho$$¯¯¯¯ -Einstein soliton on a class of almost Kenmotsu manifolds.

Abstract: In the present paper, we introduce the notion of $$*$$
-gradient $$\rho$$
-Einstein soliton on a class of almost Kenmotsu manifolds. It is shown that if a $$(2n+1)$$
-dimensional $$(k,\mu )'$$
-almost Kenmotsu manifold M admits $$*$$
-gradient $$\rho$$
-Einstein soliton with Einstein potential f, 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 Einstein potential f is harmonic or satisfies a physical Poisson’s equation. Finally, an illustrative example is presented.

3 citations

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TL;DR: In this article, the authors characterized a class of contact metric manifolds admitting a conformal Ricci soliton and showed that these manifolds are locally isometric to the Riemannian of a flat manifold of constant curvature.

Abstract: The aim of this paper is characterize a class of contact metric manifolds admitting $\ast$-conformal Ricci soliton. It is shown that if a $(2n + 1)$-dimensional $N(k)$-contact metric manifold $M$ admits $\ast$-conformal Ricci soliton or $\ast$-conformal gradient Ricci soliton, then the manifold M is $\ast$-Ricci at and locally isometric to the Riemannian of a flat $(n + 1)$-dimensional manifold and an $n$-dimensional manifold of constant curvature 4 for $n > 1$ and flat for $n = 1$. Further, for the first case, the soliton vector field is conformal and for the $\ast$-gradient case, the potential function $f$ is either harmonic or satisfy a Poisson equation. Finally, an example is presented to support the results.

1 citations

### Cites background from "Critical point equation on a class ..."

...In 2019, the notion of ∗-critical point equation [8] was introduced and further studied by the authors in [9]....

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##### References

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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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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

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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

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TL;DR: In this paper, the authors consider locally symmetric almost Kenmotsu manifold and show that the manifold is locally isometric to the Riemannian product of an n+1-dimensional manifold of constant curvature.

Abstract: We consider locally symmetric almost Kenmotsu manifolds showing that such a manifold is a Kenmotsu manifold if and only if the Lie derivative of the structure, with respect to the Reeb vector field $\xi$, vanishes. Furthermore, assuming that for a $(2n+1)$-dimensional locally symmetric almost Kenmotsu manifold such Lie derivative does not vanish and the curvature satisfies $R_{XY}\xi =0$ for any $X, Y$ orthogonal to $\xi$, we prove that the manifold is locally isometric to the Riemannian product of an $(n+1)$-dimensional manifold of constant curvature $-4$ and a flat $n$-dimensional manifold. We give an example of such a manifold.

122 citations

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TL;DR: In this paper, the authors characterize almost contact metric manifolds which are CR-integrable almost Kenmotsu, through the existence of a suitable linear connection, and give examples and completely describe the three dimensional case.

Abstract: We characterize almost contact metric manifolds which are CR-integrable almost Kenmotsu, through the existence of a suitable linear connection. We classify almost Kenmotsu manifolds satisfying a certain nullity condition, we give examples and completely describe the three dimensional case.

114 citations