Pseudo-symmetric structures on almost Kenmotsu manifolds with nullity distributions
09 Aug 2019-Vol. 23, Iss: 1, pp 13-24
TL;DR: In this paper, the authors characterized Ricci pseudosymmetric and Ricci semisymmetric almost Kenmotsu manifolds with (k, μ)-, (k; μ)′-, and generalized (k and μ)-nullity distributions.
Abstract: The object of the present paper is to characterize Ricci pseudosymmetric and Ricci semisymmetric almost Kenmotsu manifolds with (k; μ)-, (k; μ)′-, and generalized (k; μ)-nullity distributions. We also characterize (k; μ)-almost Kenmotsu manifolds satisfying the condition R ⋅ S = LꜱQ(g; S2).
Citations
More filters
Posted Content•
TL;DR: In this article, it was shown that the Ricci-yamabe soliton is locally isometric to the Riemannian product and the potential vector field is pointwise collinear with the Reeb vector field.
Abstract: The object of the present paper is to characterize two classes of almost Kenmotsu manifolds admitting Ricci-Yamabe soliton. It is shown that a $(k,\mu)'$-almost Kenmotsu manifold admitting a Ricci-Yamabe soliton or gradient Ricci-Yamabe soliton is locally isometric to the Riemannian product $\mathbb{H}^{n+1}(-4) \times \mathbb{R}^n$. For the later case, the potential vector field is pointwise collinear with the Reeb vector field. Also, a $(k,\mu)$-almost Kenmotsu manifold admitting certain Ricci-Yamabe soliton with the curvature property $Q \cdot P = 0$ is locally isometric to the hyperbolic space $\mathbb{H}^{2n+1}(-1)$ and the non-existense of the curvature property $Q \cdot R = 0$ is proved.
5 citations
Cites background from "Pseudo-symmetric structures on almo..."
...For further details on almost Kenmotsu manifolds, we refer the reader to go through the references ([2]-[4], [9])....
[...]
Posted Content•
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 "Pseudo-symmetric structures on almo..."
...10) For further details on (k, μ)-almost Kenmotsu manifolds, we refer the reader to go through the references ([5], [9], [15])....
[...]
Posted Content•
TL;DR: In this article, the authors characterized almost Kenmotsu manifolds admitting holomorphically planar conformal vector (HPCV) fields and showed that the integral manifolds of D are totally umbilical submanifolds of an almost Kaehler manifold.
Abstract: In the present paper, we characterize almost Kenmotsu manifolds admitting holomorphically planar conformal vector (HPCV) fields. We have shown that if an almost Kenmotsu manifold $M^{2n+1}$ admits a non-zero HPCV field $V$ such that $\phi V = 0$, then $M^{2n+1}$ is locally a warped product of an almost Kaehler manifold and an open interval. As a corollary of this we obtain few classifications of an almost Kenmotsu manifold to be a Kenmotsu manifold and also prove that the integral manifolds of D are totally umbilical submanifolds of $M^{2n+1}$. Further, we prove that if an almost Kenmotsu manifold with positive constant $\xi$-sectional curvature admits a non-zero HPCV field $V$, then either $M^{2n+1}$ is locally a warped product of an almost Kaehler manifold and an open interval or isometric to a sphere. Moreover, a $(k,\mu)'$-almost Kenmotsu manifold admitting a HPCV field $V$ such that $\phi V = 0$ is either locally isometric to $\mathbb{H}^{n+1}(-4) \times \mathbb{R}^n$ or $V$ is an eigenvector of $h'$. Finally, an example is presented.
2 citations
TL;DR: In this article, Ricci symmetric almost Kenmotsu manifolds were characterized under several constraints and proved that they are Einstein manifolds, and several corollaries were obtained.
Abstract: In the present paper, we characterize Ricci symmetric almost Kenmotsu manifolds under several constraints and proved that they are Einstein manifolds As a consequence, we obtain several corollaries Finally, an illustrative example is presented to verify our results
DOI•
TL;DR: In this paper , the authors introduced the notion of *-Miao-Tam critical equation on almost contact metric manifolds and studied on almost Kenmotsu manifolds with some nullity condition.
Abstract: Abstract The object of this offering article is to introduce the notion of *- Miao-Tam critical equation on almost contact metric manifolds and it is studied on almost Kenmotsu manifolds with some nullity condition. It is proved that if the metric of a (2n + 1)-dimensional (k, µ) ! -almost Kenmotsu manifold (M, g) satisfies the *-Miao-Tam critical equation, then the manifold (M, g) is *-Ricci flat and locally isometric to a product space. Finally, the result is verified by an example.
References
More filters
Book•
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
Additional excerpts
...A differentiable (2n+1)-dimensional manifold M is said to have a (φ, ξ, η)structure, or an almost contact structure, if it admits a (1, 1)-tensor field φ, a characteristic vector field ξ, and a 1-form η satisfying (see [1], [2]) φ(2) = −I + η ⊗ ξ, η(ξ) = 1, (1....
[...]
Book•
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
Additional excerpts
...A differentiable (2n+1)-dimensional manifold M is said to have a (φ, ξ, η)structure, or an almost contact structure, if it admits a (1, 1)-tensor field φ, a characteristic vector field ξ, and a 1-form η satisfying (see [1], [2]) φ(2) = −I + η ⊗ ξ, η(ξ) = 1, (1....
[...]
...where [φ, φ] is the Nijenhuis tensor of φ (see [1])....
[...]
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
"Pseudo-symmetric structures on almo..." refers background in this paper
...It is well known (see [8]) that a (2n+1)-dimensional Kenmotsu manifold M2n+1 is locally a warped product I ×f N2n, where N2n is a Kähler manifold, I is an open interval with coordinate t, and the warping function f(t) = cet for some positive constant c....
[...]
TL;DR: In this article, a study of contact metric manifolds for which the characteristic vector field of the contact structure satisfies a nullity type condition, condition (*) below, is presented.
Abstract: This paper presents a study of contact metric manifolds for which the characteristic vector field of the contact structure satisfies a nullity type condition, condition (*) below. There are a number of reasons for studying this condition and results concerning it given in the paper: There exist examples in all dimensions; the condition is invariant underD-homothetic deformations; in dimensions>5 the condition determines the curvature completely; and in dimension 3 a complete, classification is given, in particular these include the 3-dimensional unimodular Lie groups with a left invariant metric.
325 citations
Additional excerpts
...[3] introduced the notion of a (k, μ)-nullity distribution on a contact metric manifold (M2n+1, φ, ξ, η, g), which is defined for any p ∈M and k, μ ∈ R by Np(k, μ) = {Z ∈ TpM : R(X,Y )Z = k[g(Y,Z)X − g(X,Z)Y ] + μ[g(Y, Z)hX − g(X,Z)hY ]}, (1....
[...]
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
"Pseudo-symmetric structures on almo..." refers background in this paper
...Recently (see, for example, [6], [7], [9]) almost contact metric manifolds such that η is closed and dΦ = 2η∧Φ have been studied; they are called almost Kenmotsu manifolds....
[...]