Certain results on Kenmotsu pseudo-metric manifolds
TL;DR: In this article, a systematic study of Kenmotsu pseudo-metric manifolds is presented, and the Ricci solitons on these manifolds are considered, and necessary and sufficient conditions for them to have constant curvatures are provided.
Abstract: In this paper, a systematic study of Kenmotsu pseudo-metric manifolds are introduced. After studying the properties of this manifolds, we provide necessary and sufficient condition for Kenmotsu pseudo-metric manifold to have constant $\varphi$-sectional curvature, and prove the structure theorem for $\xi$-conformally flat and $\varphi$-conformally flat Kenmotsu pseudo-metric manifolds. Next, we consider Ricci solitons on this manifolds. In particular, we prove that an $\eta$-Einstein Kenmotsu pseudo-metric manifold of dimension higher than 3 admitting a Ricci soliton is Einstein, and a Kenmotsu pseudo-metric 3-manifold admitting a Ricci soliton is of constant curvature $-\varepsilon$.
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25 Sep 2020
10 citations
TL;DR: In this paper, the authors studied the Riemann soliton and gradient almost-Riemann-soliton on a certain class of almost Kenmotsu manifolds.
Abstract: The aim of this paper, is to study the Riemann soliton and gradient almost Riemann soliton on certain class of almost Kenmotsu manifolds. Also, some suitable examples of Kenmotsu and (κ,μ)′-almost ...
10 citations
TL;DR: In this article, it was shown that a Riemannian manifold equipped with a concurrent-recurrent vector field is of constant negative curvature when its metric is a Ricci soliton.
Abstract: In this paper, we initiate the study of impact of the existence of a unit vector $$
u $$
, called a concurrent-recurrent vector field, on the geometry of a Riemannian manifold. Some examples of these vector fields are provided on Riemannian manifolds, and basic geometric properties of these vector fields are derived. Next, we characterize Ricci solitons on 3-dimensional Riemannian manifolds and gradient Ricci almost solitons on a Riemannian manifold (of dimension n) admitting a concurrent-recurrent vector field. In particular, it is proved that the Riemannian 3-manifold equipped with a concurrent-recurrent vector field is of constant negative curvature $$-\alpha ^2$$
when its metric is a Ricci soliton. Further, it has been shown that a Riemannian manifold admitting a concurrent-recurrent vector field, whose metric is a gradient Ricci almost soliton, is Einstein.
8 citations
TL;DR: The main purpose of as mentioned in this paper is to study almost cosymplectic pseudo-metric manifold satisfying certain - parallel tensor fields and obtain some results related to the - parallelity of,, and.
Abstract: The main purpose of this paper is to study almost - cosymplectic pseudo metric manifold satisfying certain - parallel tensor fields. We first focus on the concept of almost - cosymplectic pseudo metric manifold and its curvature properties. Then, we obtain some results related to the - parallelity of , , and . Moreover, the deformation of almost - Kenmotsu pseudo metric structure is given. We conclude the paper with an illustrative example of almost - cosymplectic pseudo metric manifold.
2 citations
Cites background from "Certain results on Kenmotsu pseudo-..."
...In particular, the authors established necessary and sufficient conditions for Kenmotsu pseudo metric manifolds satisfying certain tensor conditions [13]....
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References
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Book•
11 Jul 2011
TL;DR: In this article, the authors introduce Semi-Riemannian and Lorenz geometries for manifold theory, including Lie groups and Covering Manifolds, as well as the Calculus of Variations.
Abstract: Manifold Theory. Tensors. Semi-Riemannian Manifolds. Semi-Riemannian Submanifolds. Riemannian and Lorenz Geometry. Special Relativity. Constructions. Symmetry and Constant Curvature. Isometries. Calculus of Variations. Homogeneous and Symmetric Spaces. General Relativity. Cosmology. Schwarzschild Geometry. Causality in Lorentz Manifolds. Fundamental Groups and Covering Manifolds. Lie Groups. Newtonian Gravitation.
3,593 citations
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
"Certain results on Kenmotsu pseudo-..." refers background in this paper
...first relation along with any one of the remaining three relations in (2.1) imply the remaining two relations. Also, for an almost contact structure, the rank of ϕ is 2n. For more details, we refer to [5]. If an almost contact manifold is endowed with a pseudo-Riemannian metric g such that g(ϕX,ϕY ) = g(X,Y) −εη(X)η(Y ), (2.2) where ε = ±1, for all X,Y ∈ TM, then (M,ϕ,ξ,η,g) is called an almost contac...
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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
"Certain results on Kenmotsu pseudo-..." refers background or methods in this paper
...On the other hand, in 1972, Kenmotsu [22] investigated a class of contact Riemannian manifolds satisfying some special conditions, and after onwards such manifolds are came to known as Kenmotsu manifolds....
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...The above corollary for Riemannian case has been proved in [22]....
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