scholarly journals Generalized Quasi-Einstein Manifolds in Contact Geometry

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1592
Author(s):  
İnan Ünal
Keyword(s):  
Constant Curvature ◽  
Ricci Tensor ◽  
Einstein Manifold ◽  
Contact Pair ◽  
Einstein Manifolds ◽  
Cyclic Parallel Ricci Tensor ◽  
Codazzi Type ◽  
Curvature Tensors ◽  
Parallel Ricci Tensor ◽  
Hopf Manifold

In this study, we investigate generalized quasi-Einstein normal metric contact pair manifolds. Initially, we deal with the elementary properties and existence of generalized quasi-Einstein normal metric contact pair manifolds. Later, we explore the generalized quasi-constant curvature of normal metric contact pair manifolds. It is proved that a normal metric contact pair manifold with generalized quasi-constant curvature is a generalized quasi-Einstein manifold. Normal metric contact pair manifolds satisfying cyclic parallel Ricci tensor and the Codazzi type of Ricci tensor are considered, and further prove that a generalized quasi-Einstein normal metric contact pair manifold does not satisfy Codazzi type of Ricci tensor. Finally, we characterize normal metric contact pair manifolds satisfying certain curvature conditions related to M-projective, conformal, and concircular curvature tensors. We show that a normal metric contact pair manifold with generalized quasi-constant curvature is locally isometric to the Hopf manifold S2n+1(1)×S1.

scholarly journals η-Ricci Solitons on Sasakian 3-Manifolds

2017 ◽  
Vol 55 (2) ◽  
pp. 143-156
Author(s):  
Pradip Majhi ◽  
Uday Chand De ◽  
Debabrata Kar
Keyword(s):  
Ricci Tensor ◽  
Ricci Soliton ◽  
Ricci Solitons ◽  
Conformally Flat ◽  
Curvature Condition ◽  
Cyclic Parallel Ricci Tensor ◽  
Codazzi Type ◽  
Parallel Ricci Tensor

AbstractIn this paper we studyη-Ricci solitons on Sasakian 3-manifolds. Among others we prove that anη-Ricci soliton on a Sasakian 3-manifold is anη-Einstien manifold. Moreover we considerη-Ricci solitons on Sasakian 3-manifolds with Ricci tensor of Codazzi type and cyclic parallel Ricci tensor. Beside these we study conformally flat andφ-Ricci symmetricη-Ricci soliton on Sasakian 3-manifolds. Alsoη-Ricci soliton on Sasakian 3-manifolds with the curvature conditionQ.R= 0 have been considered. Finally, we construct an example to prove the non-existence of properη-Ricci solitons on Sasakian 3-manifolds and verify some results.

scholarly journals SOME RESULTS ON GENERALIZED $(k,\mu)$-PARACONTACT METRIC MANIFOLDS

2018 ◽  
pp. 401
Author(s):  
Sourav Makhal
Keyword(s):  
Ricci Tensor ◽  
Structure Tensor ◽  
Cyclic Parallel Ricci Tensor ◽  
Codazzi Type ◽  
Parallel Ricci Tensor

The object of this paper is to study Codazzi type of Ricci tensor in generalized $(k,\mu )$-paracontact metric manifolds. Next we study cyclic parallel Ricci tensor in generalized $(k,\mu )$-paracontact metric manifolds. Further, we characterized generalized $(k,\mu )$-paracontact metric manifolds whose structure tensor $\phi$ is $\eta$-parallel. Finally, we investigate locally $\phi$-Ricci symmetric generalized $(k,\mu )$-paracontact metric manifolds.

scholarly journals η-Ricci Solitons on Kenmotsu 3-Manifolds

2018 ◽  
Vol 56 (1) ◽  
pp. 51-63
Author(s):  
Krishnendu De ◽  
Uday Chand De
Keyword(s):  
Ricci Tensor ◽  
Ricci Soliton ◽  
Ricci Solitons ◽  
Curvature Condition ◽  
Cyclic Parallel Ricci Tensor ◽  
Codazzi Type ◽  
Parallel Ricci Tensor

Abstract In the present paper we study η-Ricci solitons on Kenmotsu 3-manifolds. Moreover, we consider η-Ricci solitons on Kenmotsu 3-manifolds with Codazzi type of Ricci tensor and cyclic parallel Ricci tensor. Beside these, we study φ-Ricci symmetric η-Ricci soliton on Kenmotsu 3-manifolds. Also Kenmotsu 3-manifolds satisfying the curvature condition R.R = Q(S, R)is considered. Finally, an example is constructed to prove the existence of a proper η-Ricci soliton on a Kenmotsu 3-manifold.

scholarly journals ETA-RICCI SOLITONS ON LORENTZIAN PARA-KENMOTSU MANIFOLDS

2021 ◽  
pp. 419
Author(s):  
Shashikant Pandey ◽  
Abhishek Singh ◽  
Vishnu Narayan Mishra
Keyword(s):  
Ricci Tensor ◽  
Ricci Solitons ◽  
Conformally Flat ◽  
Geometric Properties ◽  
Research Article ◽  
Kenmotsu Manifolds ◽  
Cyclic Parallel Ricci Tensor ◽  
Codazzi Type ◽  
Parallel Ricci Tensor

The objective of present research article is to investigate the geometric properties of $\eta$-Ricci solitons on Lorentzian para-Kenmotsu manifolds. In this manner, we consider $\eta$-Ricci solitons on Lorentzian para-Kenmotsu manifolds satisfying $R\cdot S=0$. Further, we obtain results for $\eta$-Ricci solitons on Lorentzian para-Kenmotsu manifolds with quasi-conformally flat property. Moreover, we get results for $\eta$-Ricci solitons in Lorentzian para-Kenmotsu manifolds admitting Codazzi type of Ricci tensor and cyclic parallel Ricci tensor, $\eta$-quasi-conformally semi-symmetric, $\eta$-Ricci symmetric and quasi-conformally Ricci semi-symmetric. At last, we construct an example of a such manifold which justify the existence of proper $\eta$-Ricci solitons.

scholarly journals Some curvature tensors in N(k)-contact metric manifold

BIBECHANA ◽  
2018 ◽  
Vol 16 ◽  
pp. 55-63
Author(s):  
Riddhi Jung Shah
Keyword(s):  
Ricci Tensor ◽  
Einstein Manifold ◽  
Contact Metric Manifold ◽  
3 Dimensional ◽  
Contact Metric Manifolds ◽  
Curvature Tensors

The purpose of this paper is to study W7and W9-curvature tensors on N(k)-contact metric manifolds. We prove that a N(k)-contact metric manifold satisfying the condition W7( xi,X).W9=0 is eta-Einstein manifold. We also obtain the Ricci tensor S of type (0, 2) for phi-W9flat and divW9=0 conditions on N(k)-contact metric manifolds. Finally, we give an example of 3-dimensional N(k)-contact metric manifold.BIBECHANA 16 (2019) 55-63

Real Hypersurfaces in ℂP 2 and ℂH 2 with Cyclic Parallel ∗-Ricci Tensor

2021 ◽  
Vol 58 (3) ◽  
pp. 308-318
Author(s):  
Yaning Wang ◽  
Wenjie Wang
Keyword(s):  
Ricci Tensor ◽  
Hyperbolic Plane ◽  
Real Hypersurface ◽  
Three Dimensional ◽  
Real Hypersurfaces ◽  
Complex Projective Plane ◽  
Cyclic Parallel Ricci Tensor ◽  
Complex Projective ◽  
Parallel Ricci Tensor ◽  
Complex Hyperbolic Plane

In this paper, we prove that the ∗-Ricci tensor of a real hypersurface in complex projective plane ℂP 2 or complex hyperbolic plane ℂH 2 is cyclic parallel if and only if the hypersurface is of type (A). We find some three-dimensional real hypersurfaces having non-vanishing and non-parallel ∗-Ricci tensors which are cyclic parallel.

On Some Aspects of Geometry of Almost C(λ)-manifolds

2020 ◽  
pp. 19-24
Author(s):  
A.R. Rustanov ◽  
E.A. Polkina ◽  
S.V. Kharitonova
Keyword(s):  
Constant Curvature ◽  
Local Structure ◽  
Ricci Tensor ◽  
Einstein Manifold ◽  
Sufficient Conditions ◽  
Holomorphic Sectional Curvature ◽  
Riemannian Curvature ◽  
Riemannian Curvature Tensor ◽  
Analytic Expressions ◽  
Necessary And Sufficient

In this paper almost C(λ)-manifolds are considered. The local structure of Ricci-flat almost C(λ)-manifolds is obtained. On the space of the adjoint G-structure, necessary and sufficient conditions are obtained under which the al-most C(λ)-manifolds are manifolds of constant curvature and the structure of the Riemannian curvature tensor of an almost C(λ)-manifold of constant curvature is obtained. Relations are obtained that characterize the Einstein almost C(λ)-manifolds. It is proved that a complete almost C(λ)-Einstein manifold is either holomorphically isometrically covered by the product of a real line by a Ricciflat Kähler manifold, or is compact and has a finite fundamental group. For almost C(λ)-manifolds that are -Einstein, analytic expressions for the functions  and  characterizing these manifolds are obtained. It is shown that an almost C(λ)-manifold has an Ф-invariant Ricci tensor. We study also almost C(λ)-manifolds of pointwise constant Ф-holomorphic sectional curvature.

Pseudo B-symmetric manifolds

2017 ◽  
Vol 14 (09) ◽  
pp. 1750119
Author(s):  
Young Jin Suh ◽  
Carlo Alberto Mantica ◽  
Uday Chand De ◽  
Prajjwal Pal
Keyword(s):  
Riemannian Manifold ◽  
Explicit Form ◽  
Riemannian Manifolds ◽  
Ricci Tensor ◽  
Sufficient Condition ◽  
Conformally Flat ◽  
Harmonic Curvature ◽  
Codazzi Type ◽  
Curvature Tensors

In this paper, we introduce a new tensor named [Formula: see text]-tensor which generalizes the [Formula: see text]-tensor introduced by Mantica and Suh [Pseudo [Formula: see text] symmetric Riemannian manifolds with harmonic curvature tensors, Int. J. Geom. Methods Mod. Phys. 9(1) (2012) 1250004]. Then, we study pseudo-[Formula: see text]-symmetric manifolds [Formula: see text] which generalize some known structures on pseudo-Riemannian manifolds. We provide several interesting results which generalize the results of Mantica and Suh [Pseudo [Formula: see text] symmetric Riemannian manifolds with harmonic curvature tensors, Int. J. Geom. Methods Mod. Phys. 9(1) (2012) 1250004]. At first, we prove the existence of a [Formula: see text]. Next, we prove that a pseudo-Riemannian manifold is [Formula: see text]-semisymmetric if and only if it is Ricci-semisymmetric. After this, we obtain a sufficient condition for a [Formula: see text] to be pseudo-Ricci symmetric in the sense of Deszcz. Also, we obtain the explicit form of the Ricci tensor in a [Formula: see text] if the [Formula: see text]-tensor is of Codazzi type. Finally, we consider conformally flat pseudo-[Formula: see text]-symmetric manifolds and prove that a [Formula: see text] spacetime is a [Formula: see text]-wave under certain conditions.

Cyclic-parallel Ricci tensors on a class of almost Kenmotsu 3-manifolds

2019 ◽  
Vol 16 (06) ◽  
pp. 1950092 ◽  
Author(s):  
Yaning Wang ◽  
Xinxin Dai
Keyword(s):  
Hyperbolic Space ◽  
Lie Group ◽  
Ricci Tensor ◽  
Lie Derivative ◽  
Smooth Functions ◽  
Cyclic Parallel Ricci Tensor ◽  
Contact Distribution ◽  
Local Characterization ◽  
Parallel Ricci Tensor

In this paper, we give a local characterization for the Ricci tensor of an almost Kenmotsu [Formula: see text]-manifold [Formula: see text] to be cyclic-parallel. As an application, we prove that if [Formula: see text] has cyclic-parallel Ricci tensor and satisfies [Formula: see text], (where [Formula: see text] is the Lie derivative of [Formula: see text] along the Reeb flow and both [Formula: see text] and [Formula: see text] are smooth functions such that [Formula: see text] is invariant along the contact distribution), then [Formula: see text] is locally isometric to either the hyperbolic space [Formula: see text] or a non-unimodular Lie group equipped with a left invariant non-Kenmotsu almost Kenmotsu structure.

scholarly journals On static manifolds and related critical spaces with cyclic parallel Ricci tensor

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
H. Baltazar ◽  
A. Da Silva
Keyword(s):  
Scalar Curvature ◽  
Ricci Tensor ◽  
Critical Spaces ◽  
3 Dimensional ◽  
Critical Metrics ◽  
Volume Functional ◽  
Total Scalar Curvature ◽  
Cyclic Parallel Ricci Tensor ◽  
Scalar Curvature Functional ◽  
Parallel Ricci Tensor

Abstract We classify 3-dimensional compact Riemannian manifolds (M 3, g) that admit a non-constant solution to the equation −Δfg +Hess f − f Ric = μ Ric +λg for some special constants (μ, λ), under the assumption that the manifold has cyclic parallel Ricci tensor. Namely, the structures that we study here are: positive static triples, critical metrics of the volume functional, and critical metrics of the total scalar curvature functional. We also classify n-dimensional critical metrics of the volume functional with non-positive scalar curvature and satisfying the cyclic parallel Ricci tensor condition.

Sign in / Sign up

Export Citation Format

Share Document