Ricci Symmetric Manifolds Research Articles

The generalized weakly Ricci symmetric structure

This paper attempts to study some geometric properties along with the existence of a generalized weakly Ricci symmetric manifold and find out the reduced form of defining conditions of such a manifold. It is observed that every generalized weakly Ricci symmetric manifold is an almost generalized pseudo-Ricci symmetric manifold. We also find out the conditions for which a weakly Ricci symmetric manifold becomes a Ricci pseudosymmetric manifold in the sense of Deszcz.

Complete CSC Hypersurfaces Satisfying an Okumura-Type Inequality in Ricci Symmetric Manifolds

We investigate the spacelike hypersurface with constant scalar curvature (SCS) immersed in a Ricci symmetric manifold obeying standard curvature constraints. By supposing these hypersurfaces satisfy a suitable Okumura-type inequality recently introduced by Meléndez, which is a weaker hypothesis than to assume that they have two distinct principal curvatures, we obtain a series of umbilicity and pinching results. In particular, when the Ricci symmetric manifold is an Einstein manifold, then we further obtain some rigidity classifications of such hypersufaces.

Almost Pseudo Ricci Symmetric Manifold Admitting Schouten Tensor

The object of the present paper is to study almost pseudo Ricci symmetric manifold admitting Schouten tensor and investigated some geometric properties of this manifold.

Formula omitted]-connections in generalized geometry

We consider a family of α-connections defined by a pair of generalized dual quasi-statistical connections (∇ˆ,∇ˆ∗) on the generalized tangent bundle (TM⊕T∗M,hˇ) and determine their curvature, Ricci curvature and scalar curvature. Moreover, we provide the necessary and sufficient condition for ∇ˆ∗ to be an equiaffine connection and we prove that if h is symmetric and ∇h=0, then (TM⊕T∗M,hˇ,∇ˆ(α),∇ˆ(−α)) is a conjugate Ricci-symmetric manifold. Also, we characterize the integrability of a generalized almost product, of a generalized almost complex and of a generalized metallic structure w.r.t. the bracket defined by the α-connection. Finally we study α-connections defined by the twin metric of a pseudo-Riemannian manifold, (M,g), with a non-degenerate g-symmetric (1,1)-tensor field J such that d∇J=0, where ∇ is the Levi-Civita connection of g.

Certain results on Kenmotsu manifolds

In this paper, we focus on Kenmotsu manifolds. Firstly, we investigate almost quasi Ricci symmetric Kenmotsu manifolds. Then, we study Kenmotsu manifold admitting a Yamabe soliton. We find that if the soliton field of the Yamabe soliton is orthogonal to the characteristic vector field then it is Killing and the manifold has constant scalar curvature. Also, we deal with a Kenmotsu manifold which admits a quasi-Yamabe soliton. Finally, we give an example which verify our results.

η-Ricci Solitons on 3-dimensional Trans-Sasakian Manifolds

In this paper, we study \( \eta \)-Ricci solitons on 3-dimensional trans-Sasakian manifolds. Firstly we give conditions for the existence of these geometric structures and then observe that they provide examples of \( \eta \)-Einstein manifolds. In the case of \( \phi \)-Ricci symmetric trans-Sasakian manifolds, the η-Ricci soliton condition turns them to Einstein manifolds. Afterward, we study the implications in this geometric context of the important tensorial conditions \( R \cdot S = 0\), \(S \cdot R = 0\), \(W_2\cdot S = 0\) and \(S \cdot W_2 = 0\).

η-Ricci solitons in Kenmotsu manifolds

Abstract The object of the present paper is to study generalized weakly symmetric and generalized weakly Ricci symmetric Kenmotsu manifolds whose metric tensor is η-Ricci soliton. The paper also aims to bring out curvature conditions for which η-Ricci solitons in Kenmotsu manifolds are sometimes shrinking or expanding and some other time remain steady. The existence of each generalized weakly symmetric and generalized weakly Ricci symmetric Kenmotsu manifold is ensured by an example.

On quasi Ricci symmetric manifolds

In this paper, we study a type of Riemannian manifold, namely quasi Ricci symmetric manifold. Among others, we show that the scalar curvature of a quasi Ricci symmetric manifold is constant. In addition if the manifold is Einstein, then its Ricci tensor is zero. Also we prove that if the associated vector field of a quasi Ricci symmetric manifold is either recurrent or concurrent, then its Ricci tensor is zero.

Some Results on Weakly Symmetric Kenmotsu Manifolds

In this paper, we investigate weakly symmetric, weakly Ricci symmetric, weakly concircular symmetric and weakly concircular Ricci symmetric properties of a Kenmotsu manifold admitting a semi-symmetric metric connection. Some results on weakly -projectively symmetric Kenmotsu manifold with respect to a semi-symmetric metric connection are obtained. An example of a weakly symmetric and weakly Ricci symmetric Kenmotsu manifold with respect to this connection is constructed.

Geodesic mappings of manifolds with affine connection onto the Ricci symmetric manifolds

In the present paper we investigate geodesic mappings of manifolds with affine connection onto Ricci symmetric manifolds which are characterized by the covariantly constant Ricci tensor. We obtained a fundamental system for this problem in a form of a system of Cauchy type equations in covariant derivatives depending on no more than n(n+1) real parameters. Analogous results are obtained for geodesic mappings of manifolds with affine connection onto symmetric manifolds.

On generalized special pseudo Ricci symmetric manifolds

On generalized special pseudo Ricci symmetric manifolds

A New Class of Almost Ricci Solitons and Their Physical Interpretation.

We establish a link between a connection symmetry, called conformal collineation, and almost Ricci soliton (in particular Ricci soliton) in reducible Ricci symmetric semi-Riemannian manifolds. As a physical application, by investigating the kinematic and dynamic properties of almost Ricci soliton manifolds, we present a physical model of imperfect fluid spacetimes. This model gives a general relation between the physical quantities (u, μ, p, α, η, σ ij) of the matter tensor of the field equations and does not provide any exact solution. Therefore, we propose further study on finding exact solutions of our viscous fluid physical model for which it is required that the fluid velocity vector u be tilted. We also suggest two open problems.

On φ-pseudo symmetric LP-Sasakian manifolds with respect to quarter-symmetric non-metric connections

The object of the present paper is to study φ-pseudo symmetric and φ-pseudo Ricci symmetric LP-Sasakian manifolds with respect to Levi–Civita connections and quarter-symmetric non-metric connections. We obtain a necessary and sufficient condition for a φ-pseudo symmetric LP-Sasakian manifold with respect to a quarter symmetric non-metric connection to be φ-pseudo symmetric LP-Sasakian manifold with respect to a Levi–Civita connection.

<b>New characterization of b-m2 developable surfaces

In this paper, b-m2 developable surfaces of biharmonic b-slant helices in the special three- dimensional − φ Ricci symmetric para-Sasakian manifold P is studied. Explicit parametric equations of b-m2 developable surfaces of biharmonic b-slant helices in the special three-dimensional − φ Ricci

On pseudosymmetric manifolds

In the literature, there are two different notions of pseudosymmetric manifolds, one by Chaki [7] and other by Deszcz [16], and there are many papers related to these notions. The object of the present paper is to deduce necessary and sufficient conditions for a Chaki pseudosymmetric [7] (resp. pseudo Ricci symmetric [8]) manifold to be Deszcz pseudosymmetric (resp. Ricci pseudosymmetric). We also study the necessary and sufficient conditions for a weakly symmetric [58] (resp. weakly Ricci symmetric [59]) manifold by Tam\'assy and Binh to be Deszcz pseudosymmetric (resp. Ricci pseudosymmetric). We also obtain the reduced form of the defining condition of weakly Ricci symmetric manifolds by Tam\'assy and Binh [59]. Finally we give some examples to show the independent existence of such types of pseudosymmetry which also ensure the existence of Roter type and generalized Roter type manifolds and the manifolds with recurrent curvature 2-form ([2], [29]) associated to various curvature tensors.

A Class of Riemannian Manifolds with Special Form of Curvature Tensor

Abstract Riemannian curvature is determined by the Ricci curvature by a special formula to introduce a special class of Riemannian manifolds which called Jawarneh manifold. Some geometric properties of Jawarneh manifold have been derived and a non-trivial example is obtained to prove the existence. Keywords: Riemannian manifold; flat manifold; Einstein manifold; symmetric manifold; Ricci symmetric manifold; conformally flat manifold; cosmology. Mathematics Subject Classification (2010): 53C25 (primary); 53C21 (secondary). 1 Introduction It is well known that on a Riemannian manifold (M n , g) the curvature tensor vanishes for n = 1, and it is proportional to the metric g for n = 2. But for n = 3 the curvature tensor of a Riemannian manifold is proportional to the Ricci tensor. Hence without loss of generality we can define a special type of Riemannian manifolds by considering a Riemannian manifold (M n , g) (n>2) with non-vanishing Ricci tensor S and its curvature tensor R satisfies the relation; R(X,Y,Z) = k[S(Y,Z)X−S(X,Z)Y], (1.1) Original Research Article

English

In this paper, we investigate some curvature problems of Kenmotsu manifolds satisfying some certain conditions and we reach some classicifications. We consider -recurrent Kenmotsu manifolds and we show that -recurrent Kenmotsu manifolds are also -Einstein manifolds. Next, we study  -Ricci symmetric Kenmotsu manifolds and we find this manifolds are Einstein manifolds too. In addition, we examine locally -symmetric -Kenmotsu manifolds. Later we investigate this type manifold with quasi-conformally curvature tensor and concircular curvature tensor. In addition to these, we construct an example of Kenmotsu manifolds and we see that this example is a locally symmetric Kenmotsu manifold.   Key words: Kenmotsu manifold, -recurrent, -Ricci symmetric, locally -symmetric, concircular curvature tensor, quasi-conformally curvature tensor, -Einstein manifolds, Einstein manifolds.

On some classes of almost pseudo Ricci symmetric manifolds

On some classes of almost pseudo Ricci symmetric manifolds

On weakly symmetric and weakly Ricci symmetric warped product manifolds

This paper is concerned with some results on weakly symmetric and weakly Ricci symmetric warped product manifolds. We prove the necessary and su-ci- ent condition for a warped product manifold to be weakly symmetric and weakly Ricci symmetric. On the basis of these results two proper examples of warped product weakly symmetric and weakly Ricci symmetric manifolds are presented.

PSEUDO Z SYMMETRIC RIEMANNIAN MANIFOLDS WITH HARMONIC CURVATURE TENSORS

In this paper we introduce a new notion of Z-tensor and a new kind of Riemannian manifold that generalize the concept of both pseudo Ricci symmetric manifold and pseudo projective Ricci symmetric manifold. Here the Z-tensor is a general notion of the Einstein gravitational tensor in General Relativity. Such a new class of manifolds with Z-tensor is named pseudoZ symmetric manifold and denoted by (PZS)n. Various properties of such an n-dimensional manifold are studied, especially focusing the cases with harmonic curvature tensors giving the conditions of closeness of the associated one-form. We study (PZS)n manifolds with harmonic conformal and quasi-conformal curvature tensor. We also show the closeness of the associated 1-form when the (PZS)n manifold becomes pseudo Ricci symmetric in the sense of Deszcz (see [A. Derdzinsky and C. L. Shen, Codazzi tensor fields, curvature and Pontryagin forms, Proc. London Math. Soc.47(3) (1983) 15–26; R. Deszcz, On pseudo symmetric spaces, Bull. Soc. Math. Belg. Ser. A44 (1992) 1–34]). Finally, we study some properties of (PZS)4 spacetime manifolds.