Weyl Conformal Curvature Tensor Research Articles

On Wintgen Ideal Submanifolds Satisfying Some Pseudo-symmetry Type Curvature Conditions

Let M be a Wintgen ideal submanifold of dimension n in a real space form Rn+m(k) of dimension (n + m) and of constant curvaturek, n ≥ 4, m ≥ 1. Let g, R, Ricc, g ∧ Ricc and C be the metric tensor, the Riemann-Christoffel curvature tensor, the Ricci tensor, the Kulkarni-Nomizu product of g and Ricc, and the Weyl conformal curvature tensor of M, respectively. In this paper we study Wintgen ideal submanifolds M in real space forms Rn+m(k), n ≥ 4, m ≥ 1, satisfying the following pseudo-symmetry type curvature conditions: (i) the tensors R · C and Q(g, R) (resp., Q(g, C), Q(g, g ∧ Ricc), Q(Ricc, R) or Q(Ricc, g ∧ Ricc)) are linearly dependent; (ii) the tensors C · R and Q(g, R) (resp., Q(g, C), Q(g, g ∧ Ricc), Q(Ricc, R) or Q(Ricc, g ∧ Ricc)) are linearly dependent; (iii) the tensors R·C -C ·R and Q(g, R) (resp., Q(g, C), Q(g, g∧Ricc), Q(Ricc, R) or Q(Ricc, g∧Ricc)) are linearly dependent.

Certain results on tangent bundle endowed with generalized Tanaka Webster connection (GTWC) on Kenmotsu manifolds

<p>This work studies the complete lifts of Kenmotsu manifolds associated with the generalized Tanaka-Webster connection (GTWC) in the tangent bundle. Using the GTWC, this study explores the complete lifts of various curvature tensors and geometric structures from Kenmotsu manifolds to their tangent bundles. Specifically, it examines the complete lifts of Ricci semi-symmetry, the projective curvature tensor, $ \Phi $-projectively semi-symmetric structures, the conharmonic curvature tensor, the concircular curvature tensor, and the Weyl conformal curvature tensor. Additionally, the research delves into the complete lifts of Ricci solitons on Kenmotsu manifolds with the GTWC within the tangent bundle framework, providing new insights into their geometric properties and symmetries in the lifted space. The data on the complete lifts of the Ricci soliton in Kenmotsu manifolds associated with the GTWC in the tangent bundle are also investigated. An example of the complete lifts of a $ 5 $-dimensional Kenmotsu manifold is also included.</p>

On Semi-Riemannian Manifolds Satisfying Some Generalized Einstein Metric Conditions

The derivation-commutator $R \cdot C - C \cdot R$ of a semi-Riemannian manifold $(M,g)$, $\dim M \geq 4$, formed by its Riemann-Christoffel curvature tensor $R$ and the Weyl conformal curvature tensor $C$, under some assumptions, can be expressed as a linear combination of $(0,6)$-Tachibana tensors $Q(A,T)$, where $A$ is a symmetric $(0,2)$-tensor and $T$ a generalized curvature tensor. These conditions form a family of generalized Einstein metric conditions. In this survey paper we present recent results on manifolds and submanifolds, and in particular hypersurfaces, satisfying such conditions.

Curvature properties of metric and semi-symmetric linear connections

Considering a non-symmetric linear connection and its dual one appear 6 independent curvature tensors [15]. In the present paper we study properties of curvature tensors of semi-symmetric metric connections on a generalized Riemannian manifold. It is shown that if linear connection preserves the symmetric part of the generalized Riemannian metric with semi-symmetric torsion with an appropriate additional condition, then these six curvature tensors can be expressed by linear combination of Weyl conformal curvature tensor, Weyl projective curvature tensor and the concircular curvature tensor. In the case of the curvature tensor , we have proved that if exists a linear connection preserving the symmetric part of the generalized Riemannian metric with semi-symmetric torsion, whose curvature tensor vanishes, it is necessary and sufficient that the Riemannian metric be projectively flat. For tensor we also get a useful result. If exists a linear connection preserving the symmetric part of the generalized Riemannian metric with semi-symmetric torsion, whose curvature tensor and Ricci tensor vanish, it is necessary and sufficient that the Riemannian metric be conformally flat.

Some curvature proprieties of Kenmotsu manifolds with the Schouten-van Kampen connection

The aim of the present paper is to study the concircular curvature tensor, projective curvature tensor, Weyl conformal curvature tensor of Kenmotsu manifolds admitting Schouten-van Kampen connection and an example is given to verify our results.

Pseudosymmetry properties of generalised Wintgen ideal Legendrian submanifolds

For Legendrian submanifolds Mn in Sasakian space forms ?M2n+1(c), I. Mihai obtained an inequality relating the normalised scalar curvature (intrinsic invariant) and the squared mean curvature and the normalised scalar normal curvature of M in the ambient space ?M (extrinsic invariants) which is called the generalised Wintgen inequality, characterising also the corresponding equality case. And a Legendrian submanifold Mn in Sasakian space forms ?M2n+1(c) is said to be generalised Wintgen ideal Legendrian submanifold of ?M2n+1(c) when it realises at everyone of its points the equality in such inequality. Characterisations based on some basic intrinsic symmetries involving the Riemann-Cristoffel curvature tensor, the Ricci tensor and the Weyl conformal curvature tensor belonging to the class of pseudosymmetries in the sense of Deszcz of such generalised Wintgen ideal Legendrian submanifolds are given.

Generalized Weyl conformal curvature tensor of generalized Riemannian space

Generalized Weyl conformal curvature tensor of generalized Riemannian space

A study on killing vector fields in four-dimensional spaces

In the present study, some properties of Killing vector fields are investigated on 4-dimensional manifolds in case of the signature of the metric tensor 1 is either Lorentz or positive definite or neutral. First of all, the notation and the main object of the study are introduced on these manifolds. Later on, some special subalgebras are examined for the members of the Killing algebra when the Killing vector field vanishes at a point of the manifold admitting any of these metric signatures. The constraints of this examination to the Weyl conformal curvature tensor and the Ricci tensor are then studied and some results are obtained. Finally, some examples related to these results are given for all metric signatures.

Curvature properties of Robinson–Trautman metric

The curvature properties of Robinson–Trautman metric have been investigated. It is shown that Robinson–Trautman metric is a Roter type metric, and in a consequence, admits several kinds of pseudosymmetric type structures such as Weyl pseudosymmetric, Ricci pseudosymmetric, pseudosymmetric Weyl conformal curvature tensor etc. Moreover, it is proved that this metric is a 2-quasi-Einstein, the Ricci tensor is Riemann compatible and its Weyl conformal curvature 2-forms are recurrent. It is also shown that the energy momentum tensor of the metric is pseudosymmetric and the conditions under which such tensor is of Codazzi type and cyclic parallel have been investigated. Finally, we have made a comparison between the curvature properties of Robinson–Trautman metric and Som–Raychaudhuri metric.

On the isolation phenomena of locally conformally flat manifolds with constant scalar curvature – Submanifolds versions

In this paper, from the viewpoint of submanifold theory, we study the isolation phenomena of Riemannian manifolds with constant scalar curvature and vanishing Weyl conformal curvature tensor. Firstly, for any locally strongly convex affine hyperspheres in an (n+1)-dimensional equiaffine space Rn+1 with constant scalar curvature, we prove an inequality involving the traceless Ricci tensor, the Pick invariant and the scalar curvature. The inequality is optimal and we can further completely classify the affine hyperspheres which realize the equality case of the inequality. Secondly, and analogously, for Lagrangian minimal submanifolds of the complex projective space CPn equipped with the Fubini–Study metric, under the condition that the Weyl conformal curvature tensor vanishes, we establish a similar but reverse inequality involving the traceless Ricci tensor, the scalar curvature and the squared norm of the second fundamental form. The inequality is also optimal and we can further completely classify the submanifolds which realize the equality case of the inequality.

On Quaternionic 3 CR-Structure and Pseudo-Riemannian Metric

A CR-structure on a 2n +1-manifold gives a conformal class of Lorentz metrics on the Fefferman S1-bundle. This analogy is carried out to the quarternionic conformal 3-CR structure (a generalization of quaternionic CR- structure) on a 4n + 3 -manifold M. This structure produces a conformal class [g] of a pseudo-Riemannian metric g of type (4n + 3,3) on M × S3. Let (PSp(n +1,1), S4n+3) be the geometric model obtained from the projective boundary of the complete simply connected quaternionic hyperbolic manifold. We shall prove that M is locally modeled on (PSp(n +1,1), S4n+3) if and only if (M × S3 ,[g]) is conformally flat (i.e. the Weyl conformal curvature tensor vanishes).

On Complex $\eta $-Einstein Normal Complex Contact Metric Manifolds

The aim of this paper is focusing on $\eta$-Einstein geometry of complex contact metric manifolds. We give the definition of complex $\eta$-Einstein normal complex contact metric manifolds. In addition, we study on Weyl conformal curvature tensor $\mathcal{W}$ and concircular curvature tensor $\mathcal{Z}$ and we show that a normal complex contact metric manifold which satisfy $\mathcal{Z}\left( U,X\right) .\mathcal{W}=0$ and $\mathcal{Z}\left( V,X\right). \mathcal{W}=0$ complex $\eta$-Einstein. Also, we prove that a projectively semi-symmetric normal complex contact metric manifold is complex $\eta$-Einstein.

Some results on (k, μ)′-almost Kenmotsu manifolds

In this paper, we study the Weyl conformal curvature tensor 𝒲 and the concircular curvature tensor 𝒞 on a (k, μ)′-almost Kenmotsu manifold M2n+1 of dimension greater than 3. We obtain that if M2n+1 satisfies either R · 𝒲 = 0 or 𝒞 · 𝒞 = 0, then it is locally isometric to either the hyperbolic space ℍ2n+1 (−1) or the Riemannian product ℍn+1(−4) × ℝn.

Ricci solitons on para-Kenmotsu manifolds

The object of the present paper is to study Ricci solitons in para-Kenmotsu manifolds when the Weyl-conformal curvature tensor satisfies some geometric properties like flatness, semi-symmetry, pseudo-symmetry, Ricci-pseudosymmetry and Einstein semi-symmetry.

Covariant derivative of the curvature tensor of pseudo-Kählerian manifolds

It is well known that the curvature tensor of a pseudo-Riemannian manifold can be decomposed with respect to the pseudo-orthogonal group into the sum of the Weyl conformal curvature tensor, the traceless part of the Ricci tensor and of the scalar curvature. A similar decomposition with respect to the pseudo-unitary group exists on a pseudo-K\"ahlerian manifold; instead of the Weyl tensor one obtains the Bochner tensor. In the present paper, the known decomposition with respect to the pseudo-orthogonal group of the covariant derivative of the curvature tensor of a pseudo-Riemannian manifold is refined. A decomposition with respect to the pseudo-unitary group of the covariant derivative of the curvature tensor for pseudo-K\"ahlerian manifolds is obtained. This defines natural classes of spaces generalizing locally symmetric spaces and Einstein spaces. It is shown that the values of the covariant derivative of the curvature tensor for a non-locally symmetric pseudo-Riemannian manifold with an irreducible connected holonomy group different from the pseudo-orthogonal and pseudo-unitary groups belong to an irreducible module of the holonomy group.

On a type of almost Kenmotsu manifolds with nullity distributions

The object of the present paper is to characterize Weyl semisymmetric almost Kenmotsu manifolds with its characteristic vector field ξ belonging to the (k,μ)′-nullity distribution and (k,μ)-nullity distribution respectively. Also we characterize almost Kenmotsu manifolds satisfying the curvature condition C⋅S=0, where C and S are the Weyl conformal curvature tensor and Ricci tensor respectively with its characteristic vector field ξ belonging to the (k,μ)′-nullity distribution. As a consequence of the main results we obtain several corollaries. Finally, we present an example to verify our results.

Para-Sasakian manifolds satisfying certain curvature conditions

In this paper, we investigate P-Sasakian manifolds satisfying the conditions R(X, ξ) · C = 0 and $$C \cdot \widetilde Z = 0$$ , where C and $$\widetilde Z$$ are the Weyl conformal curvature tensor and the concircular curvature tensor respectively. Next, we study 3-dimensional P-Sasakianmanifolds. Finally, we give an example of a 3-dimensional P-Sasakian manifold.

On generalized Robertson--Walker spacetimes satisfying some curvaturecondition

We give necessary and sufficient conditions for warped product manifolds (M,g), of dimension \geqslant 4, with 1-dimensional base, and in particular, for generalized Robertson--Walker spacetimes, to satisfy some generalized Einstein metric condition. Namely, the difference tensor R . C - C . R, formed from the curvature tensor R and the Weyl conformal curvature tensor C, is expressed by the Tachibana tensor Q(S,R) formed from the Ricci tensor S and R. We also construct suitable examples of such manifolds. They are quasi-Einstein, i.e. at every point of M rank (S - a g) \leqslant 1, for some a \in R, or non-quasi-Einstein.

On a type of contact metric manifolds

The object of the present paper is to study harmonic Weyl conformal curvature tensor, conformally recurrent, cyclic parallel Ricci tensor and locally o-conformally symmetric contact metric manifolds whose characteristic vector field ξ belonging to the k-nullity distribution. Finally, we give an example to verify the result.

On generalized Sasakian-space-forms with Weyl-conformal curvature tensor

The object of the this paper is to study Weyl-pseudosymmetric generalized Sasakianspace-forms and such space-forms satisfying the conditions C(ξ,X)R = 0 and C(ξ,X)S = 0, where C is the Weyl-conformal curvature tensor, R and S are the Riemannian curvature tensor and the Ricci tensor of the space-form respectively.