Paracontact Manifolds Research Articles

A (k, μ)-Paracontact Metric Manifolds satisfying Curvature Conditions

In the present paper, we have studied the curvature tensors of $(k,\mu)$-paracontact manifold satisfying the conditions $\widetilde{Z}\cdot \widetilde{C}=0$, \ $R\cdot \widetilde{C} =0 $, \ $P\cdot \widetilde{C}=0$ and $\widetilde{C}\cdot\widetilde{C}=0.$ According these cases, $(k,\mu)$-paracontact manifolds have been characterized.

Geometric classifications of k-almost Ricci solitons admitting paracontact metrices

Abstract The prime objective of the approach is to give geometric classifications of k k -almost Ricci solitons associated with paracontact manifolds. Let M 2 n + 1 ( φ , ξ , η , g ) {M}^{2n+1}\left(\varphi ,\xi ,\eta ,g) be a paracontact metric manifold, and if a K K -paracontact metric g g represents a k k -almost Ricci soliton ( g , V , k , λ ) \left(g,V,k,\lambda ) and the potential vector field V V is Jacobi field along the Reeb vector field ξ \xi , then either k = λ − 2 n k=\lambda -2n , or g g is a k k -Ricci soliton. Next, we consider K K -paracontact manifold as a k k -almost Ricci soliton with the potential vector field V V is infinitesimal paracontact transformation or collinear with ξ \xi . We have proved that if a paracontact metric as a k k -almost Ricci soliton associated with the non-zero potential vector field V V is collinear with ξ \xi and the Ricci operator Q Q commutes with paracontact structure φ \varphi , then it is Einstein of constant scalar curvature equals to − 2 n ( 2 n + 1 ) -2n\left(2n+1) . Finally, we have deduced that a para-Sasakian manifold admitting a gradient k k -almost Ricci soliton is Einstein of constant scalar curvature equals to − 2 n ( 2 n + 1 ) -2n\left(2n+1) .

Geometry of paracontact metric as an almost Yamabe solitons

In this offering exposition, we intend to study paracontact metric manifold [Formula: see text] admitting almost Yamabe solitons. First, for a general paracontact metric manifold, it is proved that [Formula: see text] is Killing if the vector field [Formula: see text] is an infinitesimal contact transformation and that [Formula: see text] is [Formula: see text]-paracontact if [Formula: see text] is collinear with Reeb vector field. Second, we proved that a [Formula: see text]-paracontact manifold admitting a Yamabe gradient soliton is of constant curvature [Formula: see text] when [Formula: see text] and for [Formula: see text], the soliton is trivial and the manifold has constant scalar curvature. Moreover, for a paraSasakian manifold admitting a Yamabe soliton, we show that it has constant scalar curvature and [Formula: see text] is Killing when [Formula: see text]. Finally, we consider a paracontact metric [Formula: see text]-manifold with a non-trivial almost Yamabe gradient soliton. In the end, we construct two examples of paracontact metric manifolds with an almost Yamabe soliton.

Certain Curvature Conditions on $(k,\mu )$-Paracontact Metric Spaces

The aim of this paper is to classify $(k,\mu)$-paracontact metric spaces satisfying certain curvature conditions. We present the curvature tensors of (k,$\mu $)-Paracontact manifold satisfying the conditions $R\cdot W_{6}=0$, $ R\cdot W_{7}=0$, $R\cdot W_{8}=0$ and $R\cdot W_{9}=0$. According these cases, $(k,\mu)$-Paracontact manifolds have been characterized. Also, several results are obtained.

Some results on pseudosymmetric normal paracontact metric manifolds

TIn this article, the M-projective and Weyl curvature tensors on a normal paracontact metric manifold are discussed. For normal paracontact metric manifolds, pseudosymmetric cases are investigated and some interesting results are obtained. We show that a semisymmetric normal paracontact manifold is of constant sectional curvature. We also obtain that a pseudosymmetric normal paracontact metric manifold is an $\eta$-Einstein manifold. Finally, we support our topic with an example.

Conformal Ricci soliton and almost conformal Ricci soliton in paracontact geometry

In this paper, we study conformal Ricci soliton and almost conformal Ricci soliton within the framework of paracontact manifolds. Here, we have shown the characteristics of the soliton vector field and the nature of the manifold if para-Sasakian metric satisfies conformal Ricci soliton. We also demonstrate the feature of the soliton vector field V and scalar curvature when the para-Sasakian manifold admitting conformal Ricci soliton and vector field is pointwise collinear with the characteristic vector field [Formula: see text]. Next, we prove that if a K-paracontact manifold confesses a gradient conformal Ricci soliton, then it is Einstein. Next, we show that a para-Sasakian metric reveals with an almost conformal Ricci soliton that is either Einstein or [Formula: see text]-Einstein metric if the soliton vector field V is an infinitesimal contact transformation. Lastly, we decorate an example of conformal Ricci soliton on para-Sasakian manifold.

M-quasi Einstein Metric and Paracontact Geometry

The object of the upcoming article is to characterize paracontact metric manifolds conceding $m$-quasi Einstein metric. First we establish that if the metric $g$ in a $K$-paracontact manifold is the $m$-quasi Einstein metric, then the manifold is of constant scalar curvature. Furthermore, we classify $(k,\mu)$-paracontact metric manifolds whose metric is $m$-quasi Einstein metric. Finally, we construct a non-trivial example of such a manifold.

(k,μ)-Paracontact Manifolds and Their Curvature Classification

The aim of this paper is to study (k,μ)-Paracontact metric manifold. We introduce the curvature tensors of a (k,μ)-paracontact metric manifold satisfying the conditions R⋅P_*=0, R⋅L=0, R⋅W_1=0, R⋅W_0=0 and R⋅M=0. According to these cases, (k,μ)-paracontact manifolds have been characterized such as η-Einstein and Einstein. We get the necessary and sufficient conditions of a (k,μ)-paracontact metric manifold to be η-Einstein. Also, we consider new conclusions of a (k,μ)-paracontact metric manifold contribute to geometry. We think that some interesting results on a (k,μ)-paracontact metric manifold are obtained.

A NEW TYPE LORENTZIAN ALMOST PARA CONTACT MANIFOLD

The present study initially introduced a new type Lorentzian almost para contact manifold using the generalized symmetric metric connections of type $(\alpha,\beta)$. Later, some results is given about new type Lorentzian almost para contact manifold.
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A study on K-paracontact and (k,μ)-paracontact manifold admitting vanishing Cotton tensor and Bach tensor

"The object of the present paper is to study K-paracontact manifold admitting parallel Cotton tensor, vanishing Cotton tensor and to study Bach flatness on K-paracontact manifold. In that we prove for a K-paracontact metric manifold M^{2n+1} has parallel Cotton tensor if and only if M^{2n+1} is an η-Einstein manifold and r=-2n(2n+1). Further we show that if g is an η-Einstein K-paracontact metric and if g is Bach flat then g is an Einstein. Also we study vanishing Cotton tensor on (k,μ)-paracontact manifold for both k>-1 and k<-1. Finally, we prove that if M^{2n+1} is a (k,μ)-paracontact manifold for k ≠ -1 and if M^{2n+1} has vanishing Cotton tensor for μ ≠ k, then M^{2n+1} is an η-Einstein manifold."

Static perfect fluid space-time and paracontact metric geometry

The main purpose of this paper is to study and explore some characteristics of static perfect fluid space-time on paracontact metric manifolds. First, we show that if a [Formula: see text]-paracontact manifold [Formula: see text] is the spatial factor of a static perfect fluid space-time, then [Formula: see text] is of constant scalar curvature [Formula: see text] and squared norm of the Ricci operator is given by [Formula: see text]. Next, we prove that if a [Formula: see text]-paracontact metric manifold [Formula: see text] with [Formula: see text] is a spatial factor of static perfect space-time, then for [Formula: see text], [Formula: see text] is flat, and for [Formula: see text], [Formula: see text] is locally isometric to the product of a flat [Formula: see text]-dimensional manifold and an [Formula: see text]-dimensional manifold of constant negative curvature [Formula: see text]. Further, we prove that if a paracontact metric 3-manifold [Formula: see text] with [Formula: see text] is a spatial factor of static perfect space-time, then [Formula: see text] is an Einstein manifold. Finally, a suitable example has been constructed to show the existence of static perfect fluid space-time on paracontact metric manifold.

Study of Some Properties on Almost Para Contact Metric Manifolds with Certain Connection

Many differential geometers studied different types of manifolds with a semi-symmetric metric connection. In the present paper, we study semi-symmetric nonmetric connections on an almost para-contact manifold in relation to semi-symmetric nonmetric connections. In another section of the work, we have studied the curvature tensor and Nijenhuis tensor.

On a (k,mu)-paracontact manifold satisfying some conditions on the projective curvature tensor

In the present paper, we study the curvature tensors of (k,mu)-paracontact manifold satisfying the conditions P(ξ ,Y)Z=0, P(ξ ,Y)P=0, P(ξ ,Y)S =0 and (ξ,Y)R=0. According these cases, (k,μ)-paracontact manifolds have been characterized. We obtain the necessary and sufficient conditions of a (k,μ)-paracontact manifold to be η-Einsteins.

Optimal inequalities for submanifolds in an $({varepsilon})$-almost para-contact manifolds

‎The present research paper is concerned about a couple of optimal inequalities for the Casorati curvature of submanifolds in an $({varepsilon})$-almost para-contact manifolds precisely $(varepsilon)$-Kenmotsu manifolds endowed with semi-symmetric metric connection (briefly says $SSM$) by adopting the T‎. ‎Opreachr('39')s optimization technique.‎

On Lorentezian almost Para-contact Manifold

In the present paper, we focus on Lorentzian almost para-contact manifold and explain their relationship. In section 1, we have introduced the historical background of a contact manifold. Next, in section 2, we have studied the basic formulae of the Lorentzian metric manifold. Further, in section 3, we introduced a new tensor field h as well as calculated some theorems and lemma. Now in section 4, we have investigated curvature properties and their relationship with the Lorentzian almost para-contact manifold. In the end section, we have discussed the entire work

Characterizations for totally geodesic submanifolds of a $ K $-paracontact manifold

The aim of the present paper is to study pseudoparallel invariant submanifolds of a $ K $-paracontact metric manifold. We consider pseudoparallel, Ricci-generalized pseudoparallel and 2-Ricci generalized pseudo parallel invariant submanifolds of a $ K $-paracontact manifold and we obtain new results. We think contributes to providing some new and interesting results in the area of geometric structures on manifolds geometry.

A note on concircular curvature tensor in Lorentzian almost para-contact geometry

The paper deals with the notion of different classes of concircular curvature tensor on Lorentzian almost para-contact manifolds admitting a quarter-symmetric metric connection. In this paper we study Lorentzian almost para-contact manifolds with respect to the quarter-symmetric metric connection satisfying the curvature condition Z.S = 0. We also investigate the properties of ξ−concircularly flat, φ−concircularly flat and quasi-concircularly flat Lorentzian almost para-contact manifolds admitting a quarter-symmetric metric connection and it is found that in each of above cases the manifold is generalized η−Einstein manifold.

Almost quasi-Yamabe solitons and gradient almost quasi-Yamabe solitons in paracontact geometry

The purpose of the present paper is to investigate the almost quasi-Yamabe soliton and gradient almost quasi-Yamabe solitons under the framework of three-dimensional normal almost paracontact metric manifolds.

Almost Paracontact 3-Submersions

In this paper, we discuss some geometric properties of Riemannian submersions whose total space is an almost paracontact manifold with 3-structure. The study is focused on the transference of structures, the geometry of the fibres and sectional curvature tensor.

Ricci Solitons and Paracontact Geometry

In this article, first we prove that if a metric of a para-Sasakian manifold is a Ricci soliton, then either it is an Einstein (and V Killing) or a $$\eta $$-Einstein (and V leaves $$\varphi $$ invariant) manifold. Next, we prove that if a K-paracontact metric g is a gradient Ricci soliton, then it becomes a expanding soliton which is Einstein with constant scalar curvature. Further, we study the Ricci soliton where the potential vector field V is point wise collinear with the Reeb vector field on paracontact manifold. Finally, we consider the gradient Ricci soliton on $$(\kappa ,\mu )$$-paracontact manifold.