para-Sasakian Manifolds Research Articles

ON QUASI BI-SLANT RIEMANNIAN MAPS FROM LORENTZIAN PARA SASAKIAN MANIFOLDS

We first introduce quasi bi-slant Riemannian maps and study such Riemannian maps from Lorentzian para Sasakian manifolds into Riemannian manifolds. We give necessary and sufficient conditions for the integrability of the distributions which are involved in the definition of the quasi bi-slant Riemannian map and investigate their leaves. We also obtain totally geodesic conditions for such maps. Moreover, we provide some examples for this notion.

PARA-SASAKIAN MANIFOLD ADMITTING RICCI-YAMABE SOLITONS WITH QUARTER SYMMETRIC METRIC CONNECTION

In the year 2019, Guler and Crasmareanu [6] conducted an investigation into another geometric flow known as the Ricci-Yamabe map. This map is nothing but a scalar combination of the Ricci and the Yamabe flow [7]. The primary objective of the current paper is to provide a characterization of a Ricci Yamabe soliton on a para-Sasakian manifold [17]. To commence, we prove that a para-Sasakian manifold admits a nearly quasi-Einstein manifold. Moreover, we discuss whether such a manifold is shrinking, expanding, or steady. Subsequently, we generalize these findings to Ricci-Yamabe solitons on para-Sasakian manifolds equipped with a quarter symmetric metric connection. Lastly, we furnish an illustration of a three-dimensional para-Sasakian manifold admitting a Ricci-Yamabe soliton which satisfies our results.

On para-Sasakian manifold admitting Zamkovoy connection

The purpose of the present paper is to study some properties of para-Sasakian manifold admitting Zamkovoy connection. We obtain some interesting result on para-Sasakian manifold. It is shown that M-projectively flat para-Sasakian manifold is η-Einstein manifold.

Lorentzian Para-Sasakian Manifolds and *-Ricci Solitons

We study the properties of Lorentzian para-Sasakian manifolds endowed with ∗-Ricci solitons and gradient ∗-Ricci solitons. Finally, the existence of ∗-Ricci soliton on a 4-dimensional Lorentzian para-Sasakian manifold is proved by constructing a non-trivial example

Pseudoparallel Invariant Submanifolds of a Para-Sasakian Manifold

In this paper, invariant submanifolds of a para-Sasakian manifold have been studied. Some special
 submanifolds such as pseudoparallel, 2-pseudoparallel, Ricci generalized pseudoparallel, and 2-Ricci generalized pseudoparallel submanifolds of a para-Sasakian manifold have been considered. The necessary and sufficient conditions for an invariant submanifold to be totally geodesic under some conditions have been given.

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) .

Certain Results on the Lifts from an LP-Sasakian Manifold to Its Tangent Bundle Associated with a Quarter-Symmetric Metric Connection

The purpose of this study is to examine the complete lifts from the symmetric and concircular symmetric n-dimensional Lorentzian para-Sasakian manifolds (briefly, (LPS)n) to its tangent bundle TM associated with a Riemannian connection DC and a quarter-symmetric metric connection (QSMC) D¯C.

Properties of Anti-Invariant Submersions and Some Applications to Number Theory

In this article, we investigate anti-invariant Riemannian and Lagrangian submersions onto Riemannian manifolds from the Lorentzian para-Sasakian manifold. We demonstrate that, for these submersions, horizontal distributions are not integrable and their fibers are not totally geodesic. As a result, they are not totally geodesic maps. The harmonicity of such submersions is also examined. We specifically prove that they are not harmonic when the Reeb vector field is horizontal. Finally, we provide an illustration of our findings and mention some number-theoretic applications for the same submersions.

Contact pseudo-slant submanifolds of a para-Sasakian manifold according to type 1, type 2, type 3 cases

This paper aims to present work on contact pseudo-slant submanifolds of para-Sasakian manifolds. The study includes the definitions and some results on type 1, type 2, and type 3 contact pseudo-slant submanifolds. The results were interpreted by taking into account the parallelism and geodesicity of the tensors reduced to the submanifold. Additionally, minimal anti-invariant and invariant submanifolds were evaluated for the type 1, type 2, and type 3 cases of the tensors reduced to the submanifold.

On Quasi Hemi-Slant Submersions

The paper deals with the notion of quasi hemi-slant submersions from Lorentzian para Sasakian manifolds onto Riemannian manifolds. These submersions are generalization of hemi-slant submersions and semi-slant submersions. In this paper, we also study the geometry of leaves of distributions which are involved in the definition of the submersion. Further, we obtain the conditions for such distributions to be integrable and totally geodesic. Moreover, we also give the characterization theorems for proper quasi hemi-slant submersions and provide some examples of it.

Characterization of a paraSasakian manifold admitting Bach tensor

In the present article, our aim is to characterize Bach flat paraSasakian manifolds. It is established that a Bach flat paraSasakian manifold of dimension greater than three is of constant scalar curvature. Next, we prove that if the metric of a Bach flat paraSasakian manifold is a Yamabe soliton, then the soliton field becomes a Killing vector field. Finally, it is shown that a 3-dimensional Bach flat paraSasakian manifold is locally isometric to the hyperbolic space $H^{2n+1}(1)$.

On Para-Sasakian Manifold with Respect to the Schouten-van Kampen Connection

In the present paper, we have studied the curvature properties of the Schouten-van Kampen connection on the n-dimensional Para Sasakian manifold and obtained some new results. Also, we studied projective curvature tensor, concircular curvature tensor, and Nijenhuis tensor for the Para-Sasakian manifold with respect to the Schouten-van Kampen connection.

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.

Magnetic Frenet curves on para-Sasakian manifolds

The study of magnetic curves, seen as solutions of Lorentz equation, has been done mainly in 3-dimensional case, motivated by theoretical physics. Then it was extended in higher dimensions, as for instance in K?hlerian or Sasakian frame. This paper deals for the first time in literature with magnetic Frenet curves in higher dimensional paracontact context. Several classifications are provided here for different types of magnetic curves on para-Sasakian manifolds. Some relations between magnetic Frenet curves and Lorenz force are obtained on these spaces and examples of magnetic curves associated to paracontact magnetic fields are constructed. Some explicit equations of the paracontact magnetic curves on the classical para-Sasakian manifold (R2n+1, ?, ?, ?, 1) are given at the end.

Liftings from a para-Sasakian manifold to its tangent bundles

The purpose of the present paper is to study the liftings of a quarter symmetric non-metric connection from a para-Sasakian manifold to its tangent bundles. By liftings, some results of the curvature tensor, projective curvature tensor, concircular curvature tensor and conformal curvature tensor wrt a quarter symmetric non-metric connection in a P-Sasakian manifold to its tangent bundles are obtained.

Characterization of bi-slant submanifolds of paraSasakian manifold

This paper aims to present work on bi-slant submanifolds of paraSasakian manifold. The study includes the definitions and some results on Type 1, Type 2, and Type 3 slant submanifolds. We define bi-slant submanifolds and derive characterization results with some illustrative examples. Further, we also study the particular cases of bi-slant submanifolds named semi-slant and pseudo-slant submanifolds and derive some results. An example for each case is also constructed.

Certain results of κ-almost gradient Ricci-Bourguignon soliton on pseudo-Riemannian manifolds

The prime object in this article is to study κ-almost Ricci-Bourguignon soliton and κ-almost gradient Ricci-Bourguignon soliton within the framework of paracontact metric manifolds. Here, we realize some conditions under which a paracontact metric manifold admitting a κ-Ricci–Bourguignon almost soliton is Einstein (trivial) and η-Einstein. We also show that if a three dimensional para-Kenmotsu manifold M3 admitting a κ-almost gradient Ricci-Bourguignon soliton with a constant scalar curvature, then the soliton becomes an almost gradient Ricci-Bourguignon soliton whose soliton function is −Ω. We also characterize and find some notable results κ-almost gradient Ricci-Bourguignon soliton on para-Sasakian manifolds and para-cosymplectic manifolds.

Some Basic Inequalities on (ϵ)-Para Sasakian Manifold

We propose fundamental inequalities for contact pseudo-slant submanifolds of (ϵ)-para Sasakian space form employing generalized normalized δ-Casorati curvature. We characterize submanifolds for which equality cases hold and illustrate the main result with some applications. Further, we have considered a certain type of submanifold for a Ricci soliton and after computing its scalar curvature, developed an inequality to find correlations between intrinsic or extrinsic invariants.

Riemann solitons on para-Sasakian geometry

The goal of the present article is to investigate almost Riemann soliton and gradient almost Riemann soliton on 3-dimensional para-Sasakian manifolds. At first, it is proved that if $(g, Z,\lambda)$ is an almost Riemann soliton on a para-Sasakian manifold $M^3$, then it reduces to a Riemann soliton and $M^3$ is of constant sectional curvature $-1$, provided the soliton vector $Z$ has constant divergence. Besides these, we prove that if $Z$ is pointwise collinear with the characteristic vector field $\xi$, then $Z$ is a constant multiple of $\xi$ and the manifold is of constant sectional curvature $-1$. Moreover, the almost Riemann soliton is expanding. Furthermore, it is established that if a para-Sasakian manifold $M^3$ admits gradient almost Riemann soliton, then $M^3$ is locally isometric to the hyperbolic space $H^{3}(-1)$. Finally, we construct an example to justify some results of our paper.

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.