Characteristic Vector Field Research Articles

Solitons of η-Ricci–Bourguignon Type on Submanifolds in (LCS)m Manifolds

In this research article, we concentrate on the exploration of submanifolds in an (LCS)m-manifold B˜. We examine these submanifolds in the context of two distinct vector fields, namely, the characteristic vector field and the concurrent vector field. Initially, we consider some classifications of η-Ricci–Bourguignon (in short, η-RB) solitons on both invariant and anti-invariant submanifolds of B˜ employing the characteristic vector field. We establish several significant findings through this process. Furthermore, we investigate additional results by using η-RB solitons on invariant submanifolds of B˜ with concurrent vector fields, and discuss a supporting example.

Legendre magnetic flows for totally η-umbilic real hypersurfaces in a complex hyperbolic space

We study trajectories for Sasakian magnetic fields on horospheres, on geodesic spheres and on tubes around totally geodesic complex hypersurfaces in a complex hyperbolic space. Considering the subbundle formed by unit tangent vectors orthogonal to the characteristic vector field, flows associated with trajectories on this subbundle are smoothly conjugate to each other for each geodesic sphere, and are classified into two and three classes for a horosphere and for each tube, respectively.

Lightlike Hypersurfaces of an Indefinite Kaehler Manifold with an $(\ell,\,m)$-type Metric Connection

Jin introduced a non-symmetric metric connection, called an {\it $(\ell,m)$-type metric connection} \cite{Jin1, Jin2}. There are two examples of $(\ell, m)$-type: a semi-symmetric metric connection when ${\ell}=1$ and $m=0$ and a quater-symmetric connection for ${\ell}=0$ and $m=1$ . Our purpose is to investigate lightlike hypersurfaces of an indefinite (complex) Kaehler manifolds with an $(\ell,m)$-type metric connection under the tangent characteristic vector field on such hypersurfaces.

Invariant contact metric structures on tangent sphere bundles of compact symmetric spaces

A new characterization is provided for the class of compact rank-one symmetric spaces. Such spaces are the only symmetric spaces of compact type for which the standard vector field xi ^{S} on their sphere bundles is Killing with respect to some invariant Riemannian metric. The set of all these metrics is determined, as well as the set of all those invariant contact metric structures with characteristic vector field xi ^{S}. Moreover, on tangent sphere bundles of compact symmetric spaces with rank greater than or equal to two, a family of invariant contact metric structures, which contains the standard structure, is obtained.

Pseudo-symmetric almost cosymplectic 3-manifolds

In this paper, we study the semi-symmetry and pseudo-symmetry of almost cosymplectic [Formula: see text]-manifolds. First, we prove that an [Formula: see text]-almost cosymplectic [Formula: see text]-manifold [Formula: see text] is semi-symmetric if and only if it is cosymplectic. Here by an [Formula: see text]-almost cosymplectic [Formula: see text]-manifold, we mean an almost cosymplectic [Formula: see text]-manifold whose characteristic vector field [Formula: see text] is a harmonic unit vector field. If an almost cosymplectic [Formula: see text]-manifold [Formula: see text] whose fundamental endomorphism field [Formula: see text] is parallel in the direction of the characteristic vector field [Formula: see text] ([Formula: see text]), then it is [Formula: see text]-almost cosymplectic. In particular, an almost cosymplectic [Formula: see text]-manifold [Formula: see text] satisfying [Formula: see text] is semi-symmetric if and only if it is cosymplectic. Next, we prove that pseudo-symmetric [Formula: see text]-almost cosymplectic [Formula: see text]-manifolds are certain generalized almost cosymplectic [Formula: see text]-spaces.

Critical point equation within the framework of various contact metric manifolds

The aim of the present paper is to study the critical point equation (shortly CPE) conjecture within the framework of various contact metric manifolds. First we establish that Kenmotsu manifold satisfying the CPE either becomes an Einstein manifold or the derivative of potential function along characteristic vector field satisfy a certain relation on the distribution of η. Next we study CPE on (κ, μ)'-almost Kenmotsu manifold and obtain that the manifold is Einstein. Later in case of 3-dimensional trans-Sasakian manifold, we get that either the manifold becomes α-Sasakian or it becomes Einstein. Finally we give examples of 3-dimensional trans-Sasakian manifold and (κ ,μ)'-almost Kenmotsu manifold to verify our outcomes.

Real hypersurfaces foliated by totally real totally geodesic submanifolds

We bridge between submanifold geometry and curve theory. In the first half of this paper we classify real hypersurfaces in a complex projective plane and a complex hyperbolic plane all of whose integral curves γ of the characteristic vector field are totally real circles of the same curvature which is independent of the choice of γ in these planes. In the latter half, we construct real hypersurfaces which are foliated by totally real (Lagrangian) totally geodesic submanifolds in a complex hyperbolic plane, which provide one of the examples obtained in the classification.

Almost Kenmotsu Manifolds

The object of this paper is to study generalized φ-recurrent almost Kenmotsu manifolds with characteristic vector field ξ belonging to (k, µ)-nullity distribution. We have showed that these manifolds reduce to Kenmotsu manifolds with scalar curvature-1. Further we establish the relations among the associated 1-forms and proved the conditions under which gradient Ricci almost soliton reduce to gradient Ricci soliton.

Characteristics of Sasakian Manifolds Admitting Almost ∗-Ricci Solitons

This article presents some results of a geometric classification of Sasakian manifolds (SM) that admit an almost ∗-Ricci soliton (RS) structure (g,ω,X). First, we show that a complete SM equipped with an almost ∗-RS with ω≠ const is a unit sphere. Then we prove that if an SM has an almost ∗-RS structure, whose potential vector is a Jacobi vector field on the integral curves of the characteristic vector field, then the manifold is a null or positive SM. Finally, we characterize those SM represented as almost ∗-RS, which are ∗-RS, ∗-Einstein or ∗-Ricci flat.

Characterization of Biharmonic Hypersurface

The main purpose of this paper is to study biharmonic hypersurface in a quasi-paraSasakian manifold $\mathbb{Q}^{2m+1}$. Biharmonic hypersurfaces are special cases of biharmonic maps and biharmonic maps are the critical points of the bienergy functional. The condition of biharmonicity for non-degenerate hypersurfaces in $\mathbb{Q}^{2m+1}$ is investigated for both cases: either the characteristic vector field of $\mathbb{Q}^{2m+1}$ is the unit normal vector field to the hypersurface or it belongs to the tangent space of the hypersurface. Some relevant examples are also illustrated.

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.

Sasakian Structures on Tangent Sphere Bundles of Compact Rank-One Symmetric Spaces

A positive answer is given to the existence of Sasakian structures on the tangent sphere bundle of some Riemannian manifold whose sectional curvature is not constant. Among other results, it is proved that the tangent sphere bundle T_{r}(G/K), for any r> 0, of a compact rank-one symmetric space G/K, not necessarily of constant sectional curvature, admits a unique G-invariant K-contact structure whose characteristic vector field is the standard field of T(G/K). Such a structure is in fact Sasakian and it can be expressed as an induced structure from an almost Hermitian structure on the punctured tangent bundle T(G/K){setminus } {text{ zero } text{ section }}.

Generic lightlike submanifolds of an indefinite Kaehler manifold with an $(\ell, m)$-type metric connection

We study generic lightlike submanifolds $M$ of an indefinite Kaehler manifold $\bar{M}$ or an indefinite complex space form $\bar{M}(c)$ with an $(\ell, m)$-type metric connection subject such that the characteristic vector field $\zeta$ of $\bar{M}$ belongs to our screen distribution $S(TM)$ of $M$.

BAZI LİE CEBİRLERİ ÜZERİNDE HEMEN-HEMEN KONTAKT YAPILAR

In this manuscript, we show that there are no almost contact structures with parallel characteristic vector field on certain 7 dimensional Lie algebras over the real field.

CERTAIN RESULTS ON $\eta$-RICCI SOLITIONS AND ALMOST $\eta$-RICCI SOLITONS

We prove that if an $\eta$-Einstein para-Kenmotsu manifold admits a $\eta$-Ricci soliton then it is Einstein. Next, we proved that a para-Kenmotsu metric as a $\eta$-Ricci soliton is Einstein if its potential vector field $V$ is infinitesimal paracontact transformation or collinear with the Reeb vector field. Further, we prove that if a para-Kenmotsu manifold admits a gradient almost $\eta$-Ricci soliton and the Reeb vector field leaves the scalar curvature invariant then it is Einstein. We also construct an example of para-Kenmotsu manifold that admits $\eta$-Ricci soliton and satisfy our results. We also have studied $\eta$-Ricci soliton in 3-dimensional normal almost paracontact metric manifolds and we show that if in a 3-dimensional normal almost paracontact metric manifold with $\alpha, \beta $ = constant, the metric is $\eta$-Ricci soliton, where potential vector field $V$ is collinear with the characteristic vector field $\xi$, then the manifold is $\eta$-Einstein manifold.

On Quasi-Conformally Flat Generalized Sasakian-Space Forms

In this paper, we classify quasi-conformally flat generalized Sasakian-space forms under the assumption that the characteristic vector field is Killing. Also we classify quasi-conformally Weyl-symmetric generalized Sasakian-space forms.

Quotient of the Euler system on one class of curves

We consider the Euler system describing a one-dimensional inviscid flows in space along curves of a certain class. Using differential invariants for the Euler system, we obtain its quotient equation. The solutions of the quotient equation that are constant along characteristic vector field provide some solutions of the Euler system. We discuss solving the quotient using asymptotic expansions of unknown functions and virial expansion of thermodynamic state equations. Thus the quotient is reduced to a series of ODE systems.

Some Results on D-Homothetic Deformation On Almost Paracontact Metric Manifolds

In this paper, we investigate the effect of $\mathcal{D}$-homothetic deformation on almost para-contact metric manifolds. The main results of the paper are about some classes of almost paracontact metric manifolds in which the characteristic vector field is parallel. It is shown that certain classes are invariant under the $\mathcal{D}$-homothetic deformation.

A New Class of Contact Pseudo Framed Manifolds with Applications

In this paper, we introduce a new class of contact pseudo framed (CPF)-manifoldsM,g,f,λ,ξby a real tensor fieldfof type1,1, a real functionλsuch thatf3=λ2fwhereξis its characteristic vector field. We prove in our main Theorem 2 thatMadmits a closed 2-formΩifλis constant. In 1976, Blair proved that the vector fieldξof a normal contact manifold is Killing. Contrary to this, we have shown in Theorem 2 that, in general,ξof a normal CPF-manifold is non-Killing. We also have established a link of CPF-hypersurfaces with curvature, affine, conformal collineations symmetries, and almost Ricci soliton manifolds, supported by three applications. Contrary to the odd-dimensional contact manifolds, we construct several examples of even- and odd-dimensional semi-Riemannian and lightlike CPF-manifolds and propose two problems for further consideration.