Reeb Vector Field Research Articles

On curvature-adapted hypersurfaces of 𝕊2 × 𝕊2 and ℍ2 × ℍ2

In this paper, we first give the characterizations of curvature-adapted hypersurfaces in both of the two Kähler surfaces [Formula: see text] and [Formula: see text]. Then, focusing on two subclasses of the curvature-adapted hypersurfaces in [Formula: see text] and [Formula: see text], we classify the Hopf hypersurfaces with two distinct principal curvatures as well as the Hopf hypersurfaces whose Reeb vector field [Formula: see text] and Reeb function [Formula: see text] satisfying [Formula: see text].

Geodesic vector fields, induced contact structures and tightness in dimension three

In this paper, we provide new and simpler proofs of two theorems of Gluck and Harrison on contact structures induced by great circle or line fibrations. Furthermore, we prove that a geodesic vector field whose Jacobi tensor is parallel along flow lines (e.g. if the underlying manifold is locally symmetric) induces a contact structure if the ‘mixed’ sectional curvatures are nonnegative, and if a certain nondegeneracy condition holds. Additionally, we prove that in dimension three, contact structures admitting a Reeb flow which is either periodic, isometric, or free and proper, must be universally tight. In particular, we generalise an earlier result of Geiges and the author, by showing that every contact form on R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^3$$\\end{document} whose Reeb vector field spans a line fibration is necessarily tight. Furthermore, we provide a characterisation of isometric Reeb vector fields. As an application, we recover a result of Kegel and Lange on Seifert fibrations spanned by Reeb vector fields, and we classify closed contact 3-manifolds with isometric Reeb flows (also known as R-contact manifolds) up to diffeomorphism.

Generalized Bach solitons on (κ,μ)′-almost-Kenmotsu manifolds and its applications in perfect fluid spacetimes

The principal aim of this paper is to find the expression of *-Bach tensors on [Formula: see text]-almost-Kenmotsu manifolds. It is shown that if the metric of a [Formula: see text]-almost-Kenmotsu manifold admits a generalized closed *-Bach soliton, then the metric is *-Bach flat under certain restrictions. We also prove that the metric of a generalized gradient *-Bach soliton on a [Formula: see text]-almost-Kenmotsu manifold is *-Bach flat, for a soliton function which is invariant under the Reeb vector field. Further, we characterize generalized *-Bach solitons with trace-free *-Bach tensors on [Formula: see text]-almost-Kenmotsu manifolds and results are verified by an example. As an application of these solitons, we study generalized almost-Bach solitons whose metrics are associated with the Lorentzian metric of the perfect fluid spacetimes.

On K-contact metric manifolds satisfying an almost gradient Ricci–Bourguignon soliton

The main purpose of the paper is to study an almost gradient Ricci–Bourguignon soliton (RB soliton) within the framework of [Formula: see text]-contact manifolds and [Formula: see text]-contact manifolds. First, we prove that if complete [Formula: see text]-contact manifold endows a gradient RB soliton, then the manifold is compact Sasakian and isometric to unit sphere [Formula: see text]. Next, we show that if a complete contact metric satisfies an almost RB soliton with a non-zero potential vector field is collinear with the Reeb vector field [Formula: see text] and the Reeb vector field [Formula: see text] acting as an eigenvector of the Ricci operator, then it is compact Einstein Sasakian and the potential vector field is a constant multiple of the Reeb vector field [Formula: see text]. Lastly, we prove that if the metric of a non-Sasakian [Formula: see text]-contact manifold is an almost gradient RB soliton, then it is flat in dimension 3 and in higher dimensions it is locally isometric to [Formula: see text].

Conformally flat almost cosymplectic 3-manifolds

In this paper, we consider an almost cosymplectic 3-manifold M such that its scalar curvature is invariant along the Reeb vector field. We prove that if the Reeb vector field of M is harmonic, then M is conformally flat if and only if it is locally isometric to the product ℝ × N 2(c), where N 2(c) is a Kähler surface of constant sectional curvature c. Almost cosymplectic 3-manifolds on which the Reeb vector field satisfies the h-a condition together with conformal flatness are also classified.

A New Representation for Slant Curves in Sasakian 3-Manifolds

In this paper, we define a new kind of curve called $N$-slant curve whose principal normal vector field makes a constant angle with the Reeb vector field $\xi$ in Sasakian $3$-manifolds. Then, we give some characterizations of $N$-slant curves in Sasakian $3$-manifolds and we obtain some properties of the curves in $\mathbb{R}^{3}(-3)$. Moreover, we investigate the conditions of $C$-parallel and $C$-proper mean curvature vector fields along $N$-slant curves in Sasakian $3$-manifolds. Finally, we study $N$-slant curves of type $AW(k)$ where k=1,2 or 3.

Almost Ricci–Yamabe soliton on contact metric manifolds

PurposeThis paper aims to study almost Ricci–Yamabe soliton in the context of certain contact metric manifolds.Design/methodology/approachThe paper is designed as follows: In Section 3, a complete contact metric manifold with the Reeb vector field ξ as an eigenvector of the Ricci operator admitting almost Ricci–Yamabe soliton is considered. In Section 4, a complete K-contact manifold admits gradient Ricci–Yamabe soliton is studied. Then in Section 5, gradient almost Ricci–Yamabe soliton in non-Sasakian (k, μ)-contact metric manifold is assumed. Moreover, the obtained result is verified by constructing an example.FindingsWe prove that if the metric g admits an almost (α, β)-Ricci–Yamabe soliton with α ≠ 0 and potential vector field collinear with the Reeb vector field ξ on a complete contact metric manifold with the Reeb vector field ξ as an eigenvector of the Ricci operator, then the manifold is compact Einstein Sasakian and the potential vector field is a constant multiple of the Reeb vector field ξ. For the case of complete K-contact, we found that it is isometric to unit sphere S2n+1 and in the case of (k, μ)-contact metric manifold, it is flat in three-dimension and locally isometric to En+1 × Sn(4) in higher dimension.Originality/valueAll results are novel and generalizations of previously obtained results.

Some Chen Inequalities for Submanifolds in Trans-Sasakian Manifolds Admitting a Semi-Symmetric Non-Metric Connection

In the present article, we study submanifolds tangent to the Reeb vector field in trans-Sasakian manifolds. We prove Chen’s first inequality and the Chen–Ricci inequality, respectively, for such submanifolds in trans-Sasakian manifolds which admit a semi-symmetric non-metric connection. Moreover, a generalized Euler inequality for special contact slant submanifolds in trans-Sasakian manifolds endowed with a semi-symmetric non-metric connection is obtained.

A Study on Contact Metric Manifolds Admitting a Type of Solitons

The principal aim of the present article is to characterize certain properties of η‐Ricci–Bourguignon solitons on three types of contact manifolds, that are K‐contact manifolds, (κ, μ)‐contact metric manifolds, and N(κ)‐contact metric manifolds. It is shown that if a K‐contact manifold admits an η‐Ricci–Bourguignon soliton whose potential vector field is the Reeb vector field, then the scalar curvature of the manifold is constant and the manifold becomes an η‐Einstein manifold. In this regard, it is shown that if a K‐contact manifold admits a gradient η‐Ricci–Bourguignon soliton, then the scalar curvature is a constant and either the soliton reduces to gradient Ricci–Bourguignon soliton, or the potential function is a constant. We also deduce that if a (κ, μ)‐contact metric manifold admits a gradient η‐Ricci–Bourguignon soliton, then either the manifold is locally En+1 × Sn(4), or the gradient of the potential function is pointwise collinear with the Reeb vector field, or the potential function is a constant. η‐Ricci–Bourguignon solitons on three‐dimensional N(κ)‐contact metric manifolds are also considered. Finally, an example using differential equations has been constructed to verify a result.

Almost *-Ricci solitons on contact strongly pseudo-convex integrable CR-manifolds

We prove that if contact strongly pseudo-convex integrable CR-manifold admits a *-Ricci soliton where the soliton vector Z is contact, then the Reeb vector field ? is an eigenvector of the Ricci operator at each point if and only if ? is constant. Then we study contact strongly pseudo-convex integrable CR-manifold such that 1 is a almost *-Ricci soliton with potential vector field Z collinear with ?. To this end, we prove that if a 3-dimensional contact metric manifoldMwith Q? = ?Q which admits a gradient almost *-Ricci soliton, then either M is flat or f is constant.

Generic Submersions in Contact Geometry

In the present paper, we introduce a new type of Riemannian submersion in the contact framework such that the fibers of such submersion are generic submanifolds, as given in [10]. This type of submersion is a generalization of many kinds of submersion introduced before in the literature. Once the Reeb vector field \xi is tangent to the fibers, its position is given such that it should lie in the anti-invariant distribution D^0, which is given in the definition of the generic submersion. Moreover, we give an example and some results for such submersions.

Geometry of almost contact metrics as almost ∗-Ricci solitons

In this paper, we give some characterizations by considering ∗-Ricci soliton as a Kenmotsu metric. We prove that if a Kenmotsu manifold represents an almost ∗-Ricci soliton and the potential vector field [Formula: see text] is a Jacobi along the Reeb vector field, then it is a steady ∗-Ricci soliton. Next, we show that a Kenmotsu matric endowed an almost ∗-Ricci soliton which is Einstein metric if it is [Formula: see text]-Einstein or the potential vector field [Formula: see text] is collinear to the Reeb vector field or [Formula: see text] is an infinitesimal contact transformation.

Intrinsic sub-Laplacian for hypersurface in a contact sub-Riemannian manifold

We construct and study the intrinsic sub-Laplacian, defined outside the set of characteristic points, for a smooth hypersurface embedded in a contact sub-Riemannian manifold. We prove that, away from characteristic points, the intrinsic sub-Laplacian arises as the limit of Laplace–Beltrami operators built by means of Riemannian approximations to the sub-Riemannian structure using the Reeb vector field. We carefully analyse three families of model cases for this setting obtained by considering canonical hypersurfaces embedded in model spaces for contact sub-Riemannian manifolds. In these model cases, we show that the intrinsic sub-Laplacian is stochastically complete and in particular, that the stochastic process induced by the intrinsic sub-Laplacian almost surely does not hit characteristic points.

Weak Nearly Sasakian and Weak Nearly Cosymplectic Manifolds

Weak contact metric structures on a smooth manifold, introduced by V. Rovenski and R. Wolak in 2022, have provided new insight into the theory of classical structures. In this paper, we define new structures of this kind (called weak nearly Sasakian and weak nearly cosymplectic and nearly Kähler structures), study their geometry and give applications to Killing vector fields. We introduce weak nearly Kähler manifolds (generalizing nearly Kähler manifolds), characterize weak nearly Sasakian and weak nearly cosymplectic hypersurfaces in such Riemannian manifolds and prove that a weak nearly cosymplectic manifold with parallel Reeb vector field is locally the Riemannian product of a real line and a weak nearly Kähler manifold.

Classification of Killing magnetic curves in ℍ3

In this paper, we study classification of magnetic curves corresponding to Killing vector fields of [Formula: see text]. First, we solve the geodesic equation analytically. Then we calculate the trajectories generated by all the six Killing vector fields, which are considered as magnetic field vectors, by using perturbation method up to first-order with respect to the strength of the magnetic field. We present a comparison of our solution with the numerical solution for one case. We also prove that 3-dimensional [Formula: see text]-Kenmotsu manifolds cannot have any magnetic vector field in the direction of their Reeb vector fields.

On the dimension of the algebras of local infinitesimal isometries of 3-dimensional special sub-Riemannian manifolds

Suppose that we are given a contact sub-Riemannian manifold (M, H, g) of dimension 3 such that the Reeb vector field is an infinitesimal isometry (such manifolds will be referred to as special). For a point qin M denote by mathfrak {i}^*(q) the Lie algebra of germs at q of infinitesimal isometries of (M, H, g). It is proved that for a generic point qin M, dim mathfrak {i}^*(q) can only assume the values 1, 2, 4. Moreover, dim mathfrak {i}^*(q) = 4 if and only if the curvature function determined by the canonical sub-Riemannian connection is constant in a neighborhood of q. The latter case is possible if (M, H, g) is locally isometric, in a neighborhood of q, to a left-invariant sub-Riemannian structure on a 3-dimensional Lie group.

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 Lie systems: theory and applications

A Lie system is a time-dependent system of differential equations describing the integral curves of a time-dependent vector field that can be considered as a curve in a finite-dimensional Lie algebra of vector fields V. We call V a Vessiot–Guldberg Lie algebra. We define and analyse contact Lie systems, namely Lie systems admitting a Vessiot–Guldberg Lie algebra of Hamiltonian vector fields relative to a contact manifold. We also study contact Lie systems of Liouville type, which are invariant relative to the flow of a Reeb vector field. Liouville theorems, contact Marsden–Weinstein reductions, and Gromov non-squeezing theorems are developed and applied to contact Lie systems. Contact Lie systems on three-dimensional Lie groups with Vessiot–Guldberg Lie algebras of right-invariant vector fields and associated with left-invariant contact forms are classified. Our results are illustrated with examples having relevant physical and mathematical applications, e.g. Schwarz equations, Brockett systems, quantum mechanical systems, etc. Finally, a Poisson coalgebra method to derive superposition rules for contact Lie systems of Liouville type is developed.

Pairs of Associated Yamabe Almost Solitons with Vertical Potential on Almost Contact Complex Riemannian Manifolds

Almost contact complex Riemannian manifolds, also known as almost contact B-metric manifolds, are, in principle, equipped with a pair of mutually associated pseudo-Riemannian metrics. Each of these metrics is specialized as a Yamabe almost soliton with a potential collinear to the Reeb vector field. The resulting manifolds are then investigated in two important cases with geometric significance. The first is when the manifold is of Sasaki-like type, i.e., its complex cone is a holomorphic complex Riemannian manifold (also called a Kähler–Norden manifold). The second case is when the soliton potential is torse-forming, i.e., it satisfies a certain recurrence condition for its covariant derivative with respect to the Levi–Civita connection of the corresponding metric. The studied solitons are characterized. In the three-dimensional case, an explicit example is constructed, and the properties obtained in the theoretical part are confirmed.

Variational characterizations of invariant submanifolds in Sasaki manifolds

In this note, we generalize our previous results in [4] and [5] from Kähler manifold to Sasaki manifold. We prove that for a submanifold tangent to the Reeb vector field in a Sasaki manifold, the area functional, as a functional of the ambient metrics, is stationary if and only if it is an invariant submanifold. We also show that if the submanifold has dimension 3, then it is basic stable if and only if it is an invariant submanifold.