Showing papers on "Reeb vector field published in 2021"

Almost $$\eta $$ η -Ricci solitons on Kenmotsu manifolds

Abstract: We characterize the Einstein metrics in such broad classes of metrics as almost $$\eta $$ -Ricci solitons and $$\eta $$ -Ricci solitons on Kenmotsu manifolds, and generalize some known results. First, we prove that a Kenmotsu metric as an $$\eta $$ -Ricci soliton is Einstein metric if either it is $$\eta $$ -Einstein or the potential vector field V is an infinitesimal contact transformation or V is collinear to the Reeb vector field. Further, we prove that if a Kenmotsu manifold admits a gradient almost $$\eta $$ -Ricci soliton with a Reeb vector field leaving the scalar curvature invariant, then it is an Einstein manifold. Finally, we present new examples of $$\eta $$ -Ricci solitons and gradient $$\eta $$ -Ricci solitons, which illustrate our results.

Kenmotsu metric as conformal $\eta$-Ricci soliton

Abstract: The object of the present paper is to characterize the class of Kenmotsu manifolds which admits conformal $\eta$-Ricci soliton. Here, we have investigated the nature of the conformal $\eta$-Ricci soliton within the framework of Kenmotsu manifolds. It is shown that an $\eta$-Einstein Kenmotsu manifold admitting conformal $\eta$-Ricci soliton is an Einstein one. Moving further, we have considered gradient conformal $\eta$-Ricci soliton on Kenmotsu manifold and established a relation between the potential vector field and the Reeb vector field. Next, it is proved that under certain condition, a conformal $\eta$-Ricci soliton on Kenmotu manifolds under generalized D-conformal deformation remains invariant. Finally, we have constructed an example for the existence of conformal $\eta$-Ricci soliton on Kenmotsu manifold.

Para-Ricci-Like Solitons on Riemannian Manifolds with Almost Paracontact Structure and Almost Paracomplex Structure

Abstract: We introduce and study a new type of soliton with a potential Reeb vector field on Riemannian manifolds with an almost paracontact structure corresponding to an almost paracomplex structure. The special cases of para-Einstein-like, para-Sasaki-like and having a torse-forming Reeb vector field were considered. It was proved a necessary and sufficient condition for the manifold to admit a para-Ricci-like soliton, which is the structure that is para-Einstein-like. Explicit examples are provided in support of the proven statements.

The $k$-almost Yamabe solitons and contact metric manifolds

Abstract: We introduce the concept of a k-almost Yamabe soliton which extends naturally from Yamabe solitons. Our aim is to study the k-almost Yamabe soliton (g,V,k,λ) on a contact metric manifold M2n+1. Firstly, for a general contact metric manifold, it is proved that V is Killing if the potential vector field V is a contact vector field and that M is K-contact if V is collinear with Reeb vector field. Secondly, we prove that a compact K-contact manifold, admitting a k-almost Yamabe gradient soliton, is isometric to a standard unit sphere. Moreover, for a complete Sasakian manifold admitting a k-almost Yamabe soliton, we show that it is isometric to a standard unit sphere 𝕊2n+1(1) when n>1 and for n=1, M is also isometric to a standard unit sphere if it admits a closed k-almost Yamabe soliton. Finally, we consider a contact metric (κ,μ)-manifold with a nontrivial k-almost Yamabe gradient soliton and show that it is flat in dimension 3 and in higher dimension M is locally isometric to En+1×𝕊n(4). In the end, we construct two examples of contact metric manifolds with a k-almost Yamabe soliton.

Cotton solitons on almost Cokähler 3-manifolds

Abstract: Let (M3, g) be a three dimensional almost coKahler manifold such that the Reeb vector field ξ is an eigenvector field of the Ricci operator Q, i.e. Qξ = ρξ, where ρ is a smooth function on M. In th...

Ricci Curvature, Reeb Flows and Contact 3-Manifolds

Abstract: Given a contact 3-manifold, we consider the problem of when a given function can be realized as the Ricci curvature of a Reeb vector field for the contact structure. We will use topological tools to show that every admissible function can be realized as such Ricci curvature for a singular metric which is an honest compatible metric away from a measure zero set. However, we will see that resolving such singularities depends on contact topological data and is yet to be fully understood.

Sasaki–Ricci Flow and Deformations of Contact Action–Angle Coordinates on Spaces T1,1 and Yp,q

Abstract: In this paper, we are concerned with completely integrable Hamiltonian systems and generalized action–angle coordinates in the setting of contact geometry We investigate the deformations of the Sasaki–Einstein structures, keeping the Reeb vector field fixed, but changing the contact form We examine the modifications of the action–angle coordinates by the Sasaki–Ricci flow We then pass to the particular cases of the contact structures of the five-dimensional Sasaki–Einstein manifolds T1,1 and Yp,q

On non-gradient $$(m,\rho )$$-quasi-Einstein contact metric manifolds

Abstract: Many authors have studied Ricci solitons and their analogs within the framework of (almost) contact geometry. In this article, we thoroughly study the $$(m,\rho )$$ -quasi-Einstein structure on a contact metric manifold. First, we prove that if a K-contact or Sasakian manifold $$M^{2n+1}$$ admits a closed $$(m,\rho )$$ -quasi-Einstein structure, then it is an Einstein manifold of constant scalar curvature $$2n(2n+1)$$ , and for the particular case—a non-Sasakian $$(k,\mu )$$ -contact structure—it is locally isometric to the product of a Euclidean space $${\mathbb {R}}^{n+1}$$ and a sphere $$S^n$$ of constant curvature 4. Next, we prove that if a compact contact or H-contact metric manifold admits an $$(m,\rho )$$ -quasi-Einstein structure, whose potential vector field V is collinear to the Reeb vector field, then it is a K-contact $$\eta $$ -Einstein manifold.

Para-Ricci-like Solitons with Vertical Potential on Para-Sasaki-like Riemannian Π-Manifolds

Abstract: The objects of study are para-Ricci-like solitons on para-Sasaki-like, almost paracontact, almost paracomplex Riemannian manifolds, namely, Riemannian Π-manifolds. Different cases when the potential of the soliton is the Reeb vector field or pointwise collinear to it are considered. Some additional geometric properties of the constructed objects are proven. Results for a parallel symmetric second-order covariant tensor on the considered manifolds are obtained. An explicit example of dimension 5 in support of the given assertions is provided.

Conformal symplectic Weinstein conjecture and non-squeezing

Abstract: We study here, from the Gromov-Witten theory point of view, some aspects of rigidity of locally conformally symplectic manifolds, or $\lcs$ manifolds for short, which are a natural generalization of both contact and symplectic manifolds. In particular, we give a first known analogue of the classical Gromov non-squeezing in lcs geometry. Another possible version of non-squeezing related to contact non-squeezing is also discussed. In a different direction we study Gromov-Witten theory of the $\lcs$ manifold $C \times S ^{1} $ induced by a contact form $\lambda$ on $C$, and show that the extended Gromov-Witten invariant counting certain charged elliptic curves in $C \times S ^{1} $ is identified with the extended classical Fuller index of the Reeb vector field $R ^{\lambda} $, by extended we mean that these invariants can be $\pm \infty$-valued. Partly inspired by this, we conjecture existence of certain 1-d curves we call Reeb curves in certain $\lcs$ manifolds, which we call conformal symplectic Weinstein conjecture, and this is a direct extension of the classical Weinstein conjecture. Also using Gromov-Witten theory, we show that the CSW conjecture holds for a $C ^{3} $- neighborhood of the induced lcs form on $C \times S ^{1} $, for $C$ a contact manifold with contact form whose Reeb flow has non-zero extended Fuller index, e.g. $S ^{2k+1} $ with standard contact form, for which this index is $\pm \infty$. We also show that in some cases the failure of this conjecture implies existence of sky catastrophes for families of holomorphic curves in a $\lcs$ manifold. No examples of the latter phenomenon are known to exist, even in the un-tamed almost complex world, this phenomenon if it exists, would be analogous to sky catastrophes in dynamical systems discovered by Fuller.

Volume and Monge-Amp\`ere energy on polarized affine varieties

Abstract: Let $(X, \xi)$ be a polarized affine variety, i.e. an affine variety $X$ with a (possibly irrational) Reeb vector field $\xi$. We define the volume of a filtration of the coordinate ring of $X$ in terms of the asymptotics of the average of jumping numbers. When the filtration is finitely generated, it induces a Fubini-Study function $\varphi$ on the Berkovich analytification of $X$. In this case, we define the Monge-Ampere energy for $\varphi$ using the theory of forms and currents on Berkovich spaces developed by Chambert-Loir and Ducros, and show that it agrees with the volume of the filtration. In the special case when the filtration comes from a test configuration, we recover the functional defined by Collins-Szekelyhidi and Li-Xu.

Real hypersurfaces in the complex projective plane satisfying an equality involving $\delta(2)$

Abstract: It was proved in Chen's paper \cite{chen} that every real hypersurface in the complex projective plane of constant holomorphic sectional curvature $4$ satisfies $$ \delta(2)\leq \frac{9}{4}H^2+5,$$ where $H$ is the mean curvature and $\delta(2)$ is a $\delta$-invariant introduced by him. In this paper, we study non-Hopf real hypersurfaces satisfying the equality case of the inequality under the condition that the mean curvature is constant along each integral curve of the Reeb vector field. We describe how to obtain all such hypersurfaces.

Hypersurfaces of a Sasakian manifold - revisited

Abstract: We study orientable hypersurfaces in a Sasakian manifold. The structure vector field ξ of a Sasakian manifold determines a vector field v on a hypersurface that is the component of the Reeb vector field ξ tangential to the hypersurface, and it also gives rise to a smooth function σ on the hypersurface, namely the projection of ξ on the unit normal vector field N. Moreover, we have a second vector field tangent to the hypersurface, given by $\mathbf{u}=-\varphi (N)$ . In this paper, we first find a necessary and sufficient condition for a compact orientable hypersurface to be totally umbilical. Then, with the assumption that the vector field u is an eigenvector of the Laplace operator, we find a necessary condition for a compact orientable hypersurface to be isometric to a sphere. It is shown that the converse of this result holds, provided that the Sasakian manifold is the odd dimensional sphere $\mathbf{S}^{2n+1}$ . Similar results are obtained for the vector field v under the hypothesis that this is an eigenvector of the Laplace operator. Also, we use a bound on the integral of the Ricci curvature $Ric ( \mathbf{u},\mathbf{u} ) $ of the compact hypersurface to find a necessary condition for the hypersurface to be isometric to a sphere. We show that this condition is also sufficient if the Sasakian manifold is $\mathbf{S}^{2n+1}$ .

Transverse K\"ahler holonomy in Sasaki Geometry and ${\oldmathcal S}$-Stability

Abstract: We study the transverse Kahler holonomy groups on Sasaki manifolds $(M,{\oldmathcal S})$ and their stability properties under transverse holomorphic deformations of the characteristic foliation by the Reeb vector field. In particular, we prove that when the first Betti number $b_1(M)$ and the basic Hodge number $h^{0,2}_B({\oldmathcal S})$ vanish, then ${\oldmathcal S}$ is stable under deformations of the transverse Kahler flow. In addition we show that an irreducible transverse hyperkahler Sasakian structure is ${\oldmathcal S}$-unstable, whereas, an irreducible transverse Calabi-Yau Sasakian structure is ${\oldmathcal S}$-stable when $\dim M\geq 7$. Finally, we prove that the standard Sasaki join operation (transverse holonomy $U(n_1)\times U(n_2)$) as well as the fiber join operation preserve ${\oldmathcal S}$-stability.

Sasakian Structure Associated with a Second-Order ODE and Hamiltonian Dynamical Systems

Abstract: We define a contact metric structure on the manifold corresponding to a second-order ordinary differential equation $$\mathrm{d}^2y/\mathrm{d}x^2\,{=}\,f(x,y,y')$$ and show that the contact metric structure is Sasakian if and only if the 1-form $$\frac{1}{2}(\mathrm{d}p-f\mathrm{d}x)$$ defines a Poisson structure. We consider a Hamiltonian dynamical system defined by this Poisson structure and show that the Hamiltonian vector field, which is a multiple of the Reeb vector field, admits a compatible bi-Hamiltonian structure for which f can be regarded as Hamiltonian function. As a particular case, we give the compatible bi-Hamiltonian structure of the Reeb vector field such that the structure equations correspond to the Maurer–Cartan equations of an invariant coframe on the Heisenberg group and the independent variable plays the role of Hamiltonian function. We also show that the first Chern class of the normal bundle of an integral curve of a multiple of the Reeb vector field vanishes iff $$f_x+ff_p\,{=}\,\varPsi (x)$$ for some $$\varPsi $$ .

Geometry of almost contact metrics as almost $*$-Ricci solitons.

Abstract: In the present 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 with the potential vector field $V$ 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 is Einstein metric if it is $\eta$-Einstein or the potential vector field $V$ is collinear to the Reeb vector field or $V$ is an infinitesimal contact transformation.

$\eta-$Ricci solitons on contact pseudo-metric manifolds

Abstract: In this paper, we prove that a Sasakian pseudo-metric manifold which admits an $\eta-$Ricci soliton is an $\eta-$Einstein manifold, and if the potential vector field of the $\eta-$Ricci soliton is not a Killing vector field then the manifold is $\mathcal{D}-$homothetically fixed, and the vector field leaves the structure tensor field invariant. Next, we prove that a $K-$contact pseudo-metric manifold with a gradient $\eta-$Ricci soliton metric is $\eta-$Einstein. Moreover, we study contact pseudo-metric manifolds admitting an $\eta-$Ricci soliton with a potential vector field point-wise colinear with the Reeb vector field. Finally, we study gradient $\eta-$Ricci solitons on $(\kappa, \mu)$-contact pseudo-metric manifolds.

Existence of Birkhoff sections for Kupka-Smale Reeb flows of closed contact 3-manifolds.

Abstract: A Reeb vector field satisfies the Kupka-Smale condition when all its closed orbits are non-degenerate, and the stable and unstable manifolds of its hyperbolic closed orbits intersect transversely. We show that, on a closed 3-manifold, any Reeb vector field satisfying the Kupka-Smale condition admits a Birkhoff section. In particular, this implies that the Reeb vector field of a $C^\infty$-generic contact form on a closed 3-manifold admits a Birkhoff section, and that the geodesic vector field of a $C^\infty$-generic Riemannian metric on a closed surface admits a Birkhoff section.

Almost Ricci-Yamabe Soliton on Contact Metric Manifolds

Abstract: We consider almost Ricci-Yamabe soliton in the context of certain contact metric manifolds. Firstly, we prove that if the metric $g$ admits an almost $(\alpha,\beta)$-Ricci-Yamabe soliton with $\alpha eq 0$ and potential vector field collinear with the Reeb vector field $\xi$ on a complete contact metric manifold with the Reeb vector field $\xi$ 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 $\xi$. Next, if complete $K$-contact manifold admits gradient Ricci-Yamabe soliton with $\alpha eq 0$, then it is compact Sasakian and isometric to unit sphere $S^{2n+1}$. Finally, gradient almost Ricci-Yamabe soliton with $\alpha eq 0$ in non-Sasakian $(k,\mu)$-contact metric manifold is assumed and found that $M^3$ is flat and for $n>1$, $M$ is locally isometric to $E^{n+1}\times S^n(4)$ and the soliton vector field is tangential to the Euclidean factor $E^{n+1}$. An illustrative example is given to support the obtained result.

Para-Ricci-like Solitons with Vertical Potential on Para-Sasaki-Like Riemannian $\Pi$-Manifolds

Abstract: Object of study are para-Ricci-like solitons on para-Sasaki-like almost paracontact almost paracomplex Riemannian manifolds, briefly, Riemannian $\Pi$-manifolds. Different cases when the potential of the soliton is the Reeb vector field or pointwise collinear to it are considered. Some additional geometric properties of the constructed objects are proven. Results for a parallel symmetric second-order covariant tensor on the considered manifolds are obtained. Explicit example of dimension 5 in support of the given assertions is provided.

Looking at Euler flows through a contact mirror: Universality and undecidability.

Abstract: The dynamics of an inviscid and incompressible fluid flow on a Riemannian manifold is governed by the Euler equations. In recent papers [5, 6, 7, 8] several unknown facets of the Euler flows have been discovered, including universality properties of the stationary solutions to the Euler equations. The study of these universality features was suggested by Tao as a novel way to address the problem of global existence for Euler and Navier-Stokes [28]. Universality of the Euler equations was proved in [7] for stationary solutions using a contact mirror which reflects a Beltrami flow as a Reeb vector field. This contact mirror permits the use of advanced geometric techniques in fluid dynamics. On the other hand, motivated by Tao's approach relating Turing machines to Navier-Stokes equations, a Turing complete stationary Euler solution on a Riemannian $3$-dimensional sphere was constructed in [8]. Since the Turing completeness of a vector field can be characterized in terms of the halting problem, which is known to be undecidable [30], a striking consequence of this fact is that a Turing complete Euler flow exhibits undecidable particle paths [8]. In this article, we give a panoramic overview of this fascinating subject, and go one step further in investigating the undecidability of different dynamical properties of Turing complete flows. In particular, we show that variations of [8] allow us to construct a stationary Euler flow of Beltrami type (and, via the contact mirror, a Reeb vector field) for which it is undecidable to determine whether its orbits through an explicit set of points are periodic.

On Einstein hypersurfaces of a remarkable class of Sasakian manifolds

Abstract: We present a non existence result of complete, Einstein hypersurfaces tangent to the Reeb vector field of a regular Sasakian manifold which fibers onto a complex Stein manifold.

$\varepsilon\,$-contact structures and six-dimensional supergravity

Abstract: We introduce the concept of $\varepsilon\,$-contact metric structures on oriented (pseudo-)Riemannian three-manifolds, which encompasses the usual Riemannian contact metric, Lorentzian contact metric and para-contact metric structures, but which also allows the possibility for the Reeb vector field to be null. We investigate in more detail this latter case, which we call null contact structure. We observe that it is possible to extend in a natural and meaningful way both the Sasaki and K-contact conditions for null-contact structures, but we find that they are not equivalent conditions, in contradistinction to the situation for non-lightlike Reeb vector fields. Finally, we define the notion of $\varepsilon\eta\,$-Einstein structures and we discover that appropriate direct products of these structures produce solutions of six-dimensional minimal supergravity coupled to a tensor multiplet with constant dilaton.

Para-Ricci-like Solitons on Almost Paracontact Almost Paracomplex Riemannian Manifolds

Abstract: It is introduced and studied para-Ricci-like solitons with potential Reeb vector field on almost paracontact almost paracomplex Riemannian manifolds. The special cases of para-Einstein-like, para-Sasaki-like and having a torse-forming Reeb vector field have been considered. It is proved a necessary and sufficient condition the manifold to admit a para-Ricci-like soliton which is the structure to be para-Einstein-like. Explicit examples are provided in support of the proven statements.