arXiv research

The abstract proves that certain Reeb vector fields on 3-manifolds have Birkhoff sections.

We study invariant submanifolds of manifolds endowed with a normal or complex metric contact pair with decomposable endomorphism field $φ$. For the normal case, we prove that a $φ$-invariant submanifold tangent to a Reeb vector field and orthogonal to the other one is minimal. For a $φ$-invariant submanifold $N$ everyw…

The paper simplifies proofs and characterizes contact structures in 3D.

The paper studies Cotton solitons on specific geometric manifolds.

The paper characterizes Kenmotsu metrics as almost $*$-Ricci solitons.

The paper studies a new submersion type with specific conditions.

We study on which compact Sasakian 3-manifolds the Reeb field, which is a Beltrami field with eigenvalue 2, is an energy minimizer in its adjoint orbit under the action of volume preserving diffeomorphisms. This minimization property for Beltrami fields is relevant because of its connections with the phenomenon of magn…

The paper classifies K-contact forms on 3-manifolds and connects their orbits to spectral invariants.

No complete Einstein hypersurfaces found in a specific type of Sasakian manifold.

Geodesic vector fields on flat 3-manifolds are related to contact structures.

In this article, we prove that there exists at least one chord which is characteristic of Reeb vector field connecting a given Legendre submanifold in a closed contact manifold with any contact form.

The paper examines timelike conformal fields on 3-manifolds and finds they are rigidly tied to specific geometric structures.

We provide obstructions to the existence of conformally Anosov Reeb flows on a 3-manifold that partially generalize similar obstructions to Anosov Reeb flows. In particular, we show $\mathbb{S}^3$ does not admit conformally Anosov Reeb flows. We also give a Riemannian geometric condition on a metric compatible with a c…

We introduce and study $H$-paracontact metric manifolds, that is, paracontact metric manifolds whose Reeb vector field $ξ$ is harmonic. We prove that they are characterized by the condition that $ξ$ is a Ricci eigenvector. We then investigate how harmonicity of the Reeb vector field $ξ$ of a paracontact metric manifold…

The paper characterizes Einstein metrics in Kenmotsu manifolds using specific soliton types.

The paper classifies 3D spherical Sasakian manifolds using geometric and algebraic methods.

It is proved the non-existence of Hopf hypersurfaces in $G_{2}({\Bbb C}^{m+2})$, $m \geq 3$, whose normal Jacobi operator is semi-parallel, if the principal curvature of the Reeb vector field is non-vanishing and the component of the Reeb vector field in the maximal quaternionic subbundle ${\frak D}$ or its orthogonal …

I describe a general scheme which associates conjugacy classes of tori in the contactomorphism group to transverse almost complex structures on a compact contact manifold. Moreover, to tori of Reeb type whose Lie algebra contains a Reeb vector field one can associate a Sasaki cone. Thus, for contact structures of K-con…

Geometric quantization scheme for contact 3-manifolds models gravity.

We show that $φ$-invariant submanifolds of metric contact pairs with orthogonal characteristic foliations make constant angles with the Reeb vector fields. Our main result is that for the normal case such submanifolds of dimension at least $2$ are all minimal. We prove that an odd-dimensional $φ$-invariant submanifold …

The paper studies regular contact manifolds and their products.

In this paper we examine the Riemannian geometry of the group of contactomorphisms of a compact contact manifold. We compute the sectional curvature of $\mathcal{D}_θ(M)$ in the sections containing the Reeb field and show that it is non-negative. We also solve explicitly the Jacobi equation along the geodesic correspon…

In this paper we study real hypersurfaces in the complex quadric space $Q^m$ whose structure Jacobi operator commutes with their structure tensor field. We show that the Reeb curvature $α$ of such hypersurfaces is constant and if $α$ is non-zero then the hypersurface is a tube around a totally geodesic submanifold $\ma…

In this paper we study slant null curves with respect to the original parameter on 3-dimensional normal almost contact B-metric manifolds with parallel Reeb vector field. We prove that for non-geodesic such curves there exists a unique Frenet frame for which the original parameter is distinguished. Moreover, we obtain …

We define a contact metric structure on the manifold corresponding to a second order ordinary differential equation $d^2y/dx^2=f(x,y,y')$ and show that the contact metric structure is Sasakian if and only if the 1-form $\frac{1}{2}(dp-fdx)$ defines a Poisson structure. We consider a Hamiltonian dynamical system defined…

The study classifies contact metric manifolds based on Ricci-Yamabe solitons.

In this paper we give a characterization of real hypersurfaces in noncompact complex two-plane Grassmannian $SU_{2,m}/S(U_2 U_m)$, $m \geq 2$ with Reeb vector field $ξ$ belonging to the maximal quaternionic subbundle $\mathcal Q$. Then it becomes a tube over a totally real totally geodesic ${\mathbb H}H^n$, $m=2n$, in …

The paper explores how vector fields relate to volume in geometric contexts.

We study compatible toric Sasaki metrics with constant scalar curvature on co-oriented compact toric contact manifolds of Reeb type of dimension at least 5. These metrics come in rays of transversal homothety due to the possible rescaling of the Reeb vector fields. We prove that there exist Reeb vector fields for which…

On every compact, orientable, irreducible 3-manifold V which is toroidal or has torus boundary components we construct a contact 1-form whose Reeb vector field R does not have any contractible periodic orbits and is tangent to the boundary. Moreover, if bdry V is nonempty, then the Reeb vector field R is transverse to …

Sasakian manifolds provide explicit formulae of some Jacobi operators which describe the biharmonic equation of curves in Riemannian manifolds. In this paper we characterize non-geodesic biharmonic curves in Sasakian manifolds which are either tangent or normal to the Reeb vector field. In the three-dimensional case, w…

Harmonic forms and Rumin complex linked on Sasakian manifolds.

The Weinstein conjecture is extended to a new class of manifolds.

We prove that in dimension 3 every nondegenerate contact form is carried by a broken book decomposition. As an application we get that if M is a closed irreducible oriented 3-manifold that is not a graph manifold, for example a hyperbolic manifold, then every nondegenerate Reeb vector field on M has positive topologica…

Unified Jacobi coupling construction for various geometric settings.

We introduce the notion of contact Ricci flow associated with the Reeb vector field. Using it, we give a simple proof of the Poincare conjecture.

The study of Reeb dynamics on contact manifolds without periodic orbits.

We use the equivalence between embedded contact homology and Seiberg-Witten Floer homology to obtain the following improvements on the Weinstein conjecture. Let Y be a closed oriented connected 3-manifold with a stable Hamiltonian structure, and let R denote the associated Reeb vector field on Y. We prove that if Y is …

The paper classifies geometric structures of δ-almost Yamabe solitons on paracontact metric manifolds.

We introduce the notion of $\varepsilonη\,$-Einstein $\varepsilon\,$-contact metric three-manifold, which includes as particular cases $η\,$-Einstein Riemannian and Lorentzian (para) contact metric three-manifolds, but which in addition allows for the Reeb vector field to be null. We prove that the product of an $\vare…

The study examines almost Ricci-Yamabe solitons on almost Kenmotsu manifolds and their properties.

Study on Yamabe solitons on specific complex manifolds, focusing on torse-forming vector fields.

Y. J. Suh and H. Lee (Bull. Korean. Math. Soc. 47, 551-561 (2010)) characterized real hypersurfaces $M$ of type $B$ by the invariance of vector bundle $JTM^\perp$ under the shape operator and the orthogonality of $JTM^\perp$ and $\mathcal {J}TM^\perp$, where $TM^\perp$, $J$ and $\mathcal J$ are the normal bundle of $M$…

We introduce the notion of contact pair structure and the corresponding associated metrics, in the same spirit of the geometry of almost contact structures. We prove that, with respect to these metrics, the integral curves of the Reeb vector fields are geodesics and that the leaves of the Reeb action are totally geodes…

The paper explores properties of conformal vector fields on almost Kenmotsu manifolds.

In this short note it is established that there does not exist Ricci soliton with the Reeb potential vector field in an almost Kenmotsu manifold (briefly, $\mathcal{AKM}$).

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 compa…

The paper characterizes Kenmotsu manifolds with conformal η-Ricci solitons.