arXiv research

We consider the Lie algebra of all vector fields on a contact manifold as a module over the Lie subalgebra of contact vector fields. This module is split into a direct sum of two submodules: the contact algebra itself and the space of tangent vector fields. We study the geometric nature of these two modules.

Contact Lie systems analyze integral curves of Hamiltonian vector fields.

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

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

The paper studies $η-$Ricci solitons on contact pseudo-metric manifolds and their properties.

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

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

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

We provide some examples of harmonic unit vector fields as normalized gradients of isoparametric functions from a K-contact geometry setting.

The aim of this article is to study the k-almost Ricci soliton and k-almost gradient Ricci soliton on contact metric manifold. First, we prove that if a compact K-contact metric is a k-almost gradient Ricci soliton then it is isometric to a unit sphere S2n+1. Next, we extend this result on a compact k-almost Ricci soli…

Develops Hamilton-Jacobi theory for non-conservative field theories in k-contact geometry.

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

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 …

Let $M$ be an odd-dimensional Euclidean space endowed with a contact 1-form $α$. We investigate the space of symmetric contravariant tensor fields on $M$ as a module over the Lie algebra of contact vector fields, i.e. over the Lie subalgebra made up by those vector fields that preserve the contact structure. If we cons…

Study of solitons in a specific type of contact metric manifold.

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

We consider a contact manifold with a pseudo-Riemannian metric and define a contact vector field intrinsically associated to this pair of structures. We call this new differential invariant the contact Riemannian curl. On a Riemannian manifold, Killing vector fields are those that annihilate the metric; a Killing $1$-f…

The study explores $(m,ρ)$-quasi-Einstein structures on contact metric manifolds.

The paper studies regular contact manifolds and their products.

Characterizes invariant contact metric structures on tangent sphere bundles of compact symmetric spaces.

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

The study of quasi Yamabe solitons on 3D contact metric manifolds with specific curvature condition.

The study explores rigid constraints on almost Ricci-Bourguignon solitons on contact metric three-manifolds.

We study vector fields of the plane preserving the form of Liouville. We present their local models up to the natural equivalence relation, and describe local bifurcations of low codimension. To achieve that, a classification of univariate functions is given, according to a relation stricter than contact equivalence. W…

Study lightlike hypersurfaces in a specific type of manifold.

Biharmonic or polyharmonic curves and surfaces in 3-dimensional contact manifolds are investigated.

Study vector fields on hyperbolic spaces to create Ricci-Bourguignon solitons.

New contact structures extend supergravity solutions.

The paper recovers contact forms from boundary data using vector fields and Lyapunov functions.

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…

New theorem generalizes contact manifolds with symplectic properties.

In this paper, we study normal complex contact metric manifolds and we get some general results on them. Moreover, we obtained the general expression of the curvature tensor field for arbitrary vector fields. Furthermore, we show that the necessary and succient conditions to be normal a complex contact metric manifold.…

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 paper is devoted to the complete classification of all real Lie algebras of contact vector fields on the first jet space of one-dimensional submanifolds in the plane. This completes Sophus Lie's classification of all possible Lie algebras of contact symmetries for ordinary differential equations. As a main tool we …

We employ the relationship between contact structures and Beltrami fields derived in part I of this series to construct steady nonsingular solutions to the Euler equations on a Riemannian $S^3$ whose flowlines trace out closed curves of all possible knot and link types simultaneously. Using careful contact-topological …

A contact metric manifold is said to be $H$-contact, if the characteristic vector field is harmonic. We prove that the unit tangent bundle of a Riemannian manifold $M$ equipped with the standard contact metric structure is $H$-contact if and only if $M$ is $2$-stein.

The paper studies special solitons on specific contact metric manifolds.

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…

Study weak $f$-K-contact manifolds, finding Einstein-type metrics and solitons.

In this paper we proof that the Holomorphic angle for compact minimal surfaces in the sphere $S^5$ with constant Contact angle and with a parallel normal vector field must be constant.

In $N(k)$-contact metric manifolds and/or $(k,μ)$-manifolds, gradient Ricci solitons, compact Ricci solitons and Ricci solitons with $V$ pointwise collinear with the structure vector field $ξ$ are studied.

We study invariant contact p-spheres on principal circle-bundles and solve the corresponding existence problem in dimension 3. Moreover, we show that contact p-spheres can only exist on (4n-1)-dimensional manifolds and we construct examples of contact p-spheres on such manifolds. We also consider relations between taut…

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

Lecture notes introduce contact complete integrability for odd-dimensional manifolds.

It is established that the existence of non-isotropic vector field which Jacobi operator of maximal rank is an obstacle for the existence of non-trivial second-order symmetric parallel tensor field. In turns out that presence of such obstacle follows that manifold as pseudo-Riemannian manifold is locally non-reducible.…

We give a complete list of those left invariant unit vector fields on three-dimensional Lie groups with the left-invariant metric that generate a totally geodesic submanifold in the unit tangent bundle of a group with the Sasaki metric. As a result, each class of three-dimensional Lie groups admits the totally geodesic…

Study Schwarzians in the Heisenberg group, introducing new definitions and characterizing contact conformal vector fields.

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.