arXiv research

New vector fields integrate first-order ODEs.

Classifies vector fields in the kernel of a 1-form, up to equivalence.

Generalizes Frobenius theorem to quasiconformal deformations.

Analyzes vector fields in polytope decompositions, proving curve finiteness.

Study projective symmetries in Finsler spaces, showing reductions and constant flag curvature.

Contact Lie systems analyze integral curves of Hamiltonian vector fields.

Recently, the geodesibility of planar vector fields, which are algebrizable (differentiable in the sense of Lorch for some associative and commutative unital algebra), has been established. In this paper, we consider algebrizable three-dimensional vector fields, for which we give rectifications and Riemannian metrics u…

Foliate systems are those which preserve some (possibly singular) foliation of phase space, such as systems with integrals, systems with continuous symmetries, and skew product systems. We study numerical integrators which also preserve the foliation. The case in which the foliation is given by the orbits of an action …

The paper explores generalized quasi-Einstein manifolds and their properties.

Study classifies harmonic vector fields on 3-manifolds.

A vector field on a Riemannian manifold is called geodesic if its integral curves are reparametrized geodesics. We classify compact Kähler manifolds admitting nontrivial real-holomorphic geodesic gradient vector fields that satisfy an additional integrability condition. They are all biholomorphic to bundles of complex …

We show a higher order integrability theorem for distributions generated by a family of vector fields under a horizontal regularity assumption on their coefficients. We use as chart a class of almost exponential maps which we discuss in details

New proof finds three divergence-free vector fields for any 3D manifold.

Aguilar introduced isotropic almost complex structures $J_{δ, σ}$ on the tangent bundle of a Riemannian manifold $(M, g)$. In this paper, some results will be obtained on the integrability of these structures. These structures with the Liouville 1-form define a class of Riemannian metrics $g_{δ, σ}$ on $T M$ which are …

Given a compatible vector field on a compact connected almost-complex manifold, we show in this article that the multiplicities of eigenvalues among the zero point set of this vector field have intimate relations. We highlight a special case of our result and reinterpret it as a vanishing-type result in the framework o…

The study extends and generalizes a result about quasi Einstein manifolds, proving conditions for Killing vector fields.

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 generalizes relations between dynamical series and resolvents of vector fields.

The modular vector field of a Poisson-Nijenhuis Lie algebroid $A$ is defined and we prove that, in case of non-degeneracy, this vector field defines a hierarchy of bi-Hamiltonian $A$-vector fields. This hierarchy covers an integrable hierarchy on the base manifold, which may not have a Poisson-Nijenhuis structure.

We show that an invariant surface allows to construct the Jacobi vector field along a geodesic and construct the formula for the normal component of the Jacobi field. If a geodesic is the transversal intersection of two invariant surfaces (such situation we have, for example, if the geodesic is hyperbolic), then we can…

This paper describes the construction of a canonical compactification of the space of trajectories and of the unstable/stable sets of a generic gradient like vector field on a closed manifold as well as a canonical structure of a smooth manifold with corners of these spaces. As an application we discuss the geometric c…

The paper broadens the helix concept in 3D space.

In this review paper, we present several results on central extensions of the Lie algebra of symplectic (Hamiltonian) vector fields, and compare them to similar results for the Lie algebra of (exact) divergence free vector fields. In particular, we comment on universal central extensions and integrability to the group …

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

We analyse the asymptotical growth of Vassiliev invariants on non-periodic flow lines of ergodic vector fields on domains of $\R^3$. More precisely, we show that the asymptotics of Vassiliev invariants is completely determined by the helicity of the vector field. As an application, we determine the asymptotic Alexander…

In this paper we prove the following result: if two 2-dimensional 2-homogeneous rational vector fields commute, then either both vector fields can be explicitly integrated to produce rational flows with orbits being lines through the origin, or both flows can be explicitly integrated in terms of algebraic functions. In…

Researchers found non-Killing tensor fields on certain symmetric spaces.

In this paper, we discuss the geometric integration of hamiltonian systems on Poisson manifolds, in particular, in the case, when the Poisson structure is induced by a Lie algebra, that is, it is a Lie-Poisson structure. A Hamiltonian system on a Poisson manifold $(P, Π)$ is a smooth manifold $P$ equipped with a bivect…

A semigroup of annuli integrates a central extension of vector fields on S^1.

New integral estimates on substatic manifolds improve Alexandrov Theorem.

Study vector fields and flows on singular spaces like submanifolds.

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

The paper studies algebraic relations of first integrals on specific Lie groups.

We generalize the notions of the Futaki invariant and extremal vector field of a compact Kähler manifold to the general almost-Kahler case and show the periodicity of the extremal vector field when the symplectic form represents an integral cohomology class modulo torsion. We also give an explicit formula of the hermit…

Quadratic Killing tensors on Lie groups are always decomposable.

On a manifold equipped with a bivector field, we introduce for every Hamiltonian a Lagrangian on paths valued in the cotangent space whose stationary points projects onto Hamiltonian vector fields. We show that the remaining components of those stationary points tell whether the bivector field is Poisson or at least de…

For integrable Hamiltonian systems with two degrees of freedom whose Hamiltonian vector fields have incomplete flows, an analogue of the Liouville theorem is established. A canonical Liouville fibration is defined by means of an "exact" 2-parameter family of flat polygons equipped with certain pairing of sides. For the…

Let M denote a compact, orientable, 3-dimensional manifold and let a denote a contact 1-form on M; thus the wedge product of a with da is nowhere zero. This article explains how the Seiberg-Witten Floer homology groups as defined for any given Spin-C structure on M give closed, integral curves of the vector field that …

The aim of this paper is to extend the notion of commutativity of vector fields to the category of singular foliations, using Nambu structures, i.e. integrable multi-vector fields. We will classify the relationship between singular foliations and Nambu structures, and show some basic results about commuting Nambu struc…

We provide a variational description of any Liouville (i.e. volume preserving) autonomous vector fields on a smooth manifold. This is obtained via a ``maximal degree'' variational principle; critical sections for this are integral manifolds for the Liouville vector field. We work in coordinates and provide explicit for…

The purpose of this note is to establish the following theorem: Let N be a Kahler manifold, L be a compact oriented immersed minimal Lagrangian submanifold in N and V be a holomorphic vector field in a neighbourhood of L in N. Let div(V) be the (complex) divergence of V. Then the integral of div(V) over L is 0. Vice ve…

The purpose of this paper is to study anti-invariant Riemannian submersions from Kenmotsu manifolds onto Riemannian manifolds. Several fundamental results in this respect are proved. The integrability of the distributions and the geometry of foliations are investigated. We proved that there do not exist (anti-invariant…

In this article we investigate the relations between three kinds of vector fields with close connection to each other. A compact orientable manifold enables us to integrate over it, which is very different from noncompact manifolds, and this gives difference of those relationships between on compact and noncompact mani…

We consider the general nonvanishing, divergence-free vector fields defined on a domain in three space and tangent to its boundary. Based on the theory of finite type invariants, we define a family of invariants for such fields, in the style of Arnold's asymptotic linking number. Our approach is based on the configurat…

Study shows connection-preserving vector fields are equivalent to certain algebroid structures.

The study characterizes almost Kenmotsu manifolds with specific vector fields.

Starting with the most general four-dimensional spacetime possessing two commuting Killing vectors and a nontrivial Killing tensor, we analytically integrate Einstein-Yang-Mills equations for a completely arbitrary gauge group. It is assumed that the gauge field inherits the symmetries of the background and is aligned …

The paper characterizes integrability of tensors on manifolds.