Poincaré-Hopf theorem extended to Filippov vector fields on 2D manifolds.
problem Extending Poincaré-Hopf theorem to Filippov vector fields.
method Introducing new index definition for Filippov vector fields, including singularities.
result Established a variant of Hairy Ball Theorem for Filippov vector fields.
Characterizes Filippov n-algebroids using connections and formulas.
problem Generalizing Lie algebroids to Filippov n-algebroids.
method Introducing Filippov connections and transforming the Jacobi identity into the Bianchi-Filippov identity.
result Expressed the n-ary bracket using a torsion-free formula.
In this article, we study the deformations of Filippov algebroids. We define a differential graded Lie algebra (in short DGLA) for a Filippov algebroid by introducing the notion of Filippov multiderivations for a vector bundle. Later on, we discuss deformations of a Filippov algebroid in terms of low-dimensional cohomo…
We study some properties of coisotropic submanifolds of a manifold with respect to a given multivector field. Using this notion, we generalize the results of Weinstein \cite{wein} from Poisson bivector field to Nambu-Poisson tensor or more generally to any multivector field. We also introduce the notion of Nambu-Lie gr…
We discuss relations between linear Nambu-Poisson structures and Filippov algebras and define Filippov algebroids which are n-ary generalizations of Lie algebroids. We also prove results describing multiplicative Nambu- Poisson structures on Lie groups. In particular, we show that simple Lie groups do not admit multipl…
Maximal causal curves for Lipschitz metrics are either lightlike or timelike.
problem Characterizing maximal causal curves for Lipschitz metrics.
method Analyzing the parametrization and geodesic equation for maximal causal curves in terms of Filippov solutions.
result Maximal causal curves for Lipschitz metrics are either everywhere lightlike or everywhere timelike.
We prove that the geodesic equation for any semi-Riemannian metric of regularity C0,1 possesses C1-solutions in the sense of Filippov.
This paper investigates higher order generalizations of well known results for Lie algebroids and bialgebroids. It is proved that n-Lie algebroid structures correspond to n-ary generalization of Gerstenhaber algebras and are implied by n-ary generalization of linear Poisson structures on the dual bundle. A Nambu-…
The paper analyzes binary option markets with exogenous information and price sensitivity.
problem Analyzing binary option markets with exogenous information and price sensitivity.
method Derive and analyze a continuous model of binary option markets with exogenous information, using Filippov surfaces and general assumptions on purchasing rules.
result Price always converges when exogenous information is constant, and price sensitivity affects price lag vs. information.
Two proofs show that removing a loop from a plane circuit splits the plane.
problem Proving the Weak Jordan Theorem about plane circuits.
method Detailed presentation of Thomassen's and Filippov's proofs.
result The complement of any loop in a plane circuit is disconnected.
A notion of n-Lie algebra introduced by V.T. Filippov can be viewed as a generalization of a concept of binary Lie algebra to the algebras with n-ary multiplication law. A notion of Lie algebra can be extended to Z_2-graded structures giving a notion of Lie superalgebra. Analogously a notion of n-Lie algebra can be ext…
We consider geodesics in both Riemannian and Lorentzian manifolds with metrics of low regularity. We discuss existence of extremal curves for continuous metrics and present several old and new examples that highlight their subtle interrelation with solutions of the geodesic equations. Then we turn to the initial value …
The method of double extension, introduced by A.~Medina and Ph.~Revoy, is a procedure which decomposes a Lie algebra with an invariant symmetric form into elementary pieces. Such decompositions were developed for other algebras, for instance for Lie superalgebras and associative algebras, Filippov n-algebras and Jord…
Paper transforms torse-forming vector fields into simpler forms.
problem Generalizing vector fields and their transformations.
method Present techniques to transform torse-forming vector fields into simpler cases.
result Concrete examples of transformations are provided.
The paper proves that certain modified conformal vector fields are trivial on compact and non-compact manifolds.
problem Proving triviality of modified conformal vector fields on Riemannian manifolds.
method Analyzing properties of homothetic, conformal, and gradient vector fields on compact and non-compact manifolds.
result Established conditions under which m-modified conformal vector fields are trivial. The position vector field x is the most elementary and natural geometric object on a Euclidean submanifold M. The position vector field plays very important roles in mathematics as well as in physics. Similarly, the tangential component x^T of the position vector field is the most natural vector field tangent to the …
Conformal vector fields on LCP manifolds are orthogonal and Killing.
problem Understanding conformal vector fields on specific geometric manifolds.
method Analyzing properties of conformal vector fields on compact locally conformally product manifolds.
result Conformal vector fields are orthogonal to the flat distribution and Killing.
For a submanifold M in a Euclidean space, the tangential component x^T of the position vector field x of M is the most natural vector field tangent to the Euclidean submanifold, called the canonical vector field of M. In this article, first we prove that the canonical vector field of every Euclidean submanifold is alwa…
Study on vector fields on Lie groups reveals surprising algebraic coincidences.
problem Characterizing vector fields on Lie groups with Riemannian metrics.
method Algebraic and geometric analysis of left-invariant vector fields on nilpotent Lie groups.
result Spaces of Killing, one-harmonic, and conformal vector fields coincide with the center of the Lie algebra on nilpotent Lie groups.
Study biharmonic vector fields and unit vector fields on Riemannian manifolds.
problem Determine the equivalence of biharmonicity and harmonicity for vector fields and unit vector fields on Riemannian manifolds.
method Analyze biharmonic vector fields and unit vector fields on (M,g) with pseudo-Riemannian g-natural metrics on TM and T1M. result Contrary to Sasaki metric, biharmonicity and harmonicity are not equivalent for large classes of g-natural metrics on TM. This short report establishes some basic properties of smooth vector fields on product manifolds. The main results are: (i) On a product manifold there always exists a direct sum decomposition into horizontal and vertical vector fields. (ii) Horizontal and vertical vector fields are naturally isomorphic to smooth famil…
Defines quaternionic k-vector fields on quaternionic Kähler manifolds.
problem No specific problem stated; focuses on definition and properties.
method Introduced a modified Dirac operator to define quaternionic k-vector fields.
result Calculated the dimension of quaternionic k-vector fields on HPn. Study on generalized derivations in polynomial vector fields Lie algebras.
problem Understanding generalized derivations in specific Lie sub-algebras of polynomial vector fields.
method Analysis of Lie sub-algebras containing constant and Euler vector fields, under specified conditions.
result Characterization of generalized derivations in the studied Lie sub-algebras.
Study on Einstein solitons with specific vector fields and their properties.
problem Characterizing Einstein solitons with gradient, solenoidal, or concircular vector fields.
method Explicitly express the function λ by gradient vector field V and deduce geometric properties under certain curvature conditions.
result Explicit expressions for λ and geometric properties of Einstein solitons.
Investigates point spectra of vector fields and their properties.
problem Understanding the point spectra of vector fields.
method Define and study point spectra, prove properties under isometries, and analyze compactly supported fields.
result Point spectra are well-behaved under isometries and trivial for compactly supported fields.
We consider four dimensional lie groups equipped with left invariant Lorentzian Einstein metrics, and determine the harmonicity properties of vector fields on these spaces. In some cases, all these vector fields are critical points for the energy functional restricted to vector fields. We also classify vector fields de…
Examining singularities of commuting vector fields on submanifolds.
problem Understanding singularities of commuting vector fields on submanifolds.
method Analyzing real submanifolds within Kähler manifolds.
result Characterized singularities of commuting vector fields.
Vector fields invariant under Lie group action are finitely generated by polynomial fields.
problem Understanding invariant vector fields under Lie group actions.
method Analyzing the module of smooth vector fields invariant under a linear action of a compact Lie group.
result The module of invariant vector fields is finitely generated by polynomial fields.
The paper proves non-existence of torqued and anti-torqued vector fields on hyperbolic spaces.
problem Existence of torqued and anti-torqued vector fields on hyperbolic spaces.
method Analyzing the properties of conformal scalar functions and their impact on the existence of vector fields.
result Non-existence of proper torqued and anti-torqued vector fields on hyperbolic spaces.
We introduce G_2-vector fields, Rochesterian 1-forms and Rochesterian vector fields on manifolds with a closed G_2-structure as analogues of symplectic vector fields, Hamiltonian functions and Hamiltonian vector fields respectively, and we show that the spaces of G_2-vector fields and of Rochesterian vector fields are …
Enhances Hamiltonian systems stability through generalized double bracket vector fields.
problem Stabilizing already stable points in Hamiltonian systems.
method Generalized double bracket vector fields on Poisson manifolds with pseudo-Riemannian metrics.
result Enhanced equilibria stability through dissipation terms.
Study classifies harmonic vector fields on 3-manifolds.
problem Classifying harmonic unit vector fields on 3-manifolds.
method Investigates under mild curvature assumptions, classifying vector fields and manifolds.
result Classifies both vector fields and manifolds supporting them.
Braided vector fields on spatial subdomains homeomorphic to the cylinder play a crucial role in applications such as solar and plasma physics, relativistic astrophysics, fluid and vortex dynamics, elasticity, and bio-elasticity. Often the vector field's topology -- the entanglement of its field lines -- is non-trivial,…
Study proves conformal vector fields on certain Finsler manifolds are Killing fields.
problem Characterizing conformal vector fields on compact homogeneous Finsler manifolds.
method Analyzes properties of conformal vector fields and homogeneous Finsler metrics.
result Conformal vector fields on compact homogeneous Finsler manifolds are Killing fields.
Indices of vector fields and 1-forms studied for singular varieties and actions.
problem Understanding indices of vector fields and 1-forms in various contexts.
method Generalization to singular varieties and actions of finite groups.
result New insights into indices of vector fields and 1-forms.
Harmonic basis vector fields on surfaces
problem Parameterizing surfaces with harmonic vector fields
method Introducing harmonic basis vector fields and deriving conditions for their existence
result Classifying parameterizations of surfaces with harmonic basis vector fields
The paper explores how vector fields relate to volume in geometric contexts.
problem Existence of nondiffeomorphic contact forms with identical Reeb vector fields.
method Analyzes geodesible vector fields and their associated Euler classes, applying topological and geometric theorems.
result Proves the Gauss-Bonnet and Poincaré-Hopf theorems for 2D orbifolds using geodesible vector fields.
A Ricci soliton (M,g,v,λ) on a Riemannian manifold (M,g) is said to have concurrent potential field if its potential field v is a concurrent vector field. Ricci solitons arisen from concurrent vector fields on Riemannian manifolds were studied recently in \cite{CD2}. The most important concurrent vector field is …
Study shows limitations of Lie bracket commutation for nonsmooth vector fields.
problem Limitations of Lie bracket commutation for nonsmooth vector fields.
method Analysis of nonsmooth vector fields, focusing on commutation of flows and Lie bracket conditions.
result Lie bracket commutation cannot be extended to general a.e. differentiable vector fields, but holds for certain Sobolev regular fields.
Discuss various definitions of vector fields on manifolds leading to Lie algebras.
problem Defining vector fields on convenient manifolds.
method Various definitions of vector fields and their equivalence.
result Equivalent definitions of vector fields on manifolds.
Study on almost Riemann solitons with gradient or torse-forming vector fields.
problem Characterizing almost Riemann solitons with specific vector fields.
method Using Bochner formula and properties of gradient and torse-forming vector fields.
result Explicit expressions for the soliton function λ under gradient and torse-forming conditions. Study Ricci vector fields on 2D space with diagonal metrics.
problem Understanding Ricci vector fields on 2D space with specific metrics.
method Examined Ricci vector fields on R2 with a diagonal metric. result Characterized Ricci vector fields on R2 with a diagonal metric. In this paper, we define a certain "proportional volume property" for an unit vector field on a spherical domain in S3. We prove that the volume of these vector fields has an absolute minimum and this value is equal to the volume of the Hopf vector field. Some examples of such vector fields are given. We also study the…
The study classifies spaces with specific conformal vector fields.
problem Characterizing closed vacuum static spaces with non-Killing conformal vector fields.
method Provided characterizations and established an identity involving the characteristic function.
result Derived a rigidity theorem and classified spaces with the vector field.
Classifies vector field algebras in complex space.
problem Classifying vector field algebras in complex space.
method Classical results and proofs for vector fields in 2 and 3 variables, extended to arbitrary N.
result Lie's classification for vector fields in CN. Study null conformal Killing vector fields on complex surfaces.
problem Characterize pseudo-Hermitian surfaces with null vector fields.
method Analyze topological types and use vector fields to define para-hyperhermitian structures.
result Classify compact four-manifolds with orthogonal null Killing vector fields.
Characterizes magnetic unit vector fields on Lie groups.
problem Classifying magnetic unit vector fields on Lie groups.
method Characterization through critical points of Landau Hall and Dirichlet energy functionals.
result Classification of all magnetic left invariant unit vector fields on 3-dimensional Lie groups.
The aim of the present paper is to investigate intrinsically the notion of a concircular π-vector field in Finsler geometry. This generalizes the concept of a concircular vector field in Riemannian geometry and the concept of a concurrent vector field in Finsler geometry. Some properties of concircular π-vector fie…