arXiv research

Shows Euler-like vector fields come from specific embeddings.

Normal forms and isotropic embeddings via Euler-like vector fields.

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

Study vector fields on non-compact manifolds with group action.

Non-vanishing steady Euler flows and Beltrami fields found in high dimensions.

A Poincaré-Hopf theorem in the spirit of Pugh is proven for compact orbifolds with boundary. The theorem relates the index sum of a smooth vector field in generic contact with the boundary orbifold to the Euler-Satake characteristic of the orbifold and a boundary term. The boundary term is expressed as a sum of Euler c…

Euler's equations for a two-dimensional system can be written in Hamiltonian form, where the Poisson bracket is the Lie-Poisson bracket associated to the Lie algebra of divergence free vector fields. We show how to derive the Poisson brackets of 2d hydrodynamics of ideal fluids as a reduction from the one associated to…

Study on special coordinates for Dubrovin-Frobenius manifolds in low dimensions.

The paper constructs chaotic solutions to the Euler equations on high-dimensional manifolds.

The existence of a nowhere zero real vector field implies a well-known restriction on a compact manifold. But all manifolds admit nowhere zero complex vector fields. The relation between these observations is clarified.

The note answers a question about Betti numbers for 1D Euclidean space.

The Law of Vector Fields is a term coined by Gottlieb for a relative Poincaré-Hopf theorem. It was first proved by Morse and expresses the Euler characteristic of a manifold with boundary in terms of the indices of a generic vector field and the inner part of its tangential projection on the boundary. We give two diffe…

Study on generalized derivations in polynomial vector fields Lie algebras.

We determine several necessary and sufficient conditions for a closed almost-complex orbifold $Q$ with cyclic local groups to admit a nonvanishing vector field. These conditions are stated separately in terms of the orbifold Euler-Satake characteristics of $Q$ and its sectors, the Euler characteristics of the underlyin…

The first purpose of this note is to comment on a recent article of Bursztyn, Lima and Meinrenken, in which it is proved that if M is a smooth submanifold of a manifold V, then there is a bijection between germs of tubular neighborhoods of M and germs of "Euler-like" vector fields on V. We shall explain how to approach…

This paper generalizes Batchelor's theorem in $C^\infty$-superschemes.

For a manifold with boundary, the restriction of Chern's transgression form of the Euler curvature form over the boundary is closed. Its cohomology class is called the secondary Chern-Euler class and used by Sha to formulate a relative Poincaré-Hopf theorem, under the condition that the metric on the manifold is locall…

Linearizability of singular foliations is preserved under a specific equivalence relation.

Indices of singular points of a vector field or of a 1-form on a smooth manifold are closely related with the Euler characteristic through the classical Poincaré--Hopf theorem. Generalized Euler characteristics (additive topological invariants of spaces with some additional structures) are sometimes related with corres…

New proof confirms periodic orbit conjecture for Eulerisable flows.

RFM uses tangent vector fields to match data on manifolds, analyzing TV convergence for Euler discretization.

Proves a Gel'fand-Kolmogoroff type result for vector bundles and polynomial functions.

Defines discrete differential geometry concepts in homotopy type theory.

We introduce a complete obstruction to the existence of nonvanishing vector fields on a closed orbifold $Q$. Motivated by the inertia orbifold, the space of multi-sectors, and the generalized orbifold Euler characteristics, we construct for each finitely generated group $Γ$ an orbifold called the space of $Γ$-sectors o…

Researchers prove Lie algebras of differential operators and Grothendieck constructions coincide.

We extend Turaev's theory of Euler structures and torsion invariants on 3-manifolds to the case of vector fields having generic behavior on the boundary. This allows to easily define gluings of Euler structures and to develop a completely general gluing formula for Reidemeister torsion of 3-manifolds. Lastly, we descri…

We define and study the secondary Chern-Euler class for a general submanifold of a Riemannian manifold. Using this class, we define and study index for a vector field with non-isolated singularities on a submanifold. As an application, our studies give conceptual proofs of a classical result of Chern.

A $(TE)$-structure $\nabla$ over a complex manifold $M$ is a meromorphic connection defined on a holomorphic vector bundle over $\mathbb{C}\times M$, with poles of Poincaré rank one along $\{ 0 \} \times M.$ Under a mild additional condition (the so called unfolding condition), $\nabla$ induces a multiplication on $TM$…

Study stabilizes second-order systems to first-order dynamics.

Study on generalized ξ-parallel maps in Riemannian geometry.

We generalize Turaev's definition of torsion invariants of pairs (M,x), where M is a 3-dimensional manifold and x is an Euler structure on M (a non-singular vector field up to homotopy relative to bM and local modifications in int(M). Namely, we allow M to have arbitrary boundary and x to have simple (convex and/or con…

We characterize, using commuting zero-flux homologies, those volume-preserving vector fields on a $3$-manifold that are steady solutions of the Euler equations for some Riemannian metric. This result extends Sullivan's homological characterization of geodesible flows in the volume-preserving case. As an application, we…

Paper develops reduction theory for controlled Lagrangian systems with symmetry and momentum map.

We consider solutions to the complex Trkalian equation,~$ \vec{\nabla} \times \vc = \vc ,$ where~$\vc$ is a 3 component vector function with each component in the complex field, and may be expressed in the form~$ \vc = e^{ig} \vec{\nabla} F, $ with~$g$ real and~$F$ complex. We find, there are precisely two classes of s…

First we recall homology groups of prer Lie superalgebras. Then introducing double weighted chain spaces, we deal with pre Lie superalgebra of multi-vector fields with polynomial coefficients on n-dimensional number space. The bracket is Schouten bracket. We have several results about Euler number and Betti numbers of …

A natural explicit condition is given ensuring that an action of the multiplicative monoid of non-negative reals on a manifold F comes from homotheties of a vector bundle structure on F, or, equivalently, from an Euler vector field. This is used in showing that double (or higher) vector bundles present in the literatur…

We present a steady Euler flow on the round 3-sphere whose velocity vector field has the property of having two independent first integrals, being tangent to the fibres of an almost submersion onto the 2-sphere. This submersion turns out to be a critical point for the quartic Faddeev-Skyrme model with a standard potent…

The paper derives new Gauss-Bonnet formulas for frontal bundles over surfaces with boundary.

Incompressible fluid dynamics on special manifolds.

We establish a new version of the first Noether Theorem, according to which the (equivalence classes of) first integrals of given Euler-Lagrange equations in one independent variable are in exact one-to-one correspondence with the (equivalence classes of) vector fields satisfying two simple geometric conditions, namely…

A line field on a manifold is a smooth map which assigns a tangent line to all but a finite number of points of the manifold. As such, it can be seen as a generalization of vector fields. They model a number of geometric and physical properties, e.g. the principal curvature directions dynamics on surfaces or the stress…

Constructs coordinates to diagonalize Toda flow on matrices with simple spectrum.

We generalize Turaev's definition of torsion invariants of pairs $(M,ξ)$, where $M$ is a 3-dimensional manifold and $ξ$ is an Euler structure on $M$ (a non-singular vector field up to homotopy relative to the boundary of $M$ and local modifications in the interior of $M$). Namely, we allow $M$ to have arbitrary boundar…

We extend Turaev's definition of torsion invariants of 3-dimensional manifolds equipped with non-singular vector fields, by allowing (suitable) tangency circles to the boundary, and manifolds with non-zero Euler characteristic. We show that these invariants apply in particular to (the exterior of) Legendrian links in c…

We review the construction of homological evolutionary vector fields on infinite jet spaces and partial differential equations. We describe the applications of this concept in three tightly inter-related domains: the variational Poisson formalism (e.g., for equations of Korteweg-de Vries type), geometry of Liouville-ty…

Poincaré-Hopf theorem extended to Filippov vector fields on 2D manifolds.

The paper develops algorithms and topological invariants for distinguishing dynamic systems.

Affine vector fields on pseudo-Kähler manifolds are symplectic.