arXiv research

Researchers solve a 25-year-old conjecture about vector fields.

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

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 …

We generalise the average asymptotic linking number of a pair of divergence-free vector fields on homology three-spheres by considering the linking of a divergence-free vector field on a manifold of arbitrary dimension with a codimension two foliation endowed with an invariant transverse measure. We prove that the aver…

New rigidity results for quasi-Einstein metrics with non-zero divergence-free vector fields.

Proves helicity is the only regular Casimir for 3D hydrodynamics.

We study the problem of finding strain-minimising stream surfaces in a divergence-free vector field. These surfaces are generated by motions of seed curves that propagate through the field in a strain minimising manner, i.e., they move without stretching or shrinking, preserving the length of their arbitrary arc. In ge…

We prove that a compact lcK manifold with holomorphic Lee vector field is Vaisman provided that either the Lee field has constant norm or the metric is Gauduchon (i.e., the Lee field is divergence-free). We also give examples of compact lcK manifolds with holomorphic Lee vector field which are not Vaisman.

New theorem links symmetries to first integrals in plasma physics.

New differential geometry perspective on orthogonal RNNs.

Conditions found for linearizing divergence-free fields on invariant tori.

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…

Let M be a globally Riemannian symmetric space. We prove a duality estimate between pairings of vector fields with divergence zero and and in L^1 with vector fields in a critical Sobolev space on M. As a consequence we get a sharp Calderon-Zygmund estimate for solutions to Poisson's equation on M, where the right side …

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

We prove a theorem formulated by V. I. Arnold concerning a relation between the asymptotic linking number and the Hopf invariant of divergence-free vector fields. Using a modified definition for the system of short paths, we prove their existence in the general case.

A solution of a problem by V.I.Arnol'd about higher analog of the asymptotic Hopf invariant of divergence-free vector fields is presented. A higher invariant of magnetic fields, which is not expressed from the asymptotic linking numbers of magnetic lines is constructed and examples of an asymptotic invariants is constr…

The paper introduces and characterizes almost ω-Bach solitons on various product manifolds.

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 surfaces in Half-Pipe space and vector fields on hyperbolic plane.

We present a geometric construction of central extensions of covering groups of the group of volume preserving diffeomorphisms, integrating central extensions of the Lie algebra of divergence free vector fields defined by Lichnerowicz cocycles. Certain covering spaces of non-linear Grassmannians can be realized as preq…

We present an explicit realization of abelian extensions of infinite dimensional Lie groups using abelian extensions of path groups, by generalizing Mickelsson's approach to loop groups and the approach of Losev-Moore-Nekrasov-Shatashvili to current groups. We apply our method to coupled cocycles on current Lie algebra…

A {\em 2-Riemannian manifold} is a differentiable manifold exhibiting a 2-inner product on each tangent space. We first study lower dimensional 2-Riemannian manifolds by giving necessary and sufficient conditions for flatness. Afterward we associate to each 2-Riemannian manifold a unique torsion free compatible pseudoc…

We prove that any regular integral invariant of volume-preserving transformations is equivalent to the helicity. Specifically, given a functional $\mathcal I$ defined on exact divergence-free vector fields of class $C^1$ on a compact 3-manifold that is associated with a well-behaved integral kernel, we prove that $\mat…

This paper deals with the problem of describing the vector spaces of divergence-free, natural tensors on a pseudo-Riemannian manifold that are second-order; i.e., that are defined using only second derivatives of the metric. The main result establishes isomorphisms between these spaces and certain spaces of tensors (at…

We study generalized Killing spinors on the standard sphere $\mathbb{S}^3$, which turn out to be related to Lagrangian embeddings in the nearly Kähler manifold $S^3\times S^3$ and to great circle flows on $\mathbb{S}^3$. Using our methods we generalize a well known result of Gluck and Gu concerning divergence-free geod…

Incompressible fluid dynamics on special manifolds.

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

New neural network enforces mass conservation for better ice flow predictions.

The Hodge decomposition provides a very powerful mathematical method for the analysis of 2D and 3D vector fields. It states roughly that any vector field can be $L^2$-orthogonally decomposed into a curl-free, divergence-free, and a harmonic field. The harmonic field itself can be further decomposed into three component…

This thesis is divided into three parts. In the first part, we give an introduction to J. Harrison's theory of differential chains. In the second part, we apply these tools to generalize the Cauchy theorems in complex analysis. Instead of requiring a piecewise smooth path over which to integrate, we can now do so over …

Electrostatic systems with specific tensors are locally conformally flat.

We compute all 2-covariant tensors naturally constructed from a semiriemannian metric which are divergence-free and have weight greater than -2. As a consequence, it follows a characterization of the Einstein tensor as the only, up to a constant factor, 2-covariant tensor naturally constructed from a semiriemannian met…

We present a Hamiltonian framework for higher-dimensional vortex filaments (or membranes) and vortex sheets as singular 2-forms with support of codimensions 2 and 1, respectively, i.e. singular elements of the dual to the Lie algebra of divergence-free vector fields. It turns out that the localized induction approximat…

A vector field on a Riemannian manifold is called conformal Killing if it generates one-parameter group of conformal transformations. The class of conformal Killing symmetric tensor fields of an arbitrary rank is a natural generalization of the class of conformal Killing vector fields, and appears in different geometri…

Study local invariants of divergence-free webs in geometry.

In this note we first set up an analogy between spin and vorticity of a perfect 2d-fluid flow, based on the Borel-Weil contruction of the irreducible unitary representations of SU(2), and looking at the Madelung-Bohm velocity attached to the ensuing spin wave functions. We also show that, in the framework of finite dim…

Conditions for trivial gradient hyperbolic Ricci and Yamabe solitons to be Einstein or constant scalar curvature.

Upper bound found for divergence-free Killing 2-tensors on manifolds.

This article uses Cartan-Kähler theory to construct local conservation laws from covariantly closed vector valued differential forms, objects that can be given, for example, by harmonic maps between two Riemannian manifolds. We apply the article's main result to construct conservation laws for covariant divergence free…

The paper connects web theory to heavenly PDEs and Einstein metrics.

We give a generalization of the Nambu mechanics based on vector Hamiltonians theory. It is shown that any divergence-free phase flow in $\mathbb{R}^n$ can be represented as a generalized Nambu mechanics with $n-1$ integral invariants. For the case when the phase flow in $\mathbb{R}^n$ has $n-3$ or less first integrals,…

We obtain explicitly all solutions of the SU(infinity) Toda field equation with the property that the associated Einstein-Weyl space admits a 2-sphere of divergence-free shear-free geodesic congruences. The solutions depend on an arbitrary holomorphic function and give rise to new hyperKahler and selfdual Einstein metr…

This text is the extended version of a talk given at the conference Geometry, Topology, QFT and Cosmology hold from May 28 to May 30, 2008 at the Observatoire de Paris. Using exterior differential systems, I provide a positive answer to the generalized isometric embedding problem of vector bundles, and show how conserv…

Proposes a new quasi-local mass for timelike 2-surfaces in spacetimes.

New Lie-group methods preserve geometric divergence-free features on manifolds.

Study shows singular sets for certain fluid equations are negligible.

We provide a sufficient condition for the local stability of closed Einstein manifolds of positive Ricci curvature under the Ricci iteration in terms of the spectrum of the Lichnerowicz Laplacian acting on divergence-free tensor fields. We use this result to consider the stability of several Einstein manifolds under th…

We study some analytical and geometric properties of a two-dimensional nonlinear sigma model with gravitino which comes from supersymmetric string theory. When the action is critical w.r.t. variations of the various fields including the gravitino, there is a symmetric, traceless and divergence-free energy-momentum tens…