arXiv research

Vector fields invariant under Lie group action are finitely generated by polynomial fields.

The paper classifies invariant structures on complex almost Abelian groups.

New method to determine parabolic surfaces invariant under Killing fields.

We study scale invariant but not necessarily conformal invariant deformations of non-relativistic conformal field theories from the dual gravity viewpoint. We present the corresponding metric that solves the Einstein equation coupled with a massive vector field. We find that, within the class of metric we study, when w…

Study on vector fields on Lie groups reveals surprising algebraic coincidences.

New field invariant refines real spectrum and relates to absolute Galois group.

Two non-local asymptotic invariants of magnetic fields for the ideal magnetohydrodynamics are introduced. The velocity of variation of the invariants for a non-ideal magnetohydrodynamics with a small magnetic dissipation is estimated. By means of the invariants the spectra of electromagnetic fields are investigated. A …

We define an invariant of rational homology 3-spheres via vector fields. The construction of our invariant is a generalization of both that of the Kontsevich-Kuperberg-Thurston invariant and that of Watanabe's Morse homotopy invariant, which implies the equivalence of these two invariants.

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…

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…

Observable structures of a topological field theory of AKSZ type are analyzed. From a double (or multiple) complex structure of observable algebras, new topological invariants are constructed. Especially, Donaldson polynomial invariants and their generalizations are constructed from a topological field theory of AKSZ t…

Defines observer-invariant time derivatives on moving surfaces.

In this paper we consider simply connected Lie groups equipped with left invariant Randers metrics which arise from left invariant Riemannian metrics and left invariant vector fields. Then we study the intersection between automorphism and isometry groups of these spaces. Finally it has shown that for any left invarian…

Study identifies specific subvarieties in translation surfaces with quadratic field.

We call a connected Lie group endowed with a left-invariant Lorentzian flat metric Lorentzian flat Lie group. In this Note, we determine all Lorentzian flat Lie groups admitting a timelike left-invariant Killing vector field. We show that these Lie groups are 2-solvable and unimodular and hence geodesically complete. M…

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…

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…

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…

Removes singularities for Yang-Mills-Higgs fields in higher dimensions.

We study the asymptotics of a family of link invariants on the orbits of a smooth volume-preserving ergodic vector field on a compact domain of the 3-space. These invariants, called linear saddle invariants, include many concordance invariants and generate an infinite-dimensional vector space of link invariants. In con…

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

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…

We define some new invariants for 3-manifolds using the space of taut codim-1 foliations along with various techniques from noncommutative geometry. These invariants originate from our attempt to generalise Topological Quantum Field Theories in the Noncommutative geometry / topology realm.

The study explores harmonic vector fields on a specific type of Riemannian Lie group.

We determine the most general group of equivalence transformations for a family of differential equations defined by an arbitrary vector field on a manifold. We also find all invariants and differential invariants for this group up to the second order. A result on the characterization of classes of these equations by t…

Minhyong Kim introduced arithmetic Chern-Simons invariants for totally imaginary number fields as arithmetic analogues of the Chern-Simons invariants for 3-manifolds. In this paper, we extend Kim's definition for any number field, by using the modified étale cohomology groups and fundamental groups which take real plac…

Study spectral invariants over integers, discovering unboundedness and field-dependence.

We call a metric $m$-quasi-Einstein if $Ric_X^m$ (a modification of the $m$-Bakry-Emery Ricci tensor in terms of a suitable vector field $X$) is a constant multiple of the metric tensor. It is a generalization of Einstein metrics which contains Ricci solitons. In this paper, we focus on left-invariant vector fields and…

Quantum invariants from $U_h\mathfrak{sl}(2|1)$ are q-holonomic.

We consider the oscillator group equipped with a bi-invariant Lorentzian metric, and then some geometrical properties of this group i.e. homogeneous Ricci solitons and harmonicity properties of invariant vector fields are obtained. We also determine all vector fields which are critical points for the energy functional …

We study pathwise invariances of centred random fields that can be controlled through the covariance. A result involving composition operators is obtained in second-order settings, and we show that various path properties including additivity boil down to invariances of the covariance kernel. These results are extended…

The paper connects arithmetic invariants of hyperbolic 3-manifolds.

Constructs 3D topological field theories from a specific quantum group, linking to physics invariants.

Minimal vector fields on oscillator groups studied, with specific conditions for minimality.

In this paper, we investigated the behavior of left-invariant conformal vector fields on Lie groups with left-invariant pseudo-Riemannian metrics. First of all, we prove that conformal vector fields on pseudo-Riemannian unimodular Lie groups are Killing. Then we obtain a necessary condition for a pseudo-Rimennian non-u…

Given a family of Dirac operators with vanishing spectral flow we construct a thin-invariant rank-one field theory in the sense of Turner and Willerton arXiv:math.AT/0201116. Our construction of the field theory generalizes the one of the index gerbe by Lott, arXiv:math.DG/0106177, and it also complements the relation …

The paper classifies vector fields on 5D nilpotent Lie groups.

The study lists low-dimensional stratified groups and their properties.

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

A tensor field generates separation of variables for certain metrics.

Study of $k$-almost Yamabe solitons in perfect fluid spacetimes.

Study contact structures on projective spaces, proving infinite non-isotopic structures.

An odd vector field $Q$ on a supermanifold $M$ is called homological, if $Q^2=0$. The operator of Lie derivative $L_Q$ makes the algebra of smooth tensor fields on $M$ into a differential tensor algebra. In this paper, we give a complete classification of certain invariants of homological vector fields called character…

We will prove the equivariant version of Smale's transversality theorem: suppose that the compact Lie-group G acts on the compact differentiable manifold M on which an invariant Morse-function f and an invariant vector field X are given so that X is gradient-like with respect to f (i.e. X(f)<0 away from critical orbits…

A quasi-Poisson manifold is a G-manifold equipped with an invariant bivector field whose Schouten bracket is the trivector field generated by the invariant element in $\wedge^3 \g$ associated to an invariant inner product. We introduce the concept of the fusion for such manifolds, and we relate quasi-Poisson manifolds …

In 1998, R. Gompf defined a homotopy invariant $θ_G$ of oriented 2-plane fields in 3-manifolds. This invariant is defined for oriented 2-plane fields $ξ$ in a closed oriented 3-manifold $M$ when the first Chern class $c_1(ξ)$ is a torsion element of $H^2(M;\mathbb{Z})$. In this article, we define an extension of the Go…

Explains complex analytic invariants of vector fields and foliations.

Link homology theories connect to 4-manifold invariants and TQFTs.