arXiv research

New groups found in hyperbolic space with infinite fields of definition.

A new definition for vector fields extends the Jacobi set concept.

We show that degrees of the real fields of definition of arithmetic Kleinian reflection groups are bounded by 35.

Discuss various definitions of vector fields on manifolds leading to Lie algebras.

Tool uses text mining to define innovative tech fields from abstracts.

This paper derives radial fields on manifolds of symmetric positive definite matrices.

Study identifies specific subvarieties in translation surfaces with quadratic field.

The study defines fields of definition for triangle groups as Fuchsian groups.

The purpose of this paper is to put into a noncommutative context basic notions related to vector fields from classical differential geometry. The manner of exposition is an attempt to make the material as accessible as possible to classical geometers. The definition of vector field used is a specialisation of the Cart…

A graph theory approach defines curl and decomposes vector fields.

On a Hermitian manifold we construct a symmetric $(1,1)$- tensor $H$ using the torsion and the curvature of the Chern connection. On a compact balanced Hermitian manifold we find necessary and sufficient conditions in terms of the tensor $H$ for a harmonic $1$-form to be analytic and for an analytic $1$-form to be harm…

A new geometric definition of integration for differential forms.

Following the previous work of Nikulin and Agol, Belolipetsky, Storm, and Whyte it is known that there exist only finitely many (totally real) number fields that can serve as fields of definition of arithmetic hyperbolic reflection groups. We prove a new bound on the degree $n_k$ of these fields in dimension 3: $n_k$ d…

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

Paper defines and proves geometric uniqueness of Einstein field equations.

We prove that smooth 1-dimensional topological field theories over a manifold are equivalent to vector bundles with connection. The main novelty is our definition of the smooth 1-dimensional bordism category, which encodes cutting laws rather than gluing laws. We make this idea precise through a smooth version of Rezk'…

We propose a new definition for the abelian magnetic charge density of a non-abelian monopole, based on zero-modes of an associated Dirac operator. Unlike the standard definition of the charge density, this density is smooth in the core of the monopole. We show that this charge density induces a magnetic field whose ex…

Survey on embedding 3-manifolds in definite 4-manifolds, focusing on Donaldson's theorem.

Vanishing theorem for certain tensor fields on compact Hermitian manifolds.

Culler-Shalen theory uses the algebraic geometry of the SL(2,C)-character variety of a 3-manifold to construct essential surfaces in the manifold. There are module structures associated to the coordinate rings of the irreducible components of character varieties that are intimately related to essential surface construc…

Many machine learning systems make extensive use of large amounts of data regarding human behaviors. Several researchers have found various discriminatory practices related to the use of human-related machine learning systems, for example in the field of criminal justice, credit scoring and advertising. Fair machine le…

Let M be a paracompact smooth manifold, A a Weil algebra and M^A the associated Weil bundle. In this paper, we give another definition and characterization of vector field on M^A.

Log Sobolev and Michael Simon inequalities for tensor fields on curved manifolds.

We consider a family of $C^1$ vector fields satisfying a suitable higher order involutivity condition. We discuss the definition of commutators, the regularity of Sussmann's orbits and the Poincaré inequality.

The study defines divergence for multivector fields on infinite-dimensional manifolds.

Theory of Θ-positive representations for real closed fields.

We present a connection between the Killing fields that arise in the loop-group approach to integrable systems and conservation laws viewed as elements of the characteristic cohomology. We use the connection to generate the complete set of conservation laws (as elements of the characteristic cohomology) for the Tzitzei…

Study Schwarzians in the Heisenberg group, introducing new definitions and characterizing contact conformal vector fields.

For a spacelike 2-surface in spacetime, we propose a new definition of quasi-local angular momentum and quasi-local center of mass, as an element in the dual space of the Lie algebra of the Lorentz group. Together with previous defined quasi-local energy-momentum, this completes the definition of conserved quantities i…

We prove in a simple and coordinate-free way the equivalence bteween the classical definitions of the mass or the center of mass of an asymptotically flat manifold and their alternative definitions depending on the Ricci tensor and conformal Killing fields. This enables us to prove an analogous statement in the asympto…

Proves inequality for tensor fields on curved spaces.

We introduce and study covariance fields of distributions on a Riemannian manifold. At each point on the manifold, covariance is defined to be a symmetric and positive definite (2,0)-tensor. Its product with the metric tensor specifies a linear operator on the respected tangent space. Collectively, these operators form…

We introduce new partial orders on the set $S^+_n$ of positive-definite matrices of dimension $n$ derived from the homogeneous geometry of $S^+_n$ induced by the natural transitive action of the general linear group $GL(n)$. The orders are induced by affine-invariant cone fields, which arise naturally from a local anal…

Study examines null vector fields on Lorentzian manifolds.

Researchers redefine spinor field derivatives in generalized geometry.

We first present the natural definitions of the horizontal differential, the divergence (as an adjoint operator), and a $p$-harmonic form on a Finsler manifold. Next, we prove a Hodge-type theorem for a Finsler manifold in the sense that a horizontal $p$-form is harmonic if and only if the horizontal Laplacian vanishes…

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

In what follows we give a quick tour through the field of minimal submanifolds, starting at the definition and the classical results and ending up with current areas of research.

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

By resorting to Noether's Second Theorem, we relate the generalized Bianchi identities for Lagrangian field theories on gauge-natural bundles with the kernel of the associated gauge-natural Jacobi morphism. A suitable definition of the curvature of gauge-natural variational principles can be consequently formulated in …

The need to estimate smooth probability distributions (a.k.a. probability densities) from finite sampled data is ubiquitous in science. Many approaches to this problem have been described, but none is yet regarded as providing a definitive solution. Maximum entropy estimation and Bayesian field theory are two such appr…

We show that for a simple surface with boundary the attenuated ray transform in the presence of a unitary connection and a skew-Hermitian Higgs field is injective modulo the natural obstruction for functions and vector fields. We also show that the connection and the Higgs field are uniquely determined by the scatterin…

Formula derived for a magnetic line invariant.

A purely algebraic construction of super-energy tensors for arbitrary fields is presented in any dimensions. These tensors have good mathematical and physical properties, and they can be used in any theory having as basic arena an n-dimensional manifold with a metric of Lorentzian signature. In general, the completely …

New method constructs axial vector fields and defines quasi-local spin-angular momentum.

Paper reviews and proves the uniqueness of multipole moments for stationary spacetimes.

We give a derivation of the Einstein equation for gravity which employs a definition of the local energy density of the gravitational field as a symmetric second rank tensor whose value for each observer gives the trace of the spatial part of the energy-stress tensor as seen by that observer. We give a physical motivat…

In this text we combine the notions of supergeometry and supersymmetry. We construct a special class of supermanifolds whose reduced manifolds are (pseudo) Riemannian manifolds. These supermanifolds allow us to treat vector fields on the one hand and spinor fields on the other hand as equivalent geometric objects. This…