Related papers: Variance of vector fields -- Definition and proper…
Vector fields and line fields, their counterparts without orientations on tangent lines, are familiar objects in the theory of dynamical systems. Among the techniques used in their study, the Morse--Smale decomposition of a (generic) field…
This paper introduces the notions of vector field and flow on a general differentiable stack. Our main theorem states that the flow of a vector field on a compact proper differentiable stack exists and is unique up to a uniquely determined…
In our previous paper entitled "Axiomatic differential geometry -towards model categories of differential geometry-, we have given a category-theoretic framework of differential geometry. As the first part of our series of papers concerned…
Different (not only by sign) affine connections are introduced for contravariant and covariant tensor fields over a differentiable manifold by means of a non-canonical contraction operator, defining the notion dual bases and commuting with…
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…
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…
A simple proof is given for the explicit formula which allows one to recover a $C^2-$smooth vector field $A=A(x)$ in $\mathbb{R}^3$, decaying at infinity, from the knowledge of its $\nabla \times A$ and $\nabla \cdot A$. The representation…
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…
We give an argument that magnetic monopoles should not exist. It is based on the concept of the index of a vector field. The thrust of the argument is that indices of vector fields are invariants of space-time orientation and of coordinate…
We study the integrability of polynomial vector fields using Galois theory of linear differential equations when the associated foliations is reduced to a Riccati type foliation. In particular we obtain integrability results for some…
A new cohomology, induced by a vector field, is defined on pairs of differential forms ($1$--differentiable forms) in a manifold. It is proved a link with the classical de Rham cohomology and an $1$-differentable cohomology of Lichnerowicz…
We classify, up to a natural equivalence relation, vector fields of the plane which belong to the kernel of a 1--form. This form can be closed, in which case the vector fields are integrable, or not, in which case the differential of the…
The electrostatic field magnitude can play a role in a variational principle similar to the role of the index of refraction in geometrical optics, allowing the determination of the electric field lines. This was identified in the context of…
Relativistic field theory for a vector field on a curved space-time is considered assuming that the Lagrangian field density is quadratic and contains field derivatives of first order at most. By applying standard variational calculus, the…
This note proposes a new notion of a gradient-like vector field and discusses its implications for the theory of Stein and Weinstein structures.
Vector fields in the expanding Universe are considered within the multidimensional theory of General Relativity. Vector fields in general relativity form a three-parametric variety. Our consideration includes the fields with a nonzero…
Conformal fields are a recently discovered class of representations of the algebra of vector fields in $N$ dimensions. Invariant first-order differential operators (exterior derivatives) for conformal fields are constructed.
We examine the mechanism for generating a mass for a U(1) vector field introduced by Stueckelberg. First, it is shown that renormalization of the vector mass is identical to the renormalization of the vector field on account of gauge…
$Vect(N)$, the algebra of vector fields in $N$ dimensions, is studied. Some aspects of local differential geometry are formulated as $Vect(N)$ representation theory. There is a new class of modules, {\it conformal fields}, whose…
We introduce new motivic invariants of arbitrary varieties over a perfect field. These cohomological invariants take values in the category of one-motives (considered up to isogeny in positive characteristic). The algebraic definition of…