### Algebraic aspects of web geometry

Algebraic aspects of web geometry, namely its connections with the quasigroup and loop theory, the theory of local differential quasigroups and loops, and the theory of local algebras are discussed.

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

Algebraic aspects of web geometry, namely its connections with the quasigroup and loop theory, the theory of local differential quasigroups and loops, and the theory of local algebras are discussed.

The notions of left Bol and Bol-Bruck actions are introduced. A purely algebraic analogue of a Nono family (Lie triple family), the so called Sabinin-Nono family, is given. It is shown that any Sabinin-Nono family is a left Bol-Bruck action. Finally it is proved that any local Nono family is a local left Bol-Bruck action. On general matters see [L.V. Sabinin 91, 99].

In this paper we show that well-known relationships connecting the Clifford algebra on negative euclidean space, Vahlen matrices, and Möbius transformations extend to connections with the Möbius loop or gyrogroup on the open unit ball $B$ in $n$-dimensional euclidean space ${\mathbb{R}}^{n}$. One notable achievement is a compact, convenient formula for the Möbius loop operation $a*b=(a+b){(1-ab)}^{-1}$, where the operations on the right are those arising from the Clifford algebra (a formula comparable to $(w+z){(1+\overline{w}z)}^{-1}$ for the Möbius loop multiplication...

We study several linear connections (the first canonical, the Chern, the well adapted, the Levi Civita, the Kobayashi-Nomizu, the Yano, the Bismut and those with totally skew-symmetric torsion) which can be defined on the four geometric types of $({J}^{2}=\pm 1)$-metric manifolds. We characterize when such a connection is adapted to the structure, and obtain a lot of results about coincidence among connections. We prove that the first canonical and the well adapted connections define a one-parameter family of adapted...

The aim of this work is to study global $3$-webs with vanishing curvature. We wish to investigate degree $3$ foliations for which their dual web is flat. The main ingredient is the Legendre transform, which is an avatar of classical projective duality in the realm of differential equations. We find a characterization of degree $3$ foliations whose Legendre transform are webs with zero curvature.

An isotypic Kronecker web is a family of corank m foliations ${\left\{{\mathcal{F}}_{t}\right\}}_{t\in \mathbb{R}{P}^{1}}$ such that the curve of annihilators t ↦ (T x F t)⊥ ∈ Grm(T x* M) is a rational normal curve in the Grassmannian Grm(T x*M) at any point x ∈ M. For m = 1 we get Veronese webs introduced by I. Gelfand and I. Zakharevich [Gelfand I.M., Zakharevich I., Webs, Veronese curves, and bi-Hamiltonian systems, J. Funct. Anal., 1991, 99(1), 150–178]. In the present paper, we consider the problem of local classification of isotypic Kronecker webs...

The webs have been studied mainly locally, near regular points (see a short list of references on the topic in the bibliography). Let d be an integer ≥ 1. A d-web on an open set U of ℂ² is a differential equation F(x,y,y’) = 0 with $F(x,y,{y}^{\text{'}})={\sum}_{i=0}^{d}{a}_{i}(x,y){\left({y}^{\text{'}}\right)}^{d-i}$, where the coefficients ${a}_{i}$ are holomorphic functions, a₀ being not identically zero. A regular point is a point (x,y) where the d roots in y’ are distinct (near such a point, we have locally d foliations mutually transverse to each other, and caustics appear through...

Beaucoup de concepts sur les tissus n’ont été étudiés que localement. Il apparaît que certains d’entre eux se laissent globaliser, mais pas toujours de façon immédiate. Le premier objectif de cet article est de préciser à chaque fois ce qu’il en est, et de mettre en place les outils utiles à une étude globale des tissus sur une surface holomorphe $M$ arbitraire, et en particulier sur le plan projectif complexe ${\mathbb{P}}_{2}$. Certains concepts nouveaux vont alors apparaître, tels le type (ou le degré si $M={\mathbb{P}}_{2}$), la...

This paper gives a brief survey of certain recently developing aspects of the study of loops and quasigroups, focussing on some of the areas that appear to exhibit the best prospects for subsequent research and for applications both inside and outside mathematics.

We investigate real analytic Levi-flat hypersurfaces tangent to holomorphic webs. We introduce the notion of first integrals for local webs. In particular, we prove that a $k$-web with finitely many invariant subvarieties through the origin tangent to a Levi-flat hypersurface has a holomorphic first integral.

Plane $d$-webs have been studied a lot since their appearance at the turn of the 20th century. A rather recent and striking result for them is the theorem of Dufour, stating that the measurable conjugacies between 3-webs have to be analytic. Here, we show that even the set-theoretic conjugacies between two $d$-webs, $d\ge 3$ are analytic unless both webs are analytically parallelizable. Between two set-theoretically conjugate parallelizable $d$-webs, however, there always exists a nonmeasurable conjugacy; still,...