我们给出了各种结果和应用程序使用的连接$(E,\nabla)$与$d$ -Web。准确地说，我们展示了基本不变量的网络相关的微分方程的一阶，提出了网络。他们在概念上和有效地对连接及其构建投下了一些新的光芒。我们描述4和5网从这个角度来看，并显示，例如，该连接考虑到网络的线性化条件。此外，我们还得到了极大秩网的特征，如例外5-网和4-网\emph{via}，这是Poincaré定理的一个新的证明。我们建立了与行列式丛$(\det E, \det \nabla)$和抽取的3-网有关的迹公式。此外，还证明了秩的判定定理，从而给出了衡量网秩的一个明确准则。
英文摘要：
We give various results and applications using the connection $(E,\nabla)$ associated with a $d$-web. Precisely, we exhibit fundamental invariants of the web related to the differential equation of first order which presents the web. They cast some new lights on the connection and its construction, both conceptually and effectively. We describe 4 and 5-webs from this point of view and show for instance that the connection gives account for the linearizability conditions of the web. Moreover, we get characterization of maximal rank webs such as exceptional 5-webs, and 4-webs \emph{via} a new proof of the Poincaré theorem in terms of differential systems. We establish the trace formula related to the determinant bundle $(\det E, \det \nabla)$ and the extracted 3-webs. Furthermore, the theorem of determination of the rank is proved to give an explicit criterion for measuring the rank of a web.
We propose a definition of genericity for singular flat planar 3-webs formed by integral curves of implicit ODEs and give a classification of generic singularities of such webs.
In the complex setting, let $F(x,y,y')=0$ be an analytic or algebraic differential equation with $y'$-degree $d$. We deal with the qualitative study of such equations through the geometry of the planar $d$-web generated by the generic family of integral curves. Infinitesimal symmetries of these configurations are discussed by using meromorphic connections methods "à la Cartan-Spencer".
We study non-flat planar 3-webs with infinitesimal symmetries. Using multi-dimensional Schwarzian derivative we give a criterion for linearization of such webs and present a projective classification thereof. Using this classification we show that the Gronwall conjecture is true for 3-webs admitting infinitesimal symmetries.
We provide a complete classification of hexagonal singular 3-web germs in the complex plane, satisfying the following two conditions: 1) the Chern connection remains holomorphic at the singular point, 2) the web admits at least one infinitesimal symmetry at this point. As a by-product, a classification of hexagonal weighted homogeneous 3-webs is obtained.
We present a projectively invariant description of planar linear 3-webs. For a non-hexagonal 3-web, we introduce family of projective torsion-free Cartan connections, the web leaves being geodesics for each member of the family, and give a web linearization criterion. Finally, we propose an algorithm for resolving the Gronwall conjecture and illustrate this approach by proving the conjecture for 3-webs whose 2 foliations are 2 pencils of lines.
Confocal conics form an orthogonal net. Supplementing this net with one of the following: 1) the net of Cartesian coordinate lines aligned along the principal axes of conics, 2) the net of Apollonian pencils of circles whose foci coincide with the foci of conics, 3) the net of tangents to a conic of the confocal family, we get a planar 4-web. We show that each of these 4-webs is of maximal rank and characterize confocal conics from the web theory viewpoint.
A criterion for the holomorphy of the curvature of smooth planar webs and applications to dual webs of homogeneous foliations on $\mathbb{P}^{2}_{\mathbb{C}}$
We construct flat 3-webs via semi-simple geometric Frobenius manifolds of dimension three and give geometric interpretation of the Chern connection of the web. These webs turned out to be biholomorphic to the characteristic webs on the solutions of the corresponding associativity equation. We show that such webs are hexagonal and admit at least one infinitesimal symmetry at each singular point. Singularities of the web are also discussed.
本文研究了二维流形上三维网的线性化问题。通过基于论文“On the linearizability of 3-webs”（Nonlinear analysis 47，（2001）pp. 2643-2654），我们研究了3-web，其线性化在同一篇论文中被要求。我们表明，相反的声明的文件“关于Blaschke猜想为3-网”（arXiv：数学DG/0411460）和“关于线性化的平面三网和Blaschke猜想”，（C.R.Acad. Sci.巴黎，一级森林第341卷。num 3（2005）），该特定的网是可线性化的。我们明确计算的仿射变形张量和相应的平面线性连接适应于网络的线性化。
We give a geometric interpretation of weighted homogeneous solutions to the associativity equation in terms of the web theory and construct a massive Frobenius 3-fold germ via a singular 3-web germ satisfying the following conditions: 1) the web germ admits at least one infinitesimal symmetry, 2) the Chern connection form is holomorphic, 3) the curvature form vanishes identically.
We give a complete proof of the fact that the trace of the curvature of the connection associated to a planar d-web (d>3) is the sum of the Blaschke curvatures of its sub 3-webs.
According to Alain Hénaut, a planar 4-web is called Nakai's web if the cross-ratio of the tangents to the four foliations at each point is constant and if it has no hexagonal 3-subweb. We prove that Nakai's webs have rank 0 or 1. We give examples with rank 1 and present a universal way to build such examples.
Gronwall conjecture states that a planar 3-web which admits more than one distinct linearization is locally equivalent to an algebraic web. We give a partial answer to the conjecture in the affirmative for the class of planar 3-webs with the web curvature that vanishes to order three at a point. The differential relation on the third order jet of web curvature provides an explicit criterion for unique linearization.
