Hexagonal geodesic 3-webs

Geodesic Webs of Hypersurfaces

In the present paper we study geometric structures associated with webs of hypersurfaces. We prove that with any geodesic (n+2)-web on an n-dimensional manifold there is naturally associated a unique projective structure and, provided that one of web foliations is pointed, there is also associated a unique affine structure. The projective structure can be chosen by the claim that the leaves of all web foliations are totally geodesic, and the affine structure by an additional claim that one of web functions is affine. These structures allow us to determine differential invariants of geodesic we…

Geodesic Webs on a Two-Dimensional Manifold and Euler Equations

We prove that any planar 4-web defines a unique projective structure in the plane in such a way that the leaves of the foliations are geodesics of this projective structure. We also find conditions for the projective structure mentioned above to contain an affine symmetric connection, and conditions for a planar 4-web to be equivalent to a geodesic 4-web on an affine symmetric surface. Similar results are obtained for planar d-webs, d > 4, provided that additional d-4 second-order invariants vanish.

Quadratic integrals of geodesic flow, webs, and integrable billiards

We present a geometric interpretation of integrability of geodesic flow by quadratic integrals in terms of the web theory and construct integrable billiards on surfaces admitting such integrals.

On four-dimensional three-webs with integrable transversal distributions

For a four-dimensional (nonisoclinicly geodesic) three-web W (3, 2, 2), a transversal distribution $Δ$ is defined by the torsion tensor of the web. In general, this distribution is not integrable. The authors find necessary and sufficient conditions of its integrability and prove the existence theorem for webs W (3, 2, 2) with integrable distributions $Δ$. They prove that for a web W (3, 2, 2) with the integrable distribution $Δ$, the integral surfaces $V^2$ of $Δ$ are totally geodesic in an affine connection of a certain bundle of affine connections. They also consider a class of webs W (3, 2…

曲面上测地流的分式线性积分与Nakai测地4-网

Fractional-linear integrals of geodesic flows on surfaces and Nakai's geodesic 4-webs

证明了曲面上测地流的动量分式线性积分空间维数为3或5（后者对应常高斯曲率），并给出其存在的几何判据为曲面承载具有常交比的测地4-网。

On Rank Problems for Planar Webs and Projective Structures

We present old and recent results on rank problems and linearizability of geodesic planar webs.

在六边形结构与 Monge–Ampère 算子上的学习

Learning on hexagonal structures and Monge-Ampère operators

本文证明对偶平坦统计流形即 Monge–Ampère 流形，并用 Monge–Ampère 算子、可平行化 webs 与六边形结构刻画 Boltzmann 机学习。

Projective invariants of linear 3-webs and Gronwall's Conjecture

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.

Flat 3-webs via semi-simple Frobenius 3-folds

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.

由两族圆覆盖的曲面

Surfaces that are covered by two pencils of circles

本研究在莫比乌斯等价下分类了至少含两族圆的实曲面的几何特征，编码于Néron-Severi格中，还证实高维Blum猜想、解决Blaschke-Bol问题并发现新的非三维可嵌入圆六边形网格。

Chains in CR geometry as geodesics of a Kropina metric

With the help of a generalization of the Fermat principle in general relativity, we show that chains in CR geometry are geodesics of a certain Kropina metric constructed from the CR structure. We study the projective equivalence of Kropina metrics and show that if the kernel distributions of the corresponding 1-forms are non-integrable then two projectively equivalent metrics are trivially projectively equivalent. As an application, we show that sufficiently many chains determine the CR structure up to conjugacy, generalizing and reproving the main result of [J.-H. Cheng, 1988]. The correspond…

Maximum rank of a Legendrian web

We propose the Legendrian web in a contact three manifold as a second order generalization of the planar web. An Abelian relation for a Legendrian web is analogously defined as an additive equation among the first integrals of its foliations. For a class of Legendrian $\, d$-webs defined by simple second order ODE's, we give an algebraic construction of $$ ρ_d = \frac{(d-1)(d-2)(2d+3)}{6}$$ linearly independent Abelian relations. We then employ the method of local differential analysis and the theory of linear differential systems to show that $\, ρ_d$ is the maximum rank of a Legendrian $\, d…