A Canonical Frame for Nonholonomic Rank Two Distributions of Maximal Class

Canonical frames for distributions of odd rank and corank 2 with maximal first Kronecker index

We construct canonical frames and find all maximally symmetric models for a natural generic class of corank 2 distributions on manifolds of odd dimension greater or equal to 7. This class of distributions is characterized by the following two conditions: the pencil of 2-forms associated with the corresponding Pfaffian system has the maximal possible first Kronecker index and the Lie square of the subdistribution generated by the kernels of all these 2-forms is equal to the original distribution. In particular, we show that the unique, up to a local equivalence, maximally symmetric model in thi…

Nurowski's conformal structures for (2,5)-distributions via dynamics of abnormal extremals

As was shown recently by P. Nurowski, to any rank 2 maximally nonholonomic vector distribution on a 5-dimensional manifold M one can assign the canonical conformal structure of signature (3,2). His construction is based on the properties of the special 12-dimensional coframe bundle over M, which was distinguished by E. Cartan during his famous construction of the canonical coframe for this type of distributions on some 14-dimensional principal bundle over M. The natural question is how "to see" the Nurowski conformal structure of a (2,5)-distribution purely geometrically without the preliminar…

Canonical Cartan Connections on Maximally Minimal Generic Submanifolds M^5 in C^4

On a real analytic 5-dimensional CR-generic submanifold M^5 in C^4 of codimension 3, hence of CR dimension 1, which enjoys the generically satisfied nondegeneracy condition that Lie brackets up to length 3 of T^{1,0}M generate CTM, a canonical Cartan connection is constructed after reduction to a certain partially explicit e-structure of the concerned local biholomorphic equivalence problem. More advanced explorations of the incoming differential invariants due to the first and to the third authors already appeared in January 2014, hence the purpose is to show, while studying this specific Cla…

Some maximal isotropic distributions and their relation to field theory

We study the behaviour of differential forms in a manifold having at least one of their maximal isotropic local distributions endowed with the special algebraic property of being decomposable. We show that they can be represented as the sum of a form with constant coefficients and one that vanishes whenever contracted with vector fields in the former distribution, provided some simple integrability conditions are ensured. We also classify possible 'canonical coordinates' for a certain class of forms with potential applications in classical field theory.

Geometry of rank 2 distributions with nonzero Wilczynski invariants and affine control systems with one input

We demonstrate how the novel approach to the local geometry of structures of nonholonomic nature, originated by Andrei Agrachev, works in the following two situations: rank 2 distributions of maximal class in R^n with non-zero generalized Wilczynski invariants and rank 2 distributions of maximal class in R^n with additional structures such as affine control system with one input spanning these distributions, sub-(pseudo)Riemannian structures etc. The common feature of these two situations is that each abnormal extremal (of the underlying rank 2 distribution) possesses a distinguished parametri…

洛伦兹度量空间的极性和最大柯西发展

Maximality and Cauchy developments of Lorentzian length spaces

本文探讨洛伦兹度量空间的最大柯西发展问题，提出指向格罗莫夫-豪斯多夫度量定义，提供非时空例子及规范代表，强调几何流形的良定性。

显式极大全实嵌入

Explicit maximal totally real embeddings

该研究作为极大全实嵌入相关工作的续作，针对无挠实解析紧流形的典范复结构，给出了以曲率单项式对称化表示的纤维式泰勒展开及显式系数，拓展了论证适用范围并为可积方程的典范零化提供了佐证。

具有非Goursat性质的括号生成秩2分布的规范框架

Canonical Frames for Bracket Generating Rank 2 Distributions which are not Goursat

本文通过证明最大类条件自动成立，完成对括号生成秩2分布的规范绝对平行性构造，并揭示了相关分布的最大对称germs及最优控制问题中的极值轨迹特性。

Basic cohomology of canonical holomorphic foliations on complex moment-angle manifolds

We describe the basic cohomology ring of the canonical holomorphic foliation on a moment-angle manifold, LVMB-manifold or any complex manifold with a maximal holomorphic torus action. Namely, we show that the basic cohomology has a description similar to the cohomology ring of a complete simplicial toric variety due to Danilov and Jurkiewicz. This settles a question of Battaglia and Zaffran, who previously computed the basic Betti numbers for the canonical holomorphic foliation in the case of a shellable fan. Our proof uses an Eilenberg-Moore spectral sequence argument; the key ingredient is t…

Weil-Petersson类在$\mathbb{A}\mathrm{d}\mathbb{S}^{2,1}$中的最大圆盘

Maximal discs of Weil-Petersson class in $\mathbb{A}\mathrm{d}\mathbb{S}^{2,1}$

研究Weil-Petersson类在3维反de Sitter空间中的最大圆盘，证明Mess映射在Weil-Petersson万有Teichmüller空间的余切 bundle 上的辛微分同胚，并展示反共轭能量函数作为Kähler势的作用。