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首页> 外文期刊>Canadian Journal of Mathematics >Relative Darboux Theorem for Singular Manifolds and Local Contact Algebra
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Relative Darboux Theorem for Singular Manifolds and Local Contact Algebra

机译：奇异流形和局部接触代数的相对达布定理

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In 1999 V. Arnol'd introduced the local contact algebra: studying theproblem of classification of singular curves in a contact space, heshowed the existence of the ghost of the contact structure (invariantswhich are not related to the induced structure on the curve). Ourmain result implies that the only reason for existence of the localcontact algebra and the ghost is the difference between the geometricand (defined in this paper) algebraic restriction of a $1$-form to asingular submanifold. We prove that a germ of any subset $N$ of acontact manifold is well defined, up to contactomorphisms, by thealgebraic restriction to $N$ of the contact structure. This is ageneralization of the Darboux-Givental' theorem for smoothsubmanifolds of a contact manifold. Studying the difference betweenthe geometric and the algebraic restrictions gives a powerful tool forclassification of stratified submanifolds of a contact manifold. Thisis illustrated by complete solution of three classification problems,including a simple explanation of V.~Arnold's results and furtherclassification results for singular curves in a contact space. Wealso prove several results on the external geometry of a singularsubmanifold $N$ in terms of the algebraic restriction of the contactstructure to $N$. In particular, the algebraic restriction is zero ifand only if $N$ is contained in a smooth Legendrian submanifold of$M$.
机译：1999年，V。Arnol'd引入了局部接触代数：研究接触空间中奇异曲线的分类问题，他证明了接触结构的重影的存在（不变量与曲线上的诱导结构无关）。我们的主要结果表明，存在局部接触代数和重影的唯一原因是1美元形式对奇异子流形的几何和（本文定义）代数约束之间的差异。我们证明，通过代数式限制接触结构的$ N $，可以很好地定义接触流形的任何子集$ N $的细菌，直至触同。这是接触流形的光滑子流形的Darboux-Givental定理的一般化。研究几何和代数限制之间的差异为分类接触流形的分层子流形提供了一个强大的工具。完全解决三个分类问题就说明了这一点，其中包括对V.〜Arnold结果的简单解释以及对接触空间中奇异曲线的进一步分类结果。我们还将奇异子流形$ N $的外部几何形状证明为接触结构对$ N $的代数约束的几个结果。特别是，仅当$ N $包含在$ M $的光滑Legendrian子流形中时，代数限制为零。

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