Microflexiblity and local integrability of horizontal curves

IF 0.8 3区数学 Q2 MATHEMATICS

Mathematische Nachrichten Pub Date : 2024-06-16 DOI:10.1002/mana.202200306

Álvaro del Pino, Tobias Shin

{"title":"Microflexiblity and local integrability of horizontal curves","authors":"Álvaro del Pino, Tobias Shin","doi":"10.1002/mana.202200306","DOIUrl":null,"url":null,"abstract":"Let <math>\n <semantics>\n <mi>ξ</mi>\n <annotation>$\\xi$</annotation>\n </semantics></math> be an analytic bracket-generating distribution. We show that the subspace of germs that are singular (in the sense of control theory) has infinite codimension within the space of germs of smooth curves tangent to <math>\n <semantics>\n <mi>ξ</mi>\n <annotation>$\\xi$</annotation>\n </semantics></math>. We formalize this as an asymptotic statement about finite jets of tangent curves. This solves, in the analytic setting, a conjecture of Eliashberg and Mishachev regarding an earlier claim by Gromov about the microflexibility of the tangency condition.From these statements it follows, by an argument due to Gromov, that the <math>\n <semantics>\n <mi>h</mi>\n <annotation>$h$</annotation>\n </semantics></math>-principle holds for maps and immersions transverse to <math>\n <semantics>\n <mi>ξ</mi>\n <annotation>$\\xi$</annotation>\n </semantics></math>.","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 9","pages":"3252-3287"},"PeriodicalIF":0.8000,"publicationDate":"2024-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.202200306","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematische Nachrichten","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/mana.202200306","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

Let $ξ$ be an analytic bracket-generating distribution. We show that the subspace of germs that are singular (in the sense of control theory) has infinite codimension within the space of germs of smooth curves tangent to $ξ$ . We formalize this as an asymptotic statement about finite jets of tangent curves. This solves, in the analytic setting, a conjecture of Eliashberg and Mishachev regarding an earlier claim by Gromov about the microflexibility of the tangency condition.

From these statements it follows, by an argument due to Gromov, that the $h$ -principle holds for maps and immersions transverse to $ξ$ .

查看原文本刊更多论文

水平曲线的微柔性和局部可积分性

让 ξ $\xi$ 是一个解析括号生成分布。我们证明，在与 ξ $\xi$ 相切的光滑曲线的胚芽空间中，奇异胚芽的子空间（在控制论的意义上）具有无限的开方维。我们将其形式化为关于切线曲线的有限射流的渐近声明。从这些陈述中，通过格罗莫夫的论证，h $h $ 原则对于横向于 ξ $\xi$ 的映射和浸入是成立的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Mathematische Nachrichten 数学-数学

CiteScore

1.50

自引率

0.00%

发文量

157

审稿时长

4-8 weeks

期刊介绍： Mathematische Nachrichten - Mathematical News publishes original papers on new results and methods that hold prospect for substantial progress in mathematics and its applications. All branches of analysis, algebra, number theory, geometry and topology, flow mechanics and theoretical aspects of stochastics are given special emphasis. Mathematische Nachrichten is indexed/abstracted in Current Contents/Physical, Chemical and Earth Sciences; Mathematical Review; Zentralblatt für Mathematik; Math Database on STN International, INSPEC; Science Citation Index