{"title":"Simple closed geodesics in dimensions $$\\ge 3$$","authors":"","doi":"10.1007/s11784-023-01092-6","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>We show that for a generic Riemannian or reversible Finsler metric on a compact differentiable manifold <em>M</em> of dimension at least three all closed geodesics are simple and do not intersect each other. Using results by Contreras (Ann Math 2(172):761–808, 2010; in: Proceedings of International Congress Mathematicians (ICM 2010) Hyderabad, India, pp 1729–1739, 2011) this shows that for a generic Riemannian metric on a compact and simply-connected manifold all closed geodesics are simple and the number <em>N</em>(<em>t</em>) of geometrically distinct closed geodesics of length <span> <span>\\(\\le t\\)</span> </span> grows exponentially.</p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s11784-023-01092-6","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
We show that for a generic Riemannian or reversible Finsler metric on a compact differentiable manifold M of dimension at least three all closed geodesics are simple and do not intersect each other. Using results by Contreras (Ann Math 2(172):761–808, 2010; in: Proceedings of International Congress Mathematicians (ICM 2010) Hyderabad, India, pp 1729–1739, 2011) this shows that for a generic Riemannian metric on a compact and simply-connected manifold all closed geodesics are simple and the number N(t) of geometrically distinct closed geodesics of length \(\le t\) grows exponentially.
摘要 我们证明，对于至少三维的紧凑可变流形 M 上的一般黎曼或可逆芬斯勒度量，所有闭合大地线都是简单且互不相交的。利用孔特雷拉斯的结果（Ann Math 2(172):761-808, 2010; in：Proceedings of International Congress Mathematicians (ICM 2010) Hyderabad, India, pp 1729-1739, 2011）表明，对于紧凑且简单连接流形上的一般黎曼度量，所有闭大地线都是简单的，且长度为 \(\le t\) 的几何上不同的闭大地线的数量 N(t) 呈指数增长。
期刊介绍：
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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