{"title":"Upper bounds for the critical values of homology classes of loops","authors":"Hans-Bert Rademacher","doi":"10.1007/s00229-024-01541-7","DOIUrl":null,"url":null,"abstract":"<p>In this short note we discuss upper bounds for the critical values of homology classes in the based and free loop space of compact manifolds carrying a Riemannian or Finsler metric of positive Ricci curvature. In particular it follows that a shortest closed geodesic on a compact and simply-connected <i>n</i>-dimensional manifold of positive Ricci curvature <span>\\(\\text {Ric}\\ge n-1\\)</span> has length <span>\\(\\le n \\pi .\\)</span> This improves the bound <span>\\(8\\pi (n-1)\\)</span> given by Rotman (Positive Ricci curvature and the length of a shortest periodic geodesic. arXiv:2203.09492, 2022).</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00229-024-01541-7","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
In this short note we discuss upper bounds for the critical values of homology classes in the based and free loop space of compact manifolds carrying a Riemannian or Finsler metric of positive Ricci curvature. In particular it follows that a shortest closed geodesic on a compact and simply-connected n-dimensional manifold of positive Ricci curvature \(\text {Ric}\ge n-1\) has length \(\le n \pi .\) This improves the bound \(8\pi (n-1)\) given by Rotman (Positive Ricci curvature and the length of a shortest periodic geodesic. arXiv:2203.09492, 2022).
在这篇短文中,我们讨论了携带正利玛窦曲率的黎曼或芬斯勒度量的紧凑流形的基于和自由环空间中的同调类临界值的上限。这改进了罗特曼 (Positive Ricci curvature and the length of a shortest periodic geodesic. arXiv:2203.09492, 2022) 给出的边界 \(8\pi (n-1)\).