{"title":"Wide short geodesic loops on closed Riemannian manifolds","authors":"Regina Rotman","doi":"10.1142/s1793525323500486","DOIUrl":null,"url":null,"abstract":"It is not known whether or not the lenth of the shortest periodic geodesic on a closed Riemannian manifold $M^n$ can be majorized by $c(n) vol^{ 1 \\over n}$, or $\\tilde{c}(n)d$, where $n$ is the dimension of $M^n$, $vol$ denotes the volume of $M^n$, and $d$ denotes its diameter. In this paper we will prove that for each $\\epsilon >0$ one can find such estimates for the length of a geodesic loop with with angle between $\\pi-\\epsilon$ and $\\pi$ with an explicit constant that depends both on $n$ and $\\epsilon$. That is, let $\\epsilon > 0$, and let $a = \\lceil{ {1 \\over {\\sin ({\\epsilon \\over 2})}}} \\rceil+1 $. We will prove that there exists a wide (i.e. with an angle that is wider than $\\pi-\\epsilon$) geodesic loop on $M^n$ of length at most $2n!a^nd$. We will also show that there exists a wide geodesic loop of length at most $2(n+1)!^2a^{(n+1)^3} FillRad \\leq 2 \\cdot n(n+1)!^2a^{(n+1)^3} vol^{1 \\over n}$. Here $FillRad$ is the Filling Radius of $M^n$.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Topology and Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1142/s1793525323500486","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 3
Abstract
It is not known whether or not the lenth of the shortest periodic geodesic on a closed Riemannian manifold $M^n$ can be majorized by $c(n) vol^{ 1 \over n}$, or $\tilde{c}(n)d$, where $n$ is the dimension of $M^n$, $vol$ denotes the volume of $M^n$, and $d$ denotes its diameter. In this paper we will prove that for each $\epsilon >0$ one can find such estimates for the length of a geodesic loop with with angle between $\pi-\epsilon$ and $\pi$ with an explicit constant that depends both on $n$ and $\epsilon$. That is, let $\epsilon > 0$, and let $a = \lceil{ {1 \over {\sin ({\epsilon \over 2})}}} \rceil+1 $. We will prove that there exists a wide (i.e. with an angle that is wider than $\pi-\epsilon$) geodesic loop on $M^n$ of length at most $2n!a^nd$. We will also show that there exists a wide geodesic loop of length at most $2(n+1)!^2a^{(n+1)^3} FillRad \leq 2 \cdot n(n+1)!^2a^{(n+1)^3} vol^{1 \over n}$. Here $FillRad$ is the Filling Radius of $M^n$.
期刊介绍:
This journal is devoted to topology and analysis, broadly defined to include, for instance, differential geometry, geometric topology, geometric analysis, geometric group theory, index theory, noncommutative geometry, and aspects of probability on discrete structures, and geometry of Banach spaces. We welcome all excellent papers that have a geometric and/or analytic flavor that fosters the interactions between these fields. Papers published in this journal should break new ground or represent definitive progress on problems of current interest. On rare occasion, we will also accept survey papers.