封闭黎曼流形上的宽短测地线环

IF 0.5 3区 数学 Q3 MATHEMATICS
Regina Rotman
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引用次数: 3

摘要

目前尚不清楚闭合黎曼流形$M^n$上最短周期测地线的长度是否可以用$c(n) vol^{ 1 \over n}$或$\tilde{c}(n)d$来表示,其中$n$是$M^n$的尺寸,$vol$表示$M^n$的体积,$d$表示其直径。在本文中,我们将证明,对于每一个$\epsilon >0$,我们都可以找到这样的测量回路长度的估计,其角度在$\pi-\epsilon$和$\pi$之间,并具有一个显式常数,该常数依赖于$n$和$\epsilon$。也就是说,设$\epsilon > 0$,设$a = \lceil{ {1 \over {\sin ({\epsilon \over 2})}}} \rceil+1 $。我们将证明在$M^n$上存在一个长度不超过$2n!a^nd$的宽(即夹角大于$\pi-\epsilon$)测地线环。我们还将证明存在长度最多为$2(n+1)!^2a^{(n+1)^3} FillRad \leq 2 \cdot n(n+1)!^2a^{(n+1)^3} vol^{1 \over n}$的宽测地线回路。其中$FillRad$为$M^n$的填充半径。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Wide short geodesic loops on closed Riemannian manifolds
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 $M^n$.
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来源期刊
CiteScore
1.50
自引率
0.00%
发文量
13
审稿时长
>12 weeks
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.
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