商空间上的诱导Hausdorff度量
Induced Hausdorff metrics on quotient spaces
摘要：
设$G$是群，$(M,d)$是度量空间，$X$是$M$的紧致子空间，$φ:G\times M \rightarrow M$是$G$在$M$上的同胚左作用.表示为$gp=f(g,p)$。$G$相对于$X$的各向同性子群由$H_X=\{g\in G; gX=X\}$定义。在这项工作中，我们定义的诱导Hausdorff度量的$G/H_X$由$d_X(gH_X,hH_X)=d_H(gX,hX)$，其中$d_H$是Hausdorff距离$M$。令$\hat d_X$为由$d_X$诱导的本征度量。本文研究了$(G/H_X,d_X)$和$(G/H_X,\hat d_X)$的几何性质。特别地，我们证明了：如果$G$是李群，$M$是一个赋有度量$d$的可微流形，该度量局部Lipschitz等价于Finsler度量，$X$是$M$的紧子集，$φ:G \times M \rightarrow M$是$G$在$M$上的等距光滑左作用，则$(G/H_X,\hat d_X)$是$C^0$-Finsler.
英文摘要：
Let $G$ be a group, $(M,d)$ be a metric space, $X$ be a compact subspace of $M$ and $φ:G\times M \rightarrow M$ be a left action by homeomorphisms of $G$ on $M$. Denote $gp=f(g,p)$. The isotropy subgroup of $G$ with respect to $X$ is defined by $H_X=\{g\in G; gX=X\}$. In this work we define the induced Hausdorff metric on $G/H_X$ by $d_X(gH_X,hH_X)=d_H(gX,hX)$, where $d_H$ is the Hausdorff distance on $M$. Let $\hat d_X$ be the intrinsic metric induced by $d_X$. In this work we study the geometry of $(G/H_X,d_X)$ and $(G/H_X,\hat d_X)$. In particular, we prove that if $G$ is a Lie group, $M$ is a differentiable manifold endowed with a metric $d$ which is locally Lipschitz equivalent to a Finsler metric, $X$ is a compact subset of $M$ and $φ:G \times M \rightarrow M$ is a smooth left action by isometries of $G$ on $M$, then $(G/H_X,\hat d_X)$ is $C^0$-Finsler.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
