Induced Hausdorff Metrics on Quotient Spaces

Abstract

Let G be a group, (M, d) be a metric space, $X\subset M$ be a compact subset and $\varphi :G\times M\rightarrow M$ be a left action of G on M by homeomorphisms. Denote $gp=\varphi (g,p)$. 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 $d_X(g_1H_X,g_2H_X):=d_H(g_1X,g_2X)$, 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 $(G/H_X,\hat{d}_X)$ and their relationship with (M, d). In particular, we prove that if G is a Lie group, M is a differentiable manifold endowed with a metric which is locally Lipschitz equivalent to a Finsler metric, $X\subset M$ is a compact subset and $\varphi :G\times M\rightarrow M$ is a smooth left action by isometries, then $(G/H_X,\hat{d}_X)$ is a $C^0$-Finsler manifold. We also calculate the Finsler metric explicitly in some examples.