Induced Hausdorff Metrics on Quotient Spaces.

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. [ABSTRACT FROM AUTHOR]