Induced Hausdorff Metrics on Quotient Spaces

Abstract

Abstract. Let G be a group, (M, d) be a metric space, X ⊂ M be a compact subset and ϕ : G × M → M be a left action of G on M by homeomorphisms. Denote gp = ϕ(g, p). The isotropy subgroup of G with respect to X is defined by H X = {g ∈ G; gX = X}. In this work we define the induced Hausdorff, where d H is the Hausdorff distance on M . Letd 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 ,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 ⊂ M is a compact subset and ϕ : G × M → M is a smooth left action by isometries, then (G/H X ,d X ) is a C 0 -Finsler manifold. We also calculate the Finsler metric explicitly in some examples.

Sequence of Induced Hausdorff Metrics on Lie Groups

“…Therefore d 1 is locally equal to the maximum metric. (Compare with Example 8.11 in [8]). In this case we have that d 1 < d, a relationship that doesn't happen in this work.…”

“…In this section we present some definitions and results that are used in this work. They can be found in [5], [8], [11], [12], [13], [14] and [17]. For the sake of clearness, we usually don't give the definitions and results in their most general case.…”

“…Suppose that X is a compact subset in M . In Proposition 2.1 of [8] (and in [2]), we define the induced Hausdorff metric on the left coset space G/H X as d X (gH X , hH X ) = d H (gX, hX), where d H is the Hausdorff distance in M .…”

“…where the first equality is due to Theorem 3.7 of [8] and the second equality is due to the right invariance of d ∞ . Therefore F is finite.…”

“…Then there exist a unique differentiable structure on G/H X , compatible with the quotient topology, such that the natural action φ : G × G/H X → G/H X , given by φ(g, hH X ) = ghH X , is smooth. In [2] and [8], the authors define a metric d X on G/H X , which is called induced Haudorff metric, by d X (gH X , hH X ) = d H (gX, hX), where d H is the Hausdorff distance in (M, d) and gH X denote the left coset of g in G/H X . Suppose that ϕ is transitive and that there exist p ∈ M such that H p = H X .…”