Metric Structures for Riemannian and Non-Riemannian Spaces

admits algebraic approximation argument asymptotic ball of radius bi-Lipschitz equivalent boundary closed compact complete concentration condition const constant convergence convex cubes curvature curve defined definition denote diameter differential dilatation dimension dist doubling measure embedding equals Euclidean spaces example exists fact fibers finite follows fundamental group geodesic geometry given Hausdorff Hausdorff distance Heisenberg group Holder continuous homeomorphism hyperbolic hypersurface implies infimum infinite integral invariant isometric isomorphic isoperimetric dimension isoperimetric inequality Lebesgue measure Lemma length structure Lipschitz functions Lipschitz maps lower bound maximal metric space minimal nontrivial norm normalized observe parameterization path metric space proof Proposition prove quasisymmetric quasisymmetric mappings Remark Ricci Riemannian manifold Riemannian metric Rn+1 satisfies segment sequence short map simplices simply connected singular smooth sphere subset theorem topological uniform rectifiability uniformly universal cover volume zero