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In classical Riemannian geometry, one begins with a C ^∞ manifold X and then studies smooth, positive-definite sections g of the bundle S ^2 T * X . In order to introduce the fundamental notions of covariant derivative and curvature (cf. [Grl-Kl-Mey] or [Milnor], Ch. 2), use is made only of the differentiability of g and not of its positivity, as illustrated by Lorentzian geometry in general relativity. By contrast, the concepts of the length of curves in X and of the geodesic distance associated with the metric g rely only on the fact that g gives rise to a family of continuous norms on the tangent spaces T _x X of X . We will study the associated notions of length and distance for their own sake.

Throughout this chapter, M and N will denote connected, oriented, C ^∞ manifolds having the same dimension n . Additionally, M is assumed to be compact and without boundary.

The Lipschitz distance between two metric spaces X, Y , denoted d _L( X, Y ), is the infimum of the numbers $$ |\log dil(f)| + |\log dil(f^{{\text{ - 1}}} )| $$ as f varies over the set of bi-Lipschitz homeomorphisms between X and Y .

When we speak of measures μ on a metric space X , we always assume that X is a Polish space , i.e., complete with a countable base , and that “measure” means a Borel measure , where all Borel subsets in X are measurable. We are mostly concerned with finite measures where X has finite (total) mass μ -( X ) < ∞, but we also allow σ-finite measure spaces X , which are the countable unions of X _i with μ ( X _i) < ∞. For example, the ordinary n dimensional Hausdorff measure in ℝ^n is σ -finite, while the k -dimensional Hausdorff measure on ℝ^n for k < n is not σ -finite. But, we may restrict such a measure to a k -dimensional submanifold V ⊂ ℝ^n, i.e., we declare μ _k( U ) = μ _k( U ∩ V ) for all open U ⊂ ℝ^n, in which case the measure becomes σ -finite and admissible in our discussion.

In 1949, Loewner proved the following (unpublished, see [Pu] or the proofs in [Berger], [Berger], [Berger]).

In this chapter, we consider locally compact, pointed path metric spaces and the metric space structure on the collection of such spaces defined by the Hausdorff distance (or, more precisely, the uniform structure on this set defined by the family of Hausdorff distances on the balls of radius R ).

Let ( V, g ), ( W, h ) be oriented Riemannian manifolds. A mapping f : V → W is called quasiregular if it is locally Lipschitz, thus differentiable almost everywhere, and if its differential Df _x and Jacobian J ( x ) satisfy the inequality 0 < ‖ Df _x‖^n ≤ cJ ( x ) for almost all x , where c is a constant.

In Chapter 6, we introduced the notion of 2-dimensional isoperimetric rank (6.32) as the largest number p , such that each simple curve of length ℓ is bounded by a disk of area at most Cℓ ^ p / p −1. This definition only makes sense in noncompact manifolds, and we have shown that the 2-dimensional isoperimetric rank of the universal cover of a compact manifold V depends only on the fundamental group π _1( V ).

The group of diffeomorphisms Diff( V ) of a smooth manifold V naturally acts on the space of Riemannian metrics g on V ; various classes of metrics one studies in geometry are usually invariant under Diff( V ). In fact, we tend not to distinguish isometric manifolds, and a diffeomorphism f : V → V establishes an isometry between ( V, g ) and ( V, f *( g )) for each metric g . Furthermore, the geometric dictum “from local to global” suggests the study of locally defined classes of metrics on V which are moreover Diff-invariant.