By Papadopoulos A.

This ebook covers metric areas of nonpositive curvature within the feel of Busemann, that's, metric areas whose distance functionality satisfies a convexity situation. additionally contained is a scientific advent to the idea of geodesics in addition to an in depth presentation of a few points of convexity conception, that are precious within the learn of nonpositive curvature. ideas and methods mentioned within the quantity are illustrated through many examples from classical hyperbolic geometry and from the idea of TeichmÃ¼ller areas. it really is important for graduate scholars and researchers in geometry, topology and research. allotted in the Americas via the yank Mathematical Society.

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The metric of a length space is called a length metric. 36 2 Length spaces and geodesic spaces In particular, a length space is connected by rectifiable paths. We give now a list of examples of familiar length spaces (or of length metrics on familiar spaces). We also include examples of length pseudo-metrics, namely the Carathéodory, the Kobayashi and the Thurston metrics, that are not genuine metrics. We recall that a pseudo-metric on a space X is a map d : X × X → R satisfying, for all x, y and z in X, d(x, x) = 0, d(x, y) ≥ 0, d(x, y) = d(y, x) and d(x, y) ≤ d(x, y)+d(y, z).

N in D satisfying f1 (ζ1 ) = x, fn (ζn ) = y and for all j = 1, . . , n − 1, fj (ζj ) = fj +1 (ζj +1 ). The Kobayashi pseudo-distance is then equal to dK (x, y) = inf{dD (ζ1 ζ1 ) + · · · + dD (ζn ζn )} where the infimum is taken over all analytic chains joining x and y. The Kobayashi pseudo-distance is also characterized by the fact that it is the largest pseudo-distance on X such that any holomorphic map f : D → X is non-expanding (that is, f satisfies |f (x) − f (y)| ≤ |x − y|). There is also an infinitesimal version of the Kobayashi pseudo-distance, defined as follows: for any x in X and for any tangent vector v at x, the pseudo-norm of v is 42 2 Length spaces and geodesic spaces defined as v = inf { v v D} where v is a tangent vector to the disk D, v is the norm of v with respect to the metric of D, and where the infimum is taken over all vectors v in D such that there is a holomorphic maps f : D → X whose derivative sends v to v.

The Riemannian manifold M, equipped with this metric, is a length metric space. ) (vii) Hyperbolic space. For each n ≥ 2, we can define the n-dimensional hyperbolic space Hn by equipping the open unit ball B n of Rn , n B = {(x1 , . . , xn ) ∈ R : n xi2 < 1}, n i=0 with the metric whose “line element” ds is given by dx12 + · · · + dxn2 ds = 2 1− x 2 , where x denotes the Euclidean norm of the element x. This means that the length of any piecewise C 1 -path γ : [a, b] → B n is equal to b L(γ ) = a b ds = 2 a γ 21 (t) + · · · + γ 2n (t) 1 − (γ1 (t)2 + · · · + γn (t)2 ) dt, where for each i = 1, .