Download Metric spaces, convexity and nonpositive curvature by Papadopoulos A. PDF

April 4, 2017 | Science Mathematics | By admin | 0 Comments

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 Teichmüller areas. it really is important for graduate scholars and researchers in geometry, topology and research. allotted in the Americas via the yank Mathematical Society.

Show description

Read Online or Download Metric spaces, convexity and nonpositive curvature PDF

Best science & mathematics books


1m vorliegenden Bueh werden wir uns mit der Differentialgeometrie der Kurven und Flaehen im dreidimensionalen Raum besehiiftigen [2, 7]. Wir werden dabei besonderes Gewieht darauf legen, einen "ansehauliehen" Einbliek in die differentialgeometrisehen Begriffe und Satze zu gewinnen. Zu dies em Zweek werden wir, soweit sieh dies in naheliegender Weise er mogliehen lal3t, den differentialgeometrisehen Objekten elementargeome trisehe oder, wie wir dafiir aueh sagen wollen, differenzengeometrisehe Modelle gegeniiberstellen und deren elementargeometrisehe Eigensehaften mit differentialgeometrisehen Eigensehaften der Kurven und Flaehen in Be ziehung bringen.

Elements of the History of Mathematics

This paintings gathers jointly, with no vast amendment, the foremost­ ity of the ancient Notes that have seemed to date in my components de M atMmatique. purely the stream has been made self sustaining of the weather to which those Notes have been connected; they're as a result, in precept, available to each reader who possesses a legitimate classical mathematical history, of undergraduate typical.

Zero : a landmark discovery, the dreadful void, and the ultimate mind

0 shows the absence of a volume or a value. it's so deeply rooted in our psyche this day that no-one will in all probability ask "What is 0? " From the start of the very production of existence, the sensation of loss of anything or the imaginative and prescient of emptiness/void has been embedded through the author in all residing beings.

Extra info for Metric spaces, convexity and nonpositive curvature

Example text

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, .

Download PDF sample

Rated 4.50 of 5 – based on 28 votes