By Martin R. Bridson

This ebook describes the worldwide homes of simply-connected areas which are non-positively curved within the experience of A. D. Alexandrov, and the constitution of teams which act on such areas via isometries. the idea of those gadgets is built in a fashion available to someone acquainted with the rudiments of topology and crew conception: non-trivial theorems are proved by way of concatenating undemanding geometric arguments, and plenty of examples are given. half I is an advent to the geometry of geodesic areas. partly II the elemental idea of areas with higher curvature bounds is constructed. extra really expert themes, equivalent to complexes of teams, are coated partly III. The booklet is split into 3 elements, each one half is split into chapters and the chapters have quite a few subheadings. The chapters partly III are longer and for ease of reference are divided into numbered sections.

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25). Because the action of Isom(M^) is transitive on pairs of equidistant points, we may assume that [x, y] is the initial segment of a fixed geodesic segment [x, y0] of length /. Let rj0 be a constant such that if d(x, z) + d(z, yo) < d(x, yo) + rj0 then d(z, [x, y0]) < e/2. , y) + d(y, yo), which is at most d(x, y) + d(y, y0) + 28 = d(x, y0) + 28. So if 25 < 7/0, then d(m',p) < e/2 for some p e [x, y0]. ) + d(m', x) < e/2 + 8 + \d{x, y) and similarly dip, y) < e/2 + 8 + \dix, y). But/? lies on the geodesic segment [x, y0] so therefore dip, m) < e/2 + 8.

3 Length Spaces We shall now prove (2). Let a and b be distinct points of X. For every integer n > 1, there is a path cn : [0, 1] ->• X, parameterized proportional to its arc length, such that l(cn) < d(a, b) + \/n. Such a family of paths {cn} is equicontinuous; indeed for all t, f e [0, 1] we have: . )) < e i f | f — f\ < ^-|r^-. The image of each path cn is contained in the compact set B(2 d(a, b)). By the Arzela-Ascoli theorem (see below) there is a subsequence of the (cn)neN converging uniformly to a path c : [0,1] -> X.

The curve c is said to be rectifiable if its length is finite. 19 Example. An easy example of a non-rectifiable path in the metric space X = [0,1] can be constructed as follows. o < t\ < • • • < ? „ < . . „ = 1. n) = 53ft=i(~l)*+1A- This is not a rectifiable path, because its length is bounded below by the sum of the harmonic series. Let Y be the graph of c in [0, 1] x [0, 1], with the induced metric. Y is a compact, path-connected metric space, but there is no rectifiable path in Y joining (0,0) to the point (1, In 2).