 
Summary: Chapter 7
Definitions of curvature
bounded above
We begin by defining CAT spaces and spaces of curvature via (2+2)
point comparison (Section 7.1). Section 7.2 gives definitions in terms of angle
and distance comparisons for triangles. Useful definitions in terms of convex
ity of distance functions and developments are formulated in Section 7.4. A
definition using the Kirszbraun shortmap extension property may be found in
Section 8.1. In Chapter ?? we give another fourpoint definition and relate it
to the (1+3)point definition of curvature bounded below (Section 6.1) and to
Wald's original curvature condition.
Section 7.3 looks at thin triangles and their inheritance lemma. Section 7.5
discusses angles, including the first variation formula and flat triangles.
The major globalization theorems are treated in Sections 7.6, 7.7 and 7.8:
Alexandrov's patchwork theorem and the noconjugatepoint, HadamardCartan
and lifting theorems.
Reshetnyak's majorization theorem, a strong generalization of the thin
triangles definition of CAT spaces, is proved in Section 7.9.
7.1 (2+2)point comparison.
7.1.1. (2+2)point comparison. An ordered quadruple of points p1
