Summary: Chapter 6
Definitions of curvature
In Section 6.1, we will start with a global definition of Alexandrov spaces via
(1+3)-point comparison. In Section 6.2, we give a number of equivalent angle
comparison definitions. These definitions give the easiest way to adapt your
Euclidean intuition to Alexandrov's world. Later we give a few equivalent global
definitions via concavity of distance functions and development (Section 6.3).
There is yet another definition via extension of short maps on 4-point subsets
(Section 8.1). In Chapter ??, we also discuss Wald's original definition which
gives a uniform approach to definitions of spaces with lower and upper curvature
In Section 6.4, we give local versions of each definition and prove the glob-
Further, we discuss some basic corollaries of the definition: first variation
formula and non-splitting of geodesics are found in Section 6.5. Some specific
properties of spaces with positive curvature bound are given in Section 6.6.
6.1 (1+3)-point comparison.
6.1.1. Definition. A quadruple of points p, x1