 
Summary: Chapter 6
Definitions of curvature
bounded below
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 4point 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
bound.
In Section 6.4, we give local versions of each definition and prove the glob
alization theorem.
Further, we discuss some basic corollaries of the definition: first variation
formula and nonsplitting 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
, x2
