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Definitions of curvature bounded above

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 short-map extension property may be found in
Section 8.1. In Chapter ?? we give another four-point 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 no-conjugate-point, Hadamard-Cartan
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


Source: Alexander, Stephanie - Department of Mathematics, University of Illinois at Urbana-Champaign


Collections: Mathematics