As a special case, a complete CAT(0) space is also known as a Hadamard space; this is by analogy with the situation for Hadamard manifolds.

Jun 14, 2024 · In Chapters 1 and 2 we first discuss how to define the length of curves, and geodesics on (X,d), and then using these to portray the notion of ``non-positive curvature'' for a metric space. Chapter 3 concerns itself with special cases of such non-positively curved metric spaces, called CAT(0) spaces.

M ̈unster, June 22, 2004 “CAT(0)-space” is a term invented by Gromov. Also, called “Hadamard space.” Roughly, a space which is “non- positively curved” and simply connected. Theorem. Suppose. X . Then X. is contractible). The proof is based on two facts about CAT(0)-spaces. Fact 1. (Bruhat-Tits Fixed Point Theorem).

Jan 1, 2014 · Complete CAT (0) spaces are often called Hadamard spaces. CAT (0) spaces have a remarkably nice geometric structure. One can see almost immediately that in such spaces angles exist in a strong sense, the distance function is convex, one has both uniform convexity and orthogonal projection onto convex subsets, etc.

First self-contained, comprehensive introduction to CAT(0) cube complexes; Topics range from basic prerequisites to a variety of advanced topics; Suitable for use in a graduate course or for independent study

Jan 13, 2024 · A 0 0-category (or (0, 0) (0,0)-category) is, up to equivalence, the same as a set (or class).

A CAT(0) space is a geodesic metric space all of whose triple of points ( x,y,z ) ∈ X 3 satisfy the following condition: given a Euclidean comparison triangle (ˆ x,y, ˆ ˆz) in R 2 , any point p ∈ X which belongs to some geodesic

I have a question regarding the definition of CAT(0) spaces. I am using the following definition: $X$ complete metric space is CAT(0) if $\forall z,y \in X$, $\exists m \in X$ such that $\forall x...

Feb 4, 2015 · Cat(0) spaces have unique geodesics between points. Fix a base point, and along each geodesic out from that point, pull everything inwards. This is well-defined because geodesics are unique, and is continuous if you use the same map on each geodesic.

We give new examples of hyperbolic and relatively hyperbolic groups of cohomological dimension d for all d ≥ 4 (see Theorem 2.13). These examples result from applying CAT(0)/CAT(−1) filling constructions (based on singular doubly warped products) to finite volume hyperbolic manifolds with toral cusps.