CAT(k) Space - Properties of CAT(k) Spaces

Properties of CAT(k) Spaces

Let (X, d) be a CAT(k) space. Then the following properties hold:

• Given any two points x, y ∈ X (with d(x, y) < D_k if k > 0), there is a unique geodesic segment that joins x to y; moreover, this segment varies continuously as a function of its endpoints.

• Every local geodesic in X with length at most D_k is a geodesic.

• The d-balls in X of radius less than ½D_k are (geodesically) convex.

• The d-balls in X of radius less than D_k are contractible.

• Approximate midpoints are close to midpoints in the following sense: for every λ < D_k and every ε > 0, there exists a δ = δ(k, λ, ε) > 0 such that, if m is the midpoint of a geodesic segment from x to y with d(x, y) ≤ λ and

then d(m, m′) < ε.

• It follows from these properties that, for k ≤ 0, the universal cover of every CAT(k) space is contractible; in particular, the higher homotopy groups of such a space are trivial. As the example of the n-sphere Sn shows, there is, in general, no hope for a CAT(k) space to be contractible if k is strictly positive.

• An n-dimensional CAT(k) space equipped with the n-dimensional Hausdorff measure satisfies the CD condition in the sense of Lott-Villani-Sturm.