CAT(k) Space - Definitions

Definitions

For a real number k, let M_k denote the unique simply connected surface (real 2-dimensional Riemannian manifold) with constant curvature k. Denote by D_k the diameter of M_k, which is +∞ if k ≤ 0 and π/√k for k > 0.

Let (X, d) be a geodesic metric space, i.e. a metric space for which every two points x, y ∈ X can be joined by a geodesic segment, an arc length parametrized continuous curve γ : → X, γ(a) = x, γ(b) = y, whose length

is precisely d(x, y). Let Δ be a triangle in X with geodesic segments as its sides. Δ is said to satisfy the CAT(k) inequality if there is a comparison triangle Δ′ in the model space M_k, with sides of the same length as the sides of Δ, such that distances between points on Δ are less than or equal to the distances between corresponding points on Δ′.

The geodesic metric space (X, d) is said to be a CAT(k) space if every geodesic triangle Δ in X with perimeter less than 2D_k satisfies the CAT(k) inequality. A (not-necessarily-geodesic) metric space (X, d) is said to be a space with curvature ≤ k if every point of X has a geodesically convex CAT(k) neighbourhood. A space with curvature ≤ 0 may be said to have non-positive curvature.