CAT(k) Space - Examples

Examples

• Any CAT(k) space (X, d) is also a CAT(ℓ) space for all ℓ > k. In fact, the converse holds: if (X, d) is a CAT(ℓ) space for all ℓ > k, then it is a CAT(k) space.
• n-dimensional Euclidean space En with its usual metric is a CAT(0) space. More generally, any real inner product space (not necessarily complete) is a CAT(0) space; conversely, if a real normed vector space is a CAT(k) space for some real k, then it is an inner product space.
• n-dimensional hyperbolic space Hn with its usual metric is a CAT(−1) space, and hence a CAT(0) space as well.
• The n-dimensional unit sphere Sn is a CAT(1) space.
• More generally, the standard space M_k is a CAT(k) space. So, for example, regardless of dimension, the sphere of radius r (and constant curvature 1/√r) is a CAT(1/√r) space. Note that the diameter of the sphere is πr (as measured on the surface of the sphere) not 2r (as measured by going through the centre of the sphere).
• The punctured plane Π = E2 \ {0} is not a CAT(0) space since it is not geodesically convex (for example, the points (0, 1) and (0, −1) cannot be joined by a geodesic in Π with arc length 2), but every point of Π does have a CAT(0) geodesically convex neighbourhood, so Π is a space of curvature ≤ 0.
• The closed subspace X of E3 given by

equipped with the induced length metric is not a CAT(k) space for any k.

• Any product of CAT(0) spaces is CAT(0). (This does not hold for negative arguments.)