For a real number, let denote the unique complete simply connectedsurface with constant curvature. Denote by the diameter of, which is if and for. Let be a geodesic metric space, i.e. a metric space for which every two points can be joined by a geodesic segment, an arc length parametrized continuous curve, whose length is precisely. Let be a triangle in with geodesic segments as its sides. is said to satisfy the inequality if there is a comparison triangle in the model space, with sides of the same length as the sides of, such that distances between points on are less than or equal to the distances between corresponding points on. The geodesic metric space is said to be a space if every geodesic triangle in with perimeter less than satisfies the inequality. A metric space is said to be a space with curvature if every point of has a geodesically convexneighbourhood. A space with curvature may be said to have non-positive curvature.
Examples
Any space is also a space for all. In fact, the converse holds: if is a space for all, then it is a space.
More generally, the standard space is a space. So, for example, regardless of dimension, the sphere of radius is a space. Note that the diameter of the sphere is not .
The punctured plane is not a space since it is not geodesically convex, but every point of does have a geodesically convex neighbourhood, so is a space of curvature.
The closed subspace of given by
Any product of spaces is.
Hadamard spaces
As a special case, a complete CAT space is also known as a Hadamard space; this is by analogy with the situation for Hadamard manifolds. A Hadamard space is contractible and, between any two points of a Hadamard space, there is a unique geodesic segment connecting them. Most importantly, distance functions in Hadamard spaces are convex: if are two geodesics in X defined on the same interval of time I, then the function given by is convex in t.
Properties of spaces
Let be a space. Then the following properties hold:
Given any two points , there is a unique geodesic segment that joins to ; moreover, this segment varies continuously as a function of its endpoints.
Every local geodesic in with length at most is a geodesic.
The -balls in of radius less than are convex.
The -balls in of radius less than are contractible.
Approximate midpoints are close to midpoints in the following sense: for every and every there exists a such that, if is the midpoint of a geodesic segment from to with and
It follows from these properties that, for the universal cover of every space is contractible; in particular, the higher homotopy groups of such a space are trivial. As the example of the -sphere shows, there is, in general, no hope for a space to be contractible if.
