CAT(k) space

Model triangles in spaces of positive (top), negative (middle) and zero (bottom) curvature.

In a region where the curvature of the surface satisfies K ≤ 0, geodesic triangles satisfy the CAT(0) inequalities of comparison geometry, studied by Cartan, Alexandrov and Toponogov, and considered later from a different point of view by Bruhat and Tits; thanks to the vision of Gromov, this characterisation of non-positive curvature in terms of the underlying metric space has had a profound impact on modern geometry and in particular geometric group theory. Many results known for smooth surfaces and their geodesics, such as Birkhoff's method of constructing geodesics by his curve-shortening process or van Mangoldt and Hadamard's theorem that a simply connected surface of non-positive curvature is homeomorphic to the plane, are equally valid in this more general setting.

The median in the comparison triangle is always longer than the actual median.

The simplest form of the comparison inequality, first proved for surfaces by Alexandrov around 1940, states that

The distance between a vertex of a geodesic triangle and the midpoint of the opposite side is always less than the corresponding distance in the comparison triangle in the plane with the same side-lengths.

The inequality follows from the fact that if c(t) describes a geodesic parametrized by arclength and a is a fixed point, then

Taking geodesic polar coordinates with origin at a so that c(t)‖ = r(t), convexity is equivalent to

where (u,v) corresponds to the unit vector ċ(t). This follows from the inequality HrH, a consequence of the non-negativity of the derivative of the Wronskian of H and r from Sturm–Liouville theory.[1]