CAT(k) space
is a real number, is a specific type of metric space.
) are "slimmer" than corresponding "model triangles" in a standard space of constant curvature
space, the curvature is bounded from above by
A notable special case is
spaces are known as "Hadamard spaces" after the French mathematician Jacques Hadamard.
Originally, Aleksandrov called these spaces “
was coined by Mikhail Gromov in 1987 and is an acronym for Élie Cartan, Aleksandr Danilovich Aleksandrov and Victor Andreevich Toponogov (although Toponogov never explored curvature bounded above in publications).
For a real number
denote the unique complete simply connected surface (real 2-dimensional Riemannian manifold) with constant curvature
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
with geodesic segments as its sides.
inequality if there is a comparison triangle
are less than or equal to the distances between corresponding points on
The geodesic metric space
space if every geodesic triangle
A (not-necessarily-geodesic) metric space
is said to be a space with curvature
has a geodesically convex
A space with curvature
As a special case, a complete CAT(0) space is also known as a Hadamard space; this is by analogy with the situation for Hadamard manifolds.
A Hadamard space is contractible (it has the homotopy type of a single point) and, between any two points of a Hadamard space, there is a unique geodesic segment connecting them (in fact, both properties also hold for general, possibly incomplete, CAT(0) spaces).
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
Then the following properties hold: 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 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 is a convex function, i.e.
Taking geodesic polar coordinates with origin at a so that ‖c(t)‖ = r(t), convexity is equivalent to Changing to normal coordinates u, v at c(t), this inequality becomes where (u,v) corresponds to the unit vector ċ(t).
This follows from the inequality Hr ≥ H, a consequence of the non-negativity of the derivative of the Wronskian of H and r from Sturm–Liouville theory.
