Hadamard space

In an Hadamard space, a triangle is hyperbolic; that is, the middle one in the picture. In fact, any complete metric space where a triangle is hyperbolic is an Hadamard space.

In geometry, an Hadamard space, named after Jacques Hadamard, is a non-linear generalization of a Hilbert space. It is defined to be a nonempty^[1] complete metric space where, given any points x, y, there exists a point m such that for every point z,

$d(z, m)^2 + {d(x, y)^2 \over 4} \le {d(z, x)^2 + d(z, y)^2 \over 2}.$

The point m is then the midpoint of x and y: $d(x, m) = d(y, m) = d(x, y)/2$.

In a Hilbert space, the above inequality is equality (with $m = (x+y)/2$), and in general an Hadamard space is said to be flat if the above inequality is equality. A flat Hadamard space is isomorphic to a closed convex subset of a Hilbert space. In particular, a normed space is an Hadamard space if and only if it is a Hilbert space.

The geometry of Hadamard spaces resembles that of Hilbert spaces, making it a natural setting for the study of rigidity theorems. In an Hadamard space, any two points can be joined by a unique geodesic between them; in particular, it is contractible. Quite generally, if B is a bounded subset of a metric space, then the center of the closed ball of the minimum radius containing it is called the circumcenter of B.^[2] Every bounded subset of an Hadamard space is contained in the smallest closed ball (which is the same as the closure of its convex hull). If $\Gamma$ is the group of isometries of an Hadamard space leaving invariant B, then $\Gamma$ fixes the circumcenter of B. (Bruhat–Tits fixed point theorem)

The basic result for a non-positively curved manifold is the Cartan–Hadamard theorem. The analog holds for an Hadamard space: a complete, connected metric space which is locally isometric to an Hadamard space has an Hadamard space as its universal cover. Its variant applies for non-positively curved orbifolds. (cf. Lurie.)

Examples of Hadamard spaces are Hilbert spaces, the Poincaré disc, trees (e.g., Bruhat–Tits building), (p, q)-space with p, q ≥ 3 and 2pq ≥ p + q, and Riemannian manifolds of nonpositive sectional curvature (e.g., symmetric spaces). An Hadamard space is precisely a complete CAT(0) space.

Applications of Hadamard spaces are not restricted to geometry. In 1998, Dmitry Burago and Serge Ferleger ^[3] used CAT(0) geometry to solve a problem in Dynamical billiards: in a gas of hard balls, is there a uniform bound on the number of collisions? The solution begins by constructing a configuration space for the dynamical system, obtained by joining together copies of corresponding billiard table, which turns out to be an Hadamard space.

See also

• CAT(k) space

References

1. The assumption on "nonempty" has meaning: a fixed point theorem often states the set of fixed point is an Hadamard space. The main content of such an assertion is that the set is nonempty.
2. A Course in Metric Geometry, p. 334.
3. Burago D., Ferleger S. Uniform estimates on the number of collisions in semi-dispersing billiards. Ann. of Math. 147 (1998), 695-708

• Burago, Dmitri; Yuri Burago, and Sergei Ivanov. A Course in Metric Geometry. American Mathematical Society. (1984)
• Jacob Lurie: Notes on the Theory of Hadamard Spaces
• Alexander S., Kapovich V., Petrunin A. Notes on Alexandrov Geometry