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[[Image:End of universe.jpg|thumb|In an Hadamard space, a triangle is [[Hyperbolic triangle|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|geometry]], an '''Hadamard space''', named after [[Biography:Jacques Hadamard|Jacques Hadamard]], is a non-linear generalization of a [[Hilbert space]]. In the literature they are also equivalently defined as complete [[CAT(k) space|CAT(0) spaces]].

A Hadamard space is defined to be a nonempty<ref>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.</ref> complete [[Metric space|metric space]] such that, given any points <math>x</math> and <math>y,</math> there exists a point <math>m</math> such that for every point <math>z,</math>
<math display=block>d(z, m)^2 + {d(x, y)^2 \over 4} \leq {d(z, x)^2 + d(z, y)^2 \over 2}.</math>

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

In a Hilbert space, the above inequality is equality (with <math>m = (x+y)/2</math>), and in general an Hadamard space is said to be {{em|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 a Hadamard space, any two points can be joined by a unique [[Geodesic|geodesic]] between them; in particular, it is [[Contractible space|contractible]]. Quite generally, if <math>B</math> 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 <math>B.</math><ref>A Course in Metric Geometry, p.&nbsp;334.</ref> Every bounded subset of a Hadamard space is contained in the smallest closed ball (which is the same as the closure of its convex hull). If <math>\Gamma</math> is the [[Group (mathematics)|group]] of [[Isometry|isometries]] of a Hadamard space leaving invariant <math>B,</math> then <math>\Gamma</math> fixes the circumcenter of <math>B</math> ('''Bruhat–Tits fixed point theorem'''). 

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

Examples of Hadamard spaces are [[Hilbert space]]s, the Poincaré disc, complete [[Real tree|real tree]]s (for example, complete Bruhat–Tits building), <math>(p, q)</math>-space with <math>p, q \geq 3</math> and <math>2 p q \geq p + q,</math> and [[Hadamard manifold]]s, that is, complete simply-connected [[Riemannian manifold]]s of nonpositive [[Sectional curvature|sectional curvature]]. Important examples of Hadamard manifolds are simply connected nonpositively curved [[Symmetric space|symmetric space]]s.

Applications of Hadamard spaces are not restricted to geometry. In 1998, [[Biography:Dmitri Burago|Dmitri Burago]] and Serge Ferleger<ref>Burago D., Ferleger S. Uniform estimates on the number of collisions in semi-dispersing billiards. Ann. of Math. 147 (1998), 695-708</ref> used [[CAT(k) space|CAT(0) geometry]] to solve a problem in [[Dynamical billiards|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|dynamical system]], obtained by joining together copies of corresponding billiard table, which turns out to be an Hadamard space.

==See also==

* {{annotated link|CAT(k) space}}
* {{annotated link|Hadamard manifold}}

==References==

{{reflist}}

* {{citation|last1=Bridson|first1=Martin R.|last2=Haefliger|first2=André|title=Metric spaces of non-positive curvature|year=1999|publisher=Springer}}
* {{citation|last=Papadopoulos|first=Athanase|title=Metric spaces, convexity and non-positive curvature|edition=Second|series=IRMA Lectures in Mathematics and Theoretical Physics|volume=6|publisher=European Mathematical Society|year=2014|isbn=978-3-03719-132-3}}
* Burago, Dmitri; Yuri Burago, and Sergei Ivanov. ''A Course in Metric Geometry''. American Mathematical Society. (1984)
* Jacob Lurie: [http://www.math.harvard.edu/~lurie/papers/hadamard.pdf Notes on the Theory of Hadamard Spaces]
* Alexander S., Kapovich V., Petrunin A. [http://www.math.uiuc.edu/~sba/minicourse.pdf Notes on Alexandrov Geometry]

{{Functional analysis}}

[[Category:Functional analysis]]
[[Category:Geometric topology]]

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