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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.]]
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| 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<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]] where, given any points ''x'', ''y'', there exists a point ''m'' such that for every point ''z'',
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| :<math>d(z, m)^2 + {d(x, y)^2 \over 4} \le {d(z, x)^2 + d(z, y)^2 \over 2}.</math> | |
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| The point ''m'' is then the midpoint of ''x'' and ''y'': <math>d(x, m) = d(y, m) = d(x, y)/2</math>.
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| 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 ''flat'' if the above inequality is equality. A flat Hadmard 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.
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| The geometry of Hadamard spaces resembles that of Hilbert spaces, making it a natural setting for the study of [[rigidity theorem]]s. In an Hadamard space, any two points can be joined by a unique [[geodesic]] between them; in particular, it is [[contractible space|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 (metric space)|circumcenter]]'' of ''B''.<ref>A Course in Metric Geometry, p. 334.</ref> 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 <math>\Gamma</math> is the [[group (mathematics)|group]] of [[isometry|isometries]] of an Hadamard space leaving invariant ''B'', then <math>\Gamma</math> fixes the circumcenter of ''B''. ('''Bruhat–Tits fixed point theorem''')
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| 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 [[orbifold]]s. (cf. Lurie.)
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| Examples of Hadamard spaces are Hilbert spaces, the [[Poincaré disc]], trees (e.g., [[Bruhat–Tits building]]), [[Cayley graph]]s of [[hyperbolic group]]s or more generally CAT(0) groups, [[(p, q)-space|(''p'', ''q'')-space]] with ''p'', ''q'' ≥ 3 and 2''pq'' ≥ ''p'' + ''q'', and [[Riemannian manifold]]s of nonpositive [[sectional curvature]] (e.g., [[symmetric space]]s). An Hadamard space is precisely a complete [[CAT(k) space|CAT(0) space]].
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| == See also ==
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| * [[CAT(k) space]]
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| == References ==
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| {{reflist}}
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| *Burago, Dmitri; Yuri Burago, and Sergei Ivanov. ''A Course in Metric Geometry''. American Mathematical Society. (1984)
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| *[[Jacob Lurie]]: [http://www.math.harvard.edu/~lurie/papers/hadamard.pdf Notes on the Theory of Hadamard Spaces]
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| [[Category:Hilbert space]]
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