geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, an Hadamard space, named after

Jacques Hadamard Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry, and partial differential equations. Biography The son of a tea ...

, is a non-linear generalization of a

Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...

. In the literature they are also equivalently defined as complete CAT(0) spaces. A Hadamard space is defined to be a nonempty complete

metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...

such that, given any points

x

and

y,

there exists a point

m

such that for every point

z,

d(z, m)^2 +  \leq .

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 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 The Ateliers et Chantiers de France (ACF, Workshops and Shipyards of France) was a major shipyard that was established in Dunkirk, France, in 1898. The shipyard boomed in the period before World War I (1914–18), but struggled in the inter-war p ...

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 In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conn ...

between them; in particular, it is

contractible In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within t ...

. 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 In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumradius. The circumcen ...

'' of

B.

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

\Gamma

is the

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

isometries In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...

of a 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 In mathematics, the Cartan–Hadamard theorem is a statement in Riemannian geometry concerning the structure of complete Riemannian manifolds of non-positive sectional curvature. The theorem states that the universal cover of such a manifold is dif ...

. 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 In topology, a covering or covering projection is a map between topological spaces that, intuitively, locally acts like a projection of multiple copies of a space onto itself. In particular, coverings are special types of local homeomorphism ...

. Its variant applies for non-positively curved

orbifold In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space that is locally a finite group quotient of a Euclidean space. D ...

s. (cf. Lurie.) Examples of Hadamard spaces are

s, the Poincaré disc, complete real trees (for example, complete Bruhat–Tits building),

(p, q)

-space with

p, q \geq 3

and

2 p q \geq p + q,

and Hadamard manifolds, that is, complete simply-connected

Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...

s of nonpositive

sectional curvature In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a po ...

. Important examples of Hadamard manifolds are simply connected nonpositively curved

symmetric space In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of isometries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geome ...

s. Applications of Hadamard spaces are not restricted to geometry. In 1998, Dmitri Burago and Serge FerlegerBurago D., Ferleger S. Uniform estimates on the number of collisions in semi-dispersing billiards. Ann. of Math. 147 (1998), 695-708 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 In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...

, obtained by joining together copies of corresponding billiard table, which turns out to be a Hadamard space.

References

* * * Burago, Dmitri; Yuri Burago, and Sergei Ivanov. ''A Course in Metric Geometry''. American Mathematical Society. (1984) *

Jacob Lurie Jacob Alexander Lurie (born December 7, 1977) is an American mathematician who is a professor at the Institute for Advanced Study. In 2014, Lurie received a MacArthur Fellowship. Lurie's research interests are algebraic geometry, topology, and ...

Notes on the Theory of Hadamard Spaces
* Alexander S., Kapovich V., Petrunin A
Notes on Alexandrov Geometry
{{Manifolds Functional analysis Geometric topology Hilbert spaces Metric spaces

See also

References