In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, 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 teac ...
, is a non-linear generalization of a
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
. 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 together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
such that, given any points
and
there exists a point
such that for every point
The point
is then the midpoint of
and
In a Hilbert space, the above inequality is equality (with
), 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
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...
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 theorem
In mathematics, a rigid collection ''C'' of mathematical objects (for instance sets or functions) is one in which every ''c'' ∈ ''C'' is uniquely determined by less information about ''c'' than one would expect.
The above statement do ...
s. 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 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 connection. ...
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 that ...
. Quite generally, if
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 polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...
'' of
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
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 ...
of
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
then
fixes the circumcenter of
(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 A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties.
Definition
Let X be a topological space. A covering of X is a continuous map
: \pi : E \rightarrow X
such that there exists a discrete spa ...
. 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 which is locally a finite group quotient of a Euclidean space.
D ...
s. (cf. Lurie.)
Examples of Hadamard spaces are
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
s, the
Poincaré disc
Poincaré is a French surname. Notable people with the surname include:
* Henri Poincaré (1854–1912), French physicist, mathematician and philosopher of science
* Henriette Poincaré (1858-1943), wife of Prime Minister Raymond Poincaré
* Luc ...
, complete
metric tree
A metric tree is any tree data structure specialized to index data in metric spaces. Metric trees exploit properties of metric spaces such as the triangle inequality to make accesses to the data more efficient. Examples include the M-tree, vp-t ...
s (for example, complete
Bruhat–Tits building
In mathematics, a building (also Tits building, named after Jacques Tits) is a combinatorial and geometric structure which simultaneously generalizes certain aspects of flag manifolds, finite projective planes, and Riemannian symmetric spaces. Bu ...
),
-space with
and
and
Hadamard manifold In mathematics, a Hadamard manifold, named after Jacques Hadamard — more often called a Cartan–Hadamard manifold, after Élie Cartan — is a Riemannian manifold (M, g) that is complete and simply connected and has everywhere non-positive sec ...
s, that is, complete simply-connected
Riemannian manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
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 poi ...
. 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 symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, l ...
s.
Applications of Hadamard spaces are not restricted to geometry. In 1998,
Dmitri Burago
Dmitri Yurievich Burago (Дмитрий Юрьевич Бураго, born 1964) is a Russian mathematician, specializing in geometry.
He is the son of the professor of mathematics in Leningrad Yuri Dmitrievich Burago, with whom he also published a ...
and
Serge Ferleger Serge may refer to:
*Serge (fabric), a type of twill fabric
*Serge (llama) (born 2005), a llama in the Cirque Franco-Italien and internet meme
*Serge (name), a masculine given name (includes a list of people with this name)
*Serge (post), a hitchi ...
[Burago 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
A dynamical billiard is a dynamical system in which a particle alternates between free motion (typically as a straight line) and specular reflections from a boundary. When the particle hits the boundary it reflects from it without loss of speed ( ...
: 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. Examples include the mathematical models that describe the swinging of a ...
, obtained by joining together copies of corresponding billiard table, which turns out to be an Hadamard space.
See also
*
*
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. Lurie is a 2014 MacArthur Fellow.
Life
When he was a student in the Science, Mathematics, and Computer Science ...
Notes on the Theory of Hadamard Spaces* Alexander S., Kapovich V., Petrunin A
Notes on Alexandrov Geometry
{{Manifolds
Functional analysis
Geometric topology
Hilbert space
Metric spaces