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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a \mathbf(k) space, where k is a real number, is a specific type of
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 setti ...
. Intuitively,
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...
s in a \operatorname(k) space are "slimmer" than corresponding "model triangles" in a standard space of
constant curvature In mathematics, constant curvature is a concept from differential geometry. Here, curvature refers to the sectional curvature of a space (more precisely a manifold) and is a single number determining its local geometry. The sectional curvature i ...
k. In a \operatorname(k) space, the curvature is bounded from above by k. A notable special case is k=0;
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
\operatorname(0) spaces are known as " Hadamard spaces" after the
French French (french: français(e), link=no) may refer to: * Something of, from, or related to France ** French language, which originated in France, and its various dialects and accents ** French people, a nation and ethnic group identified with Franc ...
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...
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 ...
. Originally, Aleksandrov called these spaces “\mathfrak_k domain”. The terminology \operatorname(k) was coined by Mikhail Gromov in 1987 and is an
acronym An acronym is a word or name formed from the initial components of a longer name or phrase. Acronyms are usually formed from the initial letters of words, as in ''NATO'' (''North Atlantic Treaty Organization''), but sometimes use syllables, as ...
for
Élie Cartan Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometr ...
,
Aleksandr Danilovich Aleksandrov Aleksandr Danilovich Aleksandrov (russian: Алекса́ндр Дани́лович Алекса́ндров, alternative transliterations: ''Alexandr'' or ''Alexander'' (first name), and ''Alexandrov'' (last name)) (4 August 1912 – 27 July 19 ...
and Victor Andreevich Toponogov (although Toponogov never explored curvature bounded above in publications).


Definitions

For a
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
k, let M_k denote the unique complete
simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spa ...
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
(real 2-dimensional
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ...
) with constant curvature k. Denote by D_k the
diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid f ...
of M_k, which is \infty if k \leq 0 and is \frac if k>0. Let (X,d) be a geodesic metric space, i.e. a metric space for which every two points x,y\in X can be joined by a geodesic segment, an
arc length ARC may refer to: Business * Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s * Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services * ...
parametrized continuous curve \gamma\colon ,b\to X,\ \gamma(a) = x,\ \gamma(b) = y, whose length :L(\gamma) = \sup \left\ is precisely d(x,y). Let \Delta be a triangle in X with geodesic segments as its sides. \Delta is said to satisfy the \mathbf(k) inequality if there is a comparison triangle \Delta' in the model space M_k, with sides of the same length as the sides of \Delta, such that distances between points on \Delta are less than or equal to the distances between corresponding points on \Delta'. The geodesic metric space (X,d) is said to be a \mathbf(k) space if every
geodesic triangle 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 connectio ...
\Delta in X with
perimeter A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimeter has several pr ...
less than 2D_k satisfies the \operatorname(k) inequality. A (not-necessarily-geodesic) metric space (X,\,d) is said to be a space with curvature \leq k if every point of X has a
geodesically convex In mathematics — specifically, in Riemannian geometry — geodesic convexity is a natural generalization of convexity for sets and functions to Riemannian manifolds. It is common to drop the prefix "geodesic" and refer simply to "convex ...
\operatorname(k)
neighbourhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural a ...
. A space with curvature \leq 0 may be said to have non-positive curvature.


Examples

* Any \operatorname(k) space (X,d) is also a \operatorname(\ell) space for all \ell>k. In fact, the converse holds: if (X,d) is a \operatorname(\ell) space for all \ell>k, then it is a \operatorname(k) space. * The n-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
\mathbf^n with its usual metric is a \operatorname(0) space. More generally, any real
inner product space In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
(not necessarily complete) is a \operatorname(0) space; conversely, if a real
normed vector 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 ...
is a \operatorname(k) space for some real k, then it is an inner product space. * The n-dimensional hyperbolic space \mathbf^n with its usual metric is a \operatorname(-1) space, and hence a \operatorname(0) space as well. * The n-dimensional
unit sphere In mathematics, a unit sphere is simply a sphere of radius one around a given center. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A unit ...
\mathbf^n is a \operatorname(1) space. * More generally, the standard space M_k is a \operatorname(k) space. So, for example, regardless of dimension, the sphere of radius r (and constant curvature \frac) is a \operatorname\left(\frac\right) space. Note that the diameter of the sphere is \pi r (as measured on the surface of the sphere) not 2r (as measured by going through the centre of the sphere). * The punctured plane \Pi = \mathbf^2\backslash\ is not a \operatorname(0) space since it is not geodesically convex (for example, the points (0,1) and (0,-1) cannot be joined by a geodesic in \Pi with arc length 2), but every point of \Pi does have a \operatorname(0) geodesically convex neighbourhood, so \Pi is a space of curvature \leq 0. * The closed subspace X of \mathbf^3 given by X = \mathbf^ \setminus \ equipped with the induced length metric is ''not'' a \operatorname(k) space for any k. * Any product of \operatorname(0) spaces is \operatorname(0). (This does not hold for negative arguments.)


Hadamard spaces

As a special case, a complete CAT(0) space is also known as a Hadamard space; this is by analogy with the situation for Hadamard manifolds. A Hadamard space 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 th ...
(it has the
homotopy type In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
of a single point) and, between any two points of a Hadamard space, there is a unique geodesic segment connecting them (in fact, both properties also hold for general, possibly incomplete, CAT(0) spaces). Most importantly, distance functions in Hadamard spaces are
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...
: if \sigma_1, \sigma_2 are two geodesics in ''X'' defined on the same interval of time ''I'', then the function I\to \R given by :t \mapsto d \big( \sigma_ (t), \sigma_ (t) \big) is convex in ''t''.


Properties of CAT(''k'') spaces

Let (X,d) be a \operatorname(k) space. Then the following properties hold: * Given any two points x,y\in X (with d(x,y)< D_k if k> 0), there is a unique geodesic segment that joins x to y; moreover, this segment varies continuously as a function of its endpoints. * Every local geodesic in X with length at most D_k is a geodesic. * The d- balls in X of radius less than D_k/2 are (geodesically) convex. * The d-balls in X of radius less than D_k are contractible. * Approximate midpoints are close to midpoints in the following sense: for every \lambda < D_k and every \epsilon > 0 there exists a \delta = \delta(k,\lambda,\epsilon) > 0 such that, if m is the midpoint of a geodesic segment from x to y with d(x,y)\leq \lambda and \max \bigl\ \leq \frac1 d(x, y) + \delta, then d(m,m') < \epsilon. * It follows from these properties that, for k\leq 0 the universal cover of every \operatorname(k) space is contractible; in particular, the higher
homotopy group In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotop ...
s of such a space are
trivial Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense. Latin Etymology The ancient Romans used the word ''triviae'' to describe where one road split or fork ...
. As the example of the n-sphere \mathbf^n shows, there is, in general, no hope for a \operatorname(k) space to be contractible if k > 0.


Surfaces of non-positive curvature

In a region where the curvature of the surface satisfies , geodesic triangles satisfy the CAT(0) inequalities of comparison geometry, studied by Cartan, Alexandrov and Toponogov, and considered later from a different point of view by Bruhat and Tits; thanks to the vision of Gromov, this characterisation of non-positive curvature in terms of the underlying metric space has had a profound impact on modern geometry and in particular
geometric group theory Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such group (mathematics), groups and topology, topological and geometry, geometric pro ...
. Many results known for smooth surfaces and their geodesics, such as Birkhoff's method of constructing geodesics by his curve-shortening process or van Mangoldt and Hadamard's theorem that a
simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spa ...
surface of non-positive curvature is homeomorphic to the plane, are equally valid in this more general setting.


Alexandrov's comparison inequality

The simplest form of the comparison inequality, first proved for surfaces by Alexandrov around 1940, states that The inequality follows from the fact that if describes a geodesic parametrized by arclength and is a fixed point, then : is a
convex function In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigraph (the set of poi ...
, i.e. :\ddot(t) \ge 0. Taking geodesic polar coordinates with origin at so that , convexity is equivalent to : r\ddot + \dot^2 \ge 1. Changing to normal coordinates , at , this inequality becomes :, where corresponds to the unit vector . This follows from the inequality , a consequence of the non-negativity of the derivative of the Wronskian of and from
Sturm–Liouville theory In mathematics and its applications, classical Sturm–Liouville theory is the theory of ''real'' second-order ''linear'' ordinary differential equations of the form: for given coefficient functions , , and , an unknown function ''y = y''(''x'') ...
.;


See also

* Cartan–Hadamard theorem


References

* * * * * * * {{cite book , last = Hindawi , first = Mohamad A. , title = Asymptotic invariants of Hadamard manifolds , publisher = PhD thesis , location = University of Pennsylvania , url = http://www.math.upenn.edu/grad/dissertations/HindawiThesis.pdf , year = 2005 Metric geometry