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 ...

, Alexandrov spaces with curvature ≥ ''k'' form a generalization of

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 with

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 ...

≥ ''k'', where ''k'' is some real number. By definition, these spaces are

locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...

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 ...

length space In the mathematical study of metric spaces, one can consider the arclength of paths in the space. If two points are at a given distance from each other, it is natural to expect that one should be able to get from the first point to the second alon ...

s where the lower curvature bound is defined via comparison of

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 connection. ...

s in the space to geodesic triangles in standard constant-curvature Riemannian surfaces. One can show that the

Hausdorff dimension In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a ...

of an Alexandrov space with curvature ≥ ''k'' is either a non-negative integer or infinite. One can define a notion of "angle" and "tangent cone" in these spaces. Alexandrov spaces with curvature ≥ ''k'' are important as they form the limits (in the Gromov-Hausdorff metric) of sequences of Riemannian manifolds with sectional curvature ≥ ''k'', as described by Gromov's compactness theorem. Alexandrov spaces with curvature ≥ ''k'' were introduced by the Russian mathematician

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 ...

in 1948 and should not be confused with Alexandrov-discrete spaces named after the Russian topologist

Pavel Alexandrov Pavel Sergeyevich Alexandrov (russian: Па́вел Серге́евич Алекса́ндров), sometimes romanized ''Paul Alexandroff'' (7 May 1896 – 16 November 1982), was a Soviet mathematician. He wrote about three hundred papers, ma ...

. They were studied in detail by Burago,

Gromov Gromov (russian: Громов) is a Russian male surname, its feminine counterpart is Gromova (Громова). Gromov may refer to: * Alexander Georgiyevich Gromov (born 1947), Russian politician and KGB officer * Alexander Gromov (born 1959), R ...

and

Perelman Perelman ( he, פרלמן) is an Ashkenazi Jewish surname. Notable people with the surname include: * Bob Perelman (b. 1947), American poet * Chaim Perelman (1912-1984), Polish-born Belgian philosopher of law * Eliezer Ben-Yehuda () (1858-1922), ...

in 1992 and were later used in Perelman's proof of the

Poincaré conjecture In the mathematics, mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the Characterization (mathematics), characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dim ...

References

{{reflist Metric geometry Differential geometry Riemannian manifolds