In
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 ...
—specifically, in
Riemannian geometry—a Wiedersehen pair is a pair of distinct points ''x'' and ''y'' on a (usually, but not necessarily, two-dimensional)
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
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 ...
(''M'', ''g'') such that every
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. ...
through ''x'' also passes through ''y'', and the same with ''x'' and ''y'' interchanged.
For example, on an ordinary sphere where the geodesics are
great circle
In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point.
Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geomet ...
s, the Wiedersehen pairs are exactly the pairs of
antipodal points.
If every point of an
oriented
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
manifold (''M'', ''g'') belongs to a Wiedersehen pair, then (''M'', ''g'') is said to be a Wiedersehen manifold. The concept was introduced by the
Austro-Hungarian
Austria-Hungary, often referred to as the Austro-Hungarian Empire,, the Dual Monarchy, or Austria, was a constitutional monarchy and great power in Central Europe between 1867 and 1918. It was formed with the Austro-Hungarian Compromise of ...
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 ...
Wilhelm Blaschke and comes from the
German
German(s) may refer to:
* Germany (of or related to)
**Germania (historical use)
* Germans, citizens of Germany, people of German ancestry, or native speakers of the German language
** For citizens of Germany, see also German nationality law
**Ger ...
term meaning "seeing again". As it turns out, in each dimension ''n'' the only Wiedersehen manifold (up to
isometry
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 ...
) is the standard Euclidean
''n''-sphere. Initially known as the Blaschke conjecture, this result was established by combined works of
Berger,
Kazdan,
Weinstein Weinstein is a German or Yiddish surname meaning wine stone, referring to the crystals of cream of tartar (potassium bitartrate) resulting from the process of fermenting grape juice.Yang
Yang may refer to:
* Yang, in yin and yang, one half of the two symbolic polarities in Chinese philosophy
* Korean yang, former unit of currency of Korea from 1892 to 1902
* YANG, a data modeling language for the NETCONF network configuration pr ...
(odd ''n'').
See also
*
Cut locus (Riemannian manifold) In Riemannian geometry, the cut locus of a point p in a manifold is roughly the set of all other points for which there are multiple minimizing geodesics connecting them from p, but it may contain additional points where the minimizing geodesic is u ...
References
*
*
*
External links
*
*
Riemannian geometry
Equations
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