In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
—specifically, in
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...
—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, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
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 ...
(''M'', ''g'') such that every
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 ...
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.
Discussion
Any arc of a great circle is a geodesic of the sphere, so that great circles in spher ...
s, the Wiedersehen pairs are exactly the pairs of
antipodal point
In mathematics, two points of a sphere (or n-sphere, including a circle) are called antipodal or diametrically opposite if they are the endpoints of a diameter, a straight line segment between two points on a sphere and passing through its cen ...
s.
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 "anticlockwise". A space is ori ...
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, also referred to as the Austro-Hungarian Empire, the Dual Monarchy or the Habsburg Monarchy, was a multi-national constitutional monarchy in Central Europe between 1867 and 1918. A military and diplomatic alliance, it consist ...
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, mathematical structure, structure, space, Mathematica ...
Wilhelm Blaschke
Wilhelm Johann Eugen Blaschke (13 September 1885 – 17 March 1962) was an Austrian mathematician working in the fields of differential and integral geometry.
Education and career
Blaschke was the son of mathematician Josef Blaschke, who taugh ...
and comes from the
German
German(s) may refer to:
* Germany, the country of the Germans and German things
**Germania (Roman era)
* Germans, citizens of Germany, people of German ancestry, or native speakers of the German language
** For citizenship in Germany, see also Ge ...
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'' me ...
) is the standard Euclidean
''n''-sphere. Initially known as the Blaschke conjecture, this result was established by combined works of
Berger
Berger is a surname in both German language, German and French language, French, although there is no etymological connection between the names in the two languages. The French surname is an occupational name for a shepherd, from Old French ''bergi ...
,
Kazdan,
Weinstein (for even ''n''), and
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 p ...
(odd ''n'').
See also
*
Cut locus (Riemannian manifold)
In differential geometry, the cut locus of a point on a manifold is the closure of the set of all other points on the manifold that are connected to by two or more distinct shortest geodesics. More generally, the cut locus of a closed set on ...
References
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External links
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Riemannian geometry
Equations
{{Riemannian-geometry-stub
Manifolds