- Wiedersehen pair
In
mathematics — specifically, inRiemannian geometry — a Wiedersehen pair is a pair of distinct points "x" and "y" on a (usually, but not necessarily, two-dimensional) compactRiemannian manifold ("M", "g") such that everygeodesic through "x" also passes through "y" (and the same with "x" and "y" interchanged). If every point of an oriented manifold ("M", "g") belongs to a Wiedersehen pair, then ("M", "g") is said to be a Wiedersehen manifold. The concept was introduced by theAustro-Hungarian mathematician Wilhelm Blaschke and comes from the German term meaning "seeing again".Examples
* A simple (almost trivial) example of a Wiedersehen manifold is given in dimension one by any (smooth) closed curve.
* Two distinct points on the 2-sphere S2 with its usual round metric form a Wiedersehen pair
if and only if they areantipodal points . (In this case, the geodesics aregreat circle s.) Since every point of S2 has an antipode, S2 is a Wiedersehen surface.* In fact, the
Blaschke conjecture (which has been proved in the two-dimensional case) states that the only Wiedersehen surfaces are the standard round spheres.* The
unit disc "D" in theEuclidean plane E2 with its usual flat metric is "not" a Wiedersehen surface. In fact, no two distinct points "x" and "y" in "D" form a Wiedersehen pair: it is an easy matter to construct astraight line (geodesic) through "x" that does not meet "y".References
* cite book
last = Blaschke
first = Wilhelm
authorlink = Wilhelm Blaschke
title = Vorlesung über Differentialgeometrie I
location = Berlin
publisher = Springer-Verlag
year = 1921External links
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