Wiedersehen pair
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In 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 Riemannian manifold (M, g) such that every geodesic 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 the Austro-Hungarian mathematician Wilhelm Blaschke and comes from the German term meaning "seeing again".
[edit] 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 are antipodal points. (In this case, the geodesics are great circles.) 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 the Euclidean 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 a straight line (geodesic) through x that does not meet y.
[edit] References
- Blaschke, Wilhelm (1921). Vorlesung über Differentialgeometrie I. Berlin: Springer-Verlag.
