Wiedersehen pair

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. For example, on an ordinary sphere where the geodesics are great circles, the Wiedersehen pairs are exactly the pairs of antipodal points.

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". As it turns out, in each dimension n the only Wiedersehen manifold (up to isometry) is the standard Euclidean n-sphere. Initially known as the Blaschke conjecture, this result was established by combined works of Berger, Kazdan, Weinstein (for even n), and Yang (odd n).

See also

• Cut locus (Riemannian manifold)

References

• Berger, Marcel (1978). "Blaschke’s Conjecture for Sphere". Manifolds all of whose Geodesics are Closed. Berlin, Heidelberg: Springer Berlin Heidelberg. p. 236–242. doi:10.1007/978-3-642-61876-5_13. ISBN 978-3-642-61878-9. 
• Blaschke, Wilhelm (1921). Vorlesung über Differentialgeometrie I. Berlin: Springer-Verlag. 
• Kazdan, Jerry L. (1982). "An isoperimetric inequality and Wiedersehen manifolds". Seminar on Differential Geometry. (AM-102). Princeton University Press. ISBN 978-0-691-08268-4. http://www.jstor.org/stable/j.ctt1bd6kkq.9. Retrieved 2024-01-29. 
• McKay, Benjamin. "Summary of progress on the Blaschke conjecture". https://www.intlpress.com/site/pub/files/_fulltext/journals/iccm/2015/0003/0002/ICCM-2015-0003-0002-a004.pdf. 
• Weinstein, Alan (1974-01-01). "On the volume of manifolds all of whose geodesics are closed". Journal of Differential Geometry 9 (4). doi:10.4310/jdg/1214432547. ISSN 0022-040X. 
• C. T. Yang (1980). "Odd-dimensional wiedersehen manifolds are spheres". J. Differential Geom. 15 (1): 91–96. doi:10.4310/jdg/1214435386. ISSN 0022-040X. 
• Chavel, Isaac (2006). Riemannian geometry: a modern introduction. New York: Cambridge University Press. pp. 328–329. ISBN 0-521-61954-8. 

External links

• Weisstein, Eric W.. "Wiedersehen pair". http://mathworld.wolfram.com/WiedersehenPair.html. 
• Weisstein, Eric W.. "Wiedersehen surface". http://mathworld.wolfram.com/WiedersehenSurface.html.