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In mathematics, particularly in differential geometry, a Zoll surface, named after Otto Zoll, is a surface homeomorphic to the 2-sphere, equipped with a Riemannian metric all of whose geodesics are closed and of equal length. While the usual unit-sphere metric on ''S''2 obviously has this property, it also has an infinite-dimensional family of geometrically distinct deformations that are still Zoll surfaces. In particular, most Zoll surfaces do not have constant curvature. Zoll, a student of David Hilbert, discovered the first non-trivial examples.


See also


* Funk transform: The original motivation for studying the Funk transform was to describe Zoll metrics on the sphere.

References

*Besse, A.: "Manifolds all of whose geodesics are closed", ''Ergebisse Grenzgeb. Math.'', no. 93, Springer, Berlin, 1978. *Funk, P.: "Über Flächen mit lauter geschlossenen geodätischen Linien". ''Mathematische Annalen'' 74 (1913), 278–300. *Guillemin, V.: "The Radon transform on Zoll surfaces". ''Advances in Mathematics'' 22 (1976), 85–119. *LeBrun, C.; Mason, L.: "Zoll manifolds and complex surfaces". ''Journal of Differential Geometry'' 61 (2002), no. 3, 453–535. *


External links



Tannery's pear
an example of Zoll surface where all closed geodesics (up to the meridians) are shaped like a curved-figure eight. {{topology-stub Category:Surfaces