Zoll surface

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 S2 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.

A Zoll surface discovered by Zoll in 1903. A closed geodesic is pictured in red.

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, Arthur L. (1978), Manifolds all of whose geodesics are closed, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 93, Springer, Berlin, doi:10.1007/978-3-642-61876-5
  • Funk, Paul (1913), "Über Flächen mit lauter geschlossenen geodätischen Linien", Mathematische Annalen, 74: 278–300, doi:10.1007/BF01456044
  • Guillemin, Victor (1976), "The Radon transform on Zoll surfaces", Advances in Mathematics, 22 (1): 85–119, doi:10.1016/0001-8708(76)90139-0
  • LeBrun, Claude; Mason, L.J. (July 2002), "Zoll manifolds and complex surfaces", Journal of Differential Geometry, 61 (3): 453–535, doi:10.4310/jdg/1090351530
  • Zoll, Otto (March 1903). "Über Flächen mit Scharen geschlossener geodätischer Linien". Mathematische Annalen (in German). 57 (1): 108–133. doi:10.1007/bf01449019.
  • Tannery's pear, an example of Zoll surface where all closed geodesics (up to the meridians) are shaped like a curved-figure eight.
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