Zoll surface

Short description: Surface homeomorphic to a sphere

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

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.

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

• Manifolds all of whose geodesics are closed, Ergebnisse der Mathematik und ihrer Grenzgebiete, 93, Springer, Berlin, 1978, doi:10.1007/978-3-642-61876-5 
• "Über Flächen mit lauter geschlossenen geodätischen Linien", Mathematische Annalen 74: 278–300, 1913, doi:10.1007/BF01456044 
• "The Radon transform on Zoll surfaces", Advances in Mathematics 22 (1): 85–119, 1976, doi:10.1016/0001-8708(76)90139-0 
• "Zoll manifolds and complex surfaces", Journal of Differential Geometry 61 (3): 453–535, July 2002, doi:10.4310/jdg/1090351530 
• Zoll, Otto (March 1903). "Über Flächen mit Scharen geschlossener geodätischer Linien" (in German). Mathematische Annalen 57 (1): 108–133. doi:10.1007/bf01449019. http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002259133. 

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.