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.

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.
• Otto Zoll (Mar 1903). "Über Flächen mit Scharen geschlossener geodätischer Linien". Math. Ann. (in German). 57 (1): 108–133. doi:10.1007/bf01449019.