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, 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, arXiv:math/0211021, 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.
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.