A Zoll surface 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 which are still Zoll. In particular, most Zoll surfaces do not have constant curvature.

Zoll surfaces are named after David Hilbert's student Otto Zoll, who discovered the first non-trivial examples.

[edit] References

• Besse, A.: "Manifolds all of whose geodesics are closed", Ergebisse Grenzgeb. Math., no. 93, Springer, Berlin, 1978.
• Funk, P.: "Über Fläschen 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.
• Zoll, Otto; Ueber Flächen mit Scharen geschlossener geodätischer Linien. (German) Math. Ann. 57 (1903), no. 1, 108--133.