Zoll surface

In mathematics, 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.

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.
Zoll, Otto; Ueber Flächen mit Scharen geschlossener geodätischer Linien. (German) Math. Ann. 57 (1903), no. 1, 108–133.

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