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.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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