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 geodesic
s are closed
and of equal length. While the usual unit-sphere metric on ''S''2
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.
* Funk transform
: The original motivation for studying the Funk transform was to describe Zoll metrics on the sphere.
: "Manifolds all of whose geodesics are closed", ''Ergebisse Grenzgeb. Math.'', no. 93, Springer, Berlin, 1978.
: "Über Flächen mit lauter geschlossenen geodätischen Linien". ''Mathematische Annalen'' 74 (1913), 278–300.
: "The Radon transform
on Zoll surfaces". ''Advances in Mathematics
'' 22 (1976), 85–119.
; Mason, L.: "Zoll manifolds and complex surfaces". ''Journal of Differential Geometry'' 61 (2002), no. 3, 453–535.
an example of Zoll surface where all closed geodesics (up to the meridians) are shaped like a curved-figure eight.