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In mathematics, particularly in differential geometry, a Zoll surface, named after Otto Zoll, is a
Tannery's pear
an example of Zoll surface where all closed geodesics (up to the meridians) are shaped like a curved-figure eight. {{topology-stub Surfaces
In mathematics, particularly in differential geometry, a Zoll surface, named after Otto Zoll, is a surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
homeomorphic to the 2-sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ce ...
, equipped with a Riemannian metric
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space '' ...
all of whose geodesics 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.
See also
*Funk transform
In the mathematical field of integral geometry, the Funk transform (also known as Minkowski–Funk transform, Funk–Radon transform or spherical Radon transform) is an integral transform defined by integrating a function on great circles of the s ...
: The original motivation for studying the Funk transform was to describe Zoll metrics on the sphere.
References
* * * * *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. {{topology-stub Surfaces