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In mathematics, particularly in
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
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
, a Zoll surface, named after Otto Zoll
Otto is a masculine German given name and a surname. It originates as an Old High German short form (variants ''Audo'', '' Odo'', ''Udo'') of Germanic names beginning in ''aud-'', an element meaning "wealth, prosperity".
The name is recorded f ...
, 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
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
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 ''T ...
all of whose geodesic
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
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
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canonic ...
.
Zoll, a student of David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
, 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