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, Arthur L. (1978), "Manifolds all of whose geodesics are closed", Vol. 93, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer, Berlin, doi:10.1007/978-3-642-61876-5
- Funk, Paul (1913), "Über Flächen mit lauter geschlossenen geodätischen Linien", Mathematische Annalen. 74: 278–300, doi:10.1007/BF01456044
- Guillemin, Victor (1976), "The Radon transform on Zoll surfaces", Advances in Mathematics. 22 (1): 85–119, doi:10.1016/0001-8708(76)90139-0
- LeBrun, Claude & Mason, L.J. (July 2002), "Zoll manifolds and complex surfaces", Journal of Differential Geometry. 61 (3): 453–535, arXiv:math/0211021. doi:10.4310/jdg/1090351530
- Zoll, Otto (March 1903). "Über Flächen mit Scharen geschlossener geodätischer Linien" (in German). Mathematische Annalen. 57 (1): 108–133. doi:10.1007/bf01449019
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.