In mathematics, a Zoll surface, named after Otto Zoll, is a surface

Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...

 homeomorphic to the 2-sphere , equipped with a Riemannian metric all of whose geodesic

Geodesic

In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...

s are closed

Closed geodesic

In differential geometry and dynamical systems, a closed geodesic on a Riemannian manifold M is the projection of a closed orbit of the geodesic flow on M.-Examples:On the unit sphere, every great circle is an example of a closed geodesic...

 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 which are still Zoll surfaces. In particular, most Zoll surfaces do not have constant curvature.

Zoll, a student of David Hilbert

David Hilbert

David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...

, discovered the first non-trivial examples.