2009 UW-PIMS Colloquium - 02

It has been known for over a century that there are many Riemannian metrics on the 2-sphere with the property that all of their geodesics are closed. Zoll, a student of Hilbert, constructed an infinite dimensional family of surfaces of rotation with this property. Following ideas of Funk, Guillemin proved that these metrics on the 2-sphere are essentially parametrized by the odd functions on the round 2-sphere.

These ideas can also be applied to the understanding of Finsler metrics on the 2-sphere whose Finsler-Gauss curvature is constant, as will be explained in the talk. This leads, via the recent work of LeBrun and Mason, to a resolution of a basic problem in Finsler geometry: to describe such Finsler metrics on the 2-sphere.

In this talk, no knowledge of Finsler geometry will be assumed, only a very basic understanding of curves and surfaces. The history of the problem will be discussed and numerous examples will be given to illustrate the underlying ideas.