Time: 4:00 p.m.
Location: 106 McAllister Building
Name: Robert Bryant
Affiliation: Duke University
Title: The geodesic flow on Finsler 2-spheres of constant curvature
Abstract: After a review of the basics of Finsler geometry in 2-dimensions, including the definition of the Finsler-Gauss curvature, I will consider the geometry of the geodesic flow on a Finsler 2-sphere with Finsler-Gauss curvature equal to 1. Although the geodesics leaving a point on such a surface refocus at distance π to an `antipodal point', because these Finsler metrics are not reversible in general, the geodesics may not close at length 2π. In fact, examples of this phenomenon were first constructed by Katok in 1973, when he produced a 1-parameter family of such examples with mutually non-equivalent geodesic flows. I show that, in fact, for the general Finsler 2-sphere with Finsler-Gauss curvature equal to 1, its geodesic flow is conjugate to exactly one of the Katok examples. Along the way, I will explain how the Finsler 2-spheres are related to Zoll metrics and other interesting examples in 2-dimensional geometry.
