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Geodesic behavior for Finsler metrics of constant positive flag curvature on S^2

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Authors
Bryant, R
Foulon, P
Ivanov, S
Matveev, VS
Ziller, W
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Abstract
We study non-reversible Finsler metrics with constant flag curvature 1 on S^2 and show that the geodesic flow of every such metric is conjugate to that of one of Katok's examples, which form a 1-parameter family. In particular, the length of the shortest closed geodesic is a complete invariant of the geodesic flow. We also show, in any dimension, that the geodesic flow of a Finsler metrics with constant positive flag curvature is completely integrable. Finally, we give an example of a Finsler metric on S^2 with positive flag curvature such that no two closed geodesics intersect and show that this is not possible when the metric is reversible or have constant flag curvature
Type
Journal article
Subject
Finsler geometry
Geodesic flow
Permalink
https://hdl.handle.net/10161/20574
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Scholars@Duke

Bryant

Robert Bryant

Phillip Griffiths Professor of Mathematics
My research concerns problems in the geometric theory of partial differential equations.  More specifically, I work on conservation laws for PDE, Finsler geometry, projective geometry, and Riemannian geometry, including calibrations and the theory of holonomy. Much of my work involves or develops techniques for studying systems of partial differential equations that arise in geometric problems.  Because of their built-in invariance properties, these systems often have specia
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