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Geodesic behavior for Finsler metrics of constant positive flag curvature on S^2
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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 articlePermalink
https://hdl.handle.net/10161/20574Collections
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Show full item recordScholars@Duke
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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