Finsler metrics of constant curvature, and their relation to Zoll metrics and integrable systems
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Vladimir Matveev(Friedrich-Schiller-Universität Jena)
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Europe/Paris
Description
I will mostly speak about Finsler metrics of positive constant flag curvature (I explain what is it) on closed 2-dimensional surfaces.
The main result is that the geodesic flow of such a metric is conjugate to that of a Katok metric (recall that Katok metrics is are easy and well-understood examples of two-dimensional Finsler metrics of positive constant flag curvature). In particular, either all geodesics are closed, and at most two of them have length less than the generic one, or all geodesics but two are not closed; in the latter case there exists a Killing vector field.
Generalisations for the multidimensional case will be given; in particular I show that in all dimensions the topological entropy vanishes and the geodesic flow is Liouville integrable. I will also show that in all dimensions a Zermelo transformation of every metric of positive constant flag curvature has all geodesics closed.
The results are part of an almost finished paper coauthored with R. Bryant, P. Foulon, S. Ivanov and W. Ziller.
