# Browsing by Author "Matveev, VS"

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Item Open Access A solution of a problem of Sophus Lie: Normal forms of two-dimensional metrics admitting two projective vector fields(Mathematische Annalen, 2008-02-01) Bryant, RL; Manno, G; Matveev, VSWe give a complete list of normal forms for the two-dimensional metrics that admit a transitive Lie pseudogroup of geodesic-preserving transformations and we show that these normal forms are mutually non-isometric. This solves a problem posed by Sophus Lie. © 2007 Springer-Verlag.Item Open Access Geodesic behavior for Finsler metrics of constant positive flag curvature on $S^2$(2017-11-01) Bryant, RL; Foulon, P; Ivanov, SV; Matveev, VS; Ziller, WWe 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 curvatureItem Open Access Geodesic behavior for Finsler metrics of constant positive flag curvature on S^2(Journal of Differential Geometry) Bryant, R; Foulon, P; Ivanov, S; Matveev, VS; Ziller, WWe 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