Finsler Metrics and Closed Geodesics

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On one hand the connection between curvature properties of Finsler metrics and the length of the shortest closed geodesic resp. the injectivity radius and on the other hand existence results for closed geodesics on compact manifolds carrying a Finsler metric will be considered. In more detail the following problems will be investigated:1. Does equality in the length estimate given by the applicant for the shortest closed geodesic of a non-reversible Finsler metric of positive flag curvature imply that the flag curvature is constant?2. Is is possible to improve the lower bound in the estimate presented by the applicant for the injectivity radius of a compact and simply-connected Riemannian manifold with positive flag curvature? Is there a lower bound which does not depend on the reversibility?3. The existence of two closed geodesics on a compact rank one symmetric space witha bumpy non-reversible Finsler metric.