Three projective problems on Finsler surfaces
Two Finsler metrics on the same manifold are called projectively equivalent, if they have the same unparametrized, oriented geodesics. A vector field on the manifold is called projective for a Finsler metric, if its flow takes geodesics to geodesics as unparametrized curves. In this dissertation, after an introduction to the general theory of Finsler metrics and its projective aspects, results to three projective problems on Finsler metrics on surfaces are presented: Firstly. Inspired by a problem posed by Sophus Lie, it is proven that every Finsler metric, admitting three independent projective vector fields, is projectively equivalent to a Randers metric. An explicit list of such metrics is given, complete up to isometry and projective equivalence. Secondly. The problem of local, fiber-global projective metrization asks whether a given system of unparametrized, oriented curves describes the geodesics of some fiber-globally defined Finsler metric - and if yes, how unique this metric is. It is shown that the set of such metrizations is, up to the trivial freedom, in 1-to-1 correspondence with measures on the space of prescribed curves, satisfying a certain equilibrium property. Thirdly. It is proven that on surfaces of negative Euler characteristic, two real-analytic Finsler metrics can only be trivially projectively related: they are projectively equivalent, if and only if they differ by multiplication with a positive number and addition of a closed 1-form.