Navigation Problem On Finsler Manifolds And Its Applications

The indicatrixes of a Riemannian manifold are ellipsoids (centered at the originof the tangent spaces), thus it is natural to consider homothetic deformations of theindicatrixes to obtain a conformal Riemannian metric. In Finsler geometry, theindicatrixes can be any strongly convex hypersurface. Thus, besides conformaldeformation, it is also natural to consider translating each indicatrix; such a wayof constructing new Finsler metric is called a navigation problem. For example, aRiemannian metric produces a Randers metric under navigation problem.Since homothety and translation are both a?ne transformations, the affine ge-ometry on tangent spaces of a Finsler manifold is studied. In light of the classicalBlaschke immersion theory, the Finsler invariants of the indicatrix of a Minkowskispace turns out to have affine geometrical meanings. By showing that the Mat-sumoto torsion is just the affine cubic form, a direct proof of the Matsumoto-Hˉojoˉcriterion is given. That criterion of Randers metrics in dimensions no less thanthree is originally proved in a tedious algebraic way. Moreover, it is proved that themean a?ne curvature provides a new criteria of Randers metrics in any dimension.Foulon's dynamical approach and the Chern connection method in Finsler ge- ometry is proved to be equivalent. The dynamical approach is e?cient in computingthe ?ag curvature. Foulon's method is also extended to compute other quantitiessuch as the Cartan torsion, Landsberg curvature and S-curvature.Using this method in navigation problem, the relations of flag curvature, Riccicurvature and Weyl curvature of the new metrics from old ones is studied. If theunderlying vector field in navigation problem is conformal or homothetic, theserelations simplify to a significant level. As applications, some important classicalmetrics, such as Randers metrics of constant curvature and Funk metrics, are easilyderived. Besides, some properties of the conformal groups of Finsler manifolds areconsidered.The navigation problem also affects the S-curvature. It is proved that thenavigation problem on a Berwald manifold produces a metric with isotropic S-curvature if the underlying vector field is conformal. Specifically, bi-invariant Finslermetrics on Lie groups are proved to be Berwaldian. For Finsler metrics on a Liegroup which is only left invariant, an explicit formula for the S-curvature is given.A lot of Finsler metrics of scalar curvature are constructed via navigation prob-lem. They are neither Randers metrics, nor projectively ?at ones. Several Finslermetrics with vanishing S-curvature are constructed also.