The Gauss-Bonnet-Chern Formula For Finslerian Orbifolds And Related Topics

The Gauss-Bonnet-Chern formula is one of the most important results in differ-ential geometry. It discloses the intrinsic relation between the curvature, a geometric quantity, and the Euler characteristic number, a topological invariants. The intrinsic proof of S. Chern is of great significance since it can be applied to a much larger class of metrics or spaces. In this paper we will prove that the Gauss-Bonnet-Chern formula holds for some types of Finsler metrics on orbifolds. The main idea of the proof is adopted from the intrinsic proof of S. Chern for Riemannian metrics on closed mani-folds and D. Bao-S. Chern’s treatment for Landsberg metrics.Cohomogeneity one Riemannian manifolds have been studied extensively. A lot of interesting results have been obtained, such as constructing Einstein metrics, posi-tively or nonnegatively curved metrics. Randers metric is a natural generalization of Riemannian metric. Hence it is interesting to study cohomogeneity one Randers mani-folds. In this paper we also study this manifolds.In the first part of this thesis, we devote to the study of Gauss-Bonnet-Chern for-mula for Finslerian orbifolds. The formula is related to the volume function of the indicatrix. When the volume function of the indicatrix is a constant, we first give the formula for2-dimensional compact boundary-less Landsberg orbifolds. Furthermore the formula is given for any Finsler orbifold whose dimension is equal or bigger than2as well as the case of compact orbifold with boundary. Finally, we extend to the case when the volume of the indicatrix is a variable.In the second part of this thesis, we study some problems of cohomogeneity one Randers manifolds. First we completely describe the invariant vector fields on coho-mogeneity one Riemannian manifolds, and then get the cohomogeneity one Randers metrics by navigation. Second we give the construction of Killing vector fields on the regular part of cohomogeneity one Riemannian manifolds. Moreover, we use these results to give an explicit example.If a2-form satisfies the Gauss-Bonnet-Chern formula in the Landsberg manifold case, is the2-form a curvature form of a Landsberg metric? Finally, the further study of the problem is given.