Some Rigidity Theorems On Finsler Manifolds

In this paper we study some rigidity theorems on Finsler manifolds. Firstly we characterize Finsler metrics with relatively isotropic mean Landsberg curvature. It is shown that Finsler metrics with non-zero constant relatively isotropic Lands-berg curvature (resp. mean Landsberg curvature) under the conditions that the Finsler metrics are complete and the Cartan torsion (resp. mean Cartan torsion) are bounded must be Riemannian. Furthermore, a necessary condition for Finsler metrics with generalized relatively mean isotropic Landsberg curvature to be Riemn-nian are found. Secondly we consider generalized Landsberg metrics with non-zero sectional flag curvature. We prove that such kinds of Randers metrics must be Rie-mannian. We also prove that such kinds of Finsler metrics with non-zero relatively isotropic Landsberg curvature must be Riemannian. Thirdly we characterize gen-eralized L-reducible Finsler metrics. We investigate some necessary conditions that a generalized L-reducible Finsler metric is a Randers metric. Finally we consider compact Finsler manifolds with negative flag curvature. We obtain some rigidity re-sults that a compact Finsler manifold with negative flag curvature satisfying certain Riemann curvature condition must be a Riemannian manifold.