Some Topics On Complex Finsler Manifold

Complex Finsler manifold is a manifold endowed with Complex Finsler metrics. It is the generalization of the usual Hermitian manifold. It is well known, that the Kobayshi metric and the Caratheodory metric (not necessarily smooth) are the famous Finsler metrics on the Complex manifold. In this paper , we do the research on the complex manifold endowed with smooth Finsler metrics, study Projectively Related Complex Finsler Metrics , compare the Cartan connection with the Chern-Finsler connection under the weakly Kahler-Finsler metrics, and also have studied the most important complex Finsler metrics—Randers metrics.The whole dissertation includes four chapters. In the second chapter and the forth chapter, we discusse respectively Projectively Related Finsler Metrics and Randers metrics on the complex condition, in the third chapter, we compare the Cartan connection with the Chern-Finsler connection under the weakly Kahler-Finsler metrics.In the first chapter, we introduce some definitions and notations of the Complex Finsler manifold, including Complex Finsler manifold, Horizontal and Vertical bundles, non-linear Complex connections, Cartan connection.Furthermore there is the most important results and so on.In the second chapter, we introduce the difinition of Projectively Related complex Finsler Metrics on complex Finsler manifold and give the conditions for two Projectively Related Complex Finsler Metrics having the same geodesics as point sets, i.e.projectively equivalent conditions.In the third chapter, complex Finsler Metrics F induce a real Finsler Metric F° by using isomorphism °, they have same geodesies and induce the same distance function . On the other hand, the Cartan connection induced by F° and the Chern-Finsler connection induced by F are different. We use the isomorphism ° to get their comparisions.In the forth chapter, introducing Randers metric on complex Finsler manifold, and giving its strong Kahler conditions.