(α, β)-complex Finsler Metrics And Product Complex Finsler Metrics

As is known, complex Finsler metrics such as Kobayashi and Carathédory metrics play an important role in the classification of bounded domains up to biholomorphism in Cⁿ. Both of the Kobayashi and Caratheédory metrics are complex Finsler metrics. In general they don't have well regularities. However, on strongly convex domains in Cⁿ they agree and both of them are smooth metrics in sense of Finsler geometry.The purpose of this paper is to study two classes of complex Finsler manifolds, i.e., complex (α,β)-metric and product complex Finsler metric. We first obtain their metric tensors and their determinants respectively, and then obtain a necessary and sufficient conditions for these metrics to be strongly pseudoconvex Finsler metrics. After geting the inverse matrix of their metric tensor matrixes, we obtain the explicit expression of the Christoffel symbolsΓ_;μ^αassociated to the complex (α,β)-Finsler metrics and the product complex Finsler metrics, respectively, and furthermore obtain a necessary and sufficient condition for them to be strongly Kaehler Finsler metrics. At the end of this paper, we obtain an explicit expression of the holomorphic curvature of the product Finsler manifold.The whole dissertation includes three chapters. In chapter 1, we introduce some basic concepts of strongly pseudoconvex complex Finsler manifold, including complex horizontal bundle, or equivalently complex non-linear connection, the Chern-Finsler connection and its holomorphic curvature. In chapter 2, we introduce a class of complex (α,β)-metrics. By calculating the coefficient of the complex non-linear connection associated to the complex (α,β)-Finsler metrics, we obtain the necessary and sufficient conditions for the complex (α,β) Finsler manifold to be complex Berwald and strongly Kaehler Finsler manifolds. In chapter 3, we introduce a product complex manifolds with product complex Finsler metrics. By geting the coefficients of the complex non-linear conections associated to the product complex Finsler metric, we obtain the holomorphic curvature of the product complex Finsler manifold.