Conformal Complex Finsler Metrics

Let M be a complex manifold of dimension n, denote byÏ€: T^1,0M→M the holomorphic tangent bundle of M, and (?) = T^1,0M - {0}. Let F : T^1,0M→R⁺ be a strongly pseudoconvex Finsler metric on M[1], the pair (M, F) is called a strongly pseudoconvex complex Finsler manifold in sense of Abate and Patrizio[1]. Let (?) = e^σ(z)F : T^1,0M→R⁺ be a conformal transformation of the given strongly pseudoconvex complex Finsler metric F , whereσ: T^1,0M→R is a smooth real function on T^1,0M, then the pair (M, (?)) is also called a conformal transformation of (M, F). Let the complex Finsler connection be the complex Rund connection associated to (M, F) [18], [19]. The purpose of this paper is to study some geometric properties of the conformal complex Finsler manifold (M, (?)).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 complex Rund connection and its curvature and torsions, and some lemmas which will be used in this paper.In chapter 2, we obtain various kinds of commutation formulae of Hermtian tensor covariantly derivatived with respect to the complex Rund connection associated to (M, F) and (M, (?)), respectively.In chapter 3, we discuss two kind of conformal transformations of com-plex Finsler metric F, that is, the conformal transformations F→(?) = e^σF whereσ: M→R and F→(?) = e^σF whereσ: T^1,0M→R. With respect to these conformal transformations, we obtain the local expression of the adapted frames {δ_μ,(?)} and the complex non-linear connection coefficientsΓ_;μ^αassociatedto (M, (?)). respectively. After this we consider the conformal invariance of the torsion, the horizontal curvature, the vertical curvature, the holomorphic curvature and the flag curvature, with respect to the complex Rund connection associated to (M, (?)). Under the assumption that (M, F) be a Kaehler-Finsler (weakly Kaehler-Finsler) manifold, we get the necessary and sufficient condi-tions for (M, (?)) to be a Kaehler-Finsler (weakly Kaehler-Finsler) manifold.In chapter 4, as an application of conformal transformation, we study a special complex (α,β) manifold, that is, the complex Kropina manifold.