The Szabó Metric On Product Of Complex Manifolds

Let (M₁, α), (M₂, β) be Hermitian manifolds, z= (z₁,z₂) ∈ M₁ × M₂,v = (v¹, … , vⁿ, vⁿ⁺¹, … , v^n+m) : = y₁ (?) y₂ ∈ T_z1M₁ (?) T_z2M₂, then we can define complex Szabó metric on the product M₁ × M₂whereBy directly computing the Chern connection coefficients, we have proofed that F_ε is a Berwald metrc (that is , the Chern connection coefficients Γ_β;μ^α have no v-dependence). Moreover ,F_ε is strongly Ka|¨hler-Finsler iff α,β are both Ka|¨hler metrics and also we obtain the concrete formula of its holomorphic curvature.The paper comprises of two sections:Section one mainly introducts some elementary conception and setting knowledge about complex Finsler metre, which includes definition and some examples of the complex Finsler metric , the complex vertical connection, the curvature, the Chern-Finsler connection ,the holomorphic curvature and so on. Section two gives the main results and its proofs.