The Curvature Properties Of Finsler Warped Product Metrics

Finsler warped product metrics are the natural extension of Riemannian warped product metrics.Finsler warped product metrics is a new kind of Finsler metrics including spherically symmetric Finsler metrics and belong to the generalized(α,β)Finsler metrics.Riemannian quantities are the natural extension of the corresponding geometric quantities in Riemann geometry in Finsler geometry,such as the Riemann curvature,the flag curvature,the Ricci curvature and the Weyl curvature.Beside the Riemannian quantities,there are several important non-Riemannian quantities in Finsler geometry.Non-Riemannian quantities are unique in Finsler geometry which disappear in Riemann geometry,such as the S curvature,the Douglas curvature and the(?)curvature.There are many relationships between non-Riemannian quantities and between non-Riemannian and Riemannian quantities.One of the fundamental problems in Finsler geometry is to study Finsler metrics of the constant flag curvature and Einstein Finsler metrics.In[12],there has obtained two PDEs that characterize Finsler warped product metrics of scalar flag curvature.In this paper,we improve this result.In particular,we find equations to characterize Finsler warped product metrics of constant flag curvature.Then we improve the result in[12]on characterizing Einstein Finsler warped product metrics.As its application we construct some new warped product Douglas metrics of constant Ricci curvature by using known locally projectively flat spherically metrics of constant flag curvature.This paper consists of four parts:In the first part,we illustrate the backgrounds of our research and the related definitions,theorems and conclusions.In the second part,we discuss Finsler warped product metrics of scalar flag curvature by using the Weyl curvature.In the third part,we discuss Finsler warped product metrics of constant flag curvature by using the(?)curvature.In the fourth part,we discuss Finsler warped product metrics with isotropic Ricci curvature and construct some new examples.