On Finsler Warped Product Metrics Of Isotropic S-curvature

Finsler warped product metrics are the natural extension of Riemannian warped product metrics,it is a new kind of Finsler metrics including spherically symmetric Finsler metrics Riemannian quantities are the natural extension of the corresponding geometric quantities in Riemann geometry in Finsler geometry,such as the flag curvature,the Ricci curvature.Beside the Riemannian quantities,there are several important non-Riemannian quantities in Finsler geometry,such as the S curvature,the Douglas curvature.At present,there is relatively little research content on the aspect of Finsler warped product metrics.There are many topics to be developed and further researched in this field,so the choice of research content in this paper has great value and research significance.Recently,people have found the differential equation that characterizes Finsler warped product metrics with vanishing Douglas curvature,and found the necessary and sufficient conditions of the non-spherically symmetric Finsler warped product metrics with scalar flag curvature.We know that there is a great relationship between S curvature and flag curvature,so it is important to study the properties of S curvature of Finsler warped product metrics.This article mainly studies the Finsler warped product metrics with isotropic S curvature,and gives the necessary and sufficient conditions of the Finsler warped product metrics with this curvature property.Using this result,we further give a complete classification of the Douglas type Finsler warped product metrics with isotropic S curvature,and prove that they must be Randers metrics or Berwald metricsThis 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 introduce two volume forms of Finsler warped product metrics.In the third part,we discuss the S-curvatur of Finsler warped product metrics.In the fourth part,we give the proof of main results and study Randers Finsler warped product metrics.