The Curvature Properties Of Finsler Warped Product Metrics

In recent years,the non-Riemannian curvature properties of the Finsler metric have become the focus of research.In this paper,we studied the warped product manifold M:=I ×M,where I is an interval of R,M is an(n-1)-dimensional manifold equipped with a Riemannian metric α,F(u,v)=α(u,v)φ(r,s).A sufficient and necessary condition for it to have weakly isotropic S curvature is obtained,and a series of examples with vanishing S curvature are given.At the same time,the conditions of the wapred product Finsler metric with vanishing x curvature are described,and a class of examples are given.Firstly,in the second chapter of this paper,by using tensor analysis and theories of PDEs,we studied Finsler metrics with weakly isotropic S curvature,and its characterization equation is obtained.Based on it,a series of examples of Douglas Finsler warped product metrics with vanishing S curvature are obtained.Secondly,in the third chapter of this paper,we obtained the sufficient and necessary condition for the Finsler warped metrics with vanishing χ curvature.Furthermore,we constructed a series of the Finsler warped product metrics with vanishing x curvature.