On a class of almost regular Landsberg metrics

作     者：Shasha Zhou Jiayue Wang Benling Li 

作者机构：Department of Mathematics Ningbo University 

出 版 物：《Science China Mathematics》 (中国科学：数学（英文版）)

年 卷 期：2019年第62卷第5期

页      面：935-960页

核心收录：

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

基　　金：supported by Zhejiang Provincial Natural Science Foundation of China (ZPNSFC) (Grant No. R18A010002) National Natural Science Foundation of China (Grant No. 11371209) K.C. Wong Magna Fund in Ningbo University 

主　　题：Finsler metric Landsberg metric Berwald metric (α,β)-metric 

摘      要：There is a long existing unicorn problem in Finsler geometry: whether or not any Landsberg metric is a Berwald metric? Some classes of metrics were studied in the past and no regular non-Berwaldian Landsberg metric was found. However, if the metric is almost regular(allowed to be singular in some directions),some non-Berwaldian Landsberg metrics were found in the past years. All of them are composed by Riemannian metrics and 1-forms. This motivates us to ?nd more almost regular non-Berwaldian Landsberg metrics in the class of general(α, β)-metrics. In this paper, we ?rst classify almost regular Landsberg general(α, β)-metrics into three cases and prove that those regular metrics must be Berwald metrics. By solving some nonlinear PDEs,some new almost regular Landsberg metrics are constructed which have not been described before.