闵可夫斯基积芬斯勒度量的道格拉斯曲率和外尔曲率  

Douglas Curvature and Weyl Curvature of the Minkowskian Product Finsler Metric

作　　者：张娜 卢维娜[1] 韩江慧 加依达尔·里扎别克 ZHANG Na;LU Wei-na;HAN Jiang-hui;JIAYIDAER·Lizhabieke(College of Mathematical Sciences,Xinjiang Normal University,Urumqi,Xinjiang,830017,China)

机构地区：[1]新疆师范大学数学科学学院,新疆乌鲁木齐830017

出　　处：《新疆师范大学学报（自然科学版）》2023年第1期67-74,共8页Journal of Xinjiang Normal University(Natural Sciences Edition)

基　　金：国家自然科学基金(12261088,11761069)。

摘　　要：设F_(1)和F_(2)分别是流形M_(1)和M_(2)上的芬斯勒度量,F_(1)和F_(2)关于积函数f的闵可夫斯基积是在乘积流形M=M_(1)×M_(2)上赋予的芬斯勒度量F=f(K,H),其中K=F^(2)_(1),H=F^(2)_(2).文章首先推导出闵可夫斯基积芬斯勒度量F的道格拉斯曲率和外尔曲率的表达式;其次,给出道格拉斯度量F_(1)和F_(2)的闵可夫斯基积是道格拉斯度量的充要条件,并证明了两个具有消失的平均贝瓦尔德曲率的道格拉斯度量的闵可夫斯基积仍是道格拉斯度量;最后,给出外尔度量F_(1)和F_(2)的闵可夫斯基积是外尔度量的充要条件,并得到了两个具有消失的射影外尔曲率的芬斯勒度量的闵可夫斯基积仍具有消失的射影外尔曲率的充要条件。Let F_(1) and F_(2) be the Finsler metrics on the manifolds M_(1) and M_(2),respectively.The Minkowskian product Finsler metric of F_(1) and F_(2) with respect to the product function f is the Finsler metric F=f(K,H)endowed on the product manifold M=M_(1)×M_(2),where K=F_(1)^(2),H=F_(2)^(2).In this paper,firstly,we derive the Douglas curvature and Weyl curvature expression of the Minkowskian product Finsler metric F,respectively.Secondly of all,under the conditions that F_(1) and F_(2) are Douglas metrics or Weyl metrics,we obtain the necessary and sufficient conditions for the Minkowskian product Finsler metric F to be Douglas metric or Weyl metric,respectively.Finally,we prove that the Minkowskian product of two Douglas metrics with vanishing mean Berwald curvature is still a Douglas metric and obtain a necessary and sufficient conditions for the Minkowskian product of two Finsler metrics with vanishing projective Weyl curvature to still have vanishing projective Weyl curvature.

关 键 词：芬斯勒度量 闵可夫斯基积 道格拉斯曲率 外尔曲率 

分 类 号：O186.1[理学—数学]