黎曼流形上导航术问题的推广  

Generalizations of Navigation Problems on Riemann Manifolds

作　　者：程新跃 瞿秋红 CHENG Xin-yue;QU Qiu-hong(School of Mathematical Sciences,Chongqing Normal University,Chongqing 401331,China)

机构地区：[1]重庆师范大学数学科学学院,重庆401331

出　　处：《西南大学学报（自然科学版）》2021年第4期85-91,共7页Journal of Southwest University(Natural Science Edition)

基　　金：国家自然科学基金项目(11871126);重庆师范大学研究基金项目(17XLB022).

摘　　要：黎曼流形上的导航术问题在芬斯勒几何中扮演着非常重要的角色.Randers度量和Kropina度量都可以由黎曼流形(M,h)上具有向量场W的导航术问题的解来刻画,其中‖W‖≤1.论文首先揭示了芬斯勒流形上的导航术问题与流形的单位切球的几何之间的重要关系.当芬斯勒流形(M,Φ)上的向量场V=V(x)满足条件Φ(x,-V x)<1时,证明了由导航数据(Φ,V)确定的芬斯勒度量F是一个正则的芬斯勒度量;当Φ(x,-V x)=1时,证明了F是一个锥芬斯勒度量.进一步,研究了Kropina流形和Randers流形上的导航术问题.当F是流形M上的Kropina度量,且向量场V满足F(x,-V x)≤1时,证明了由导航数据(F,V)确定的导航术问题的解F必然是Randers度量或Kropina度量;当F为Randers度量,且向量场V满足F(x,-V x)=1时,证明了由导航数据(F,V)确定的导航术问题的解F必然是Kropina度量.The navigation problem on Riemannian manifolds plays a very important role in Finsler geometry.Both Randers metric and Kropina metric can be characterized by the solution of the navigation problem on a Riemannian manifold(M,h)with a vector field W satisfying‖W‖≤1.First,the important relationships between navigation problems and the geometry of indicatrix on Finsler manifolds are revealed in this paper.When the vector field V=V(x)on Finsler manifold(M,Φ)satisfiesΦ(x,-V x)<1,it is proved that the Finsler metric F determined by navigation data(Φ,V)is a regular Finsler metric;and that whenΦ(x,-V x)=1,F is a conic Finsler metric.Next,the navigation problems on Kropina manifolds and Randers manifolds are studied.When F is a Kropina metric on a manifold M and vector field V satisfies F(x,-V x)≤1,it is proved that the solution F of the navigation problem with the navigation data(F,V)is either a Randers metric or a Kropina metric.Further,when F is a Randers metric and the vector field V satisfies F(x,-V x)=1,it is proved that the solution F of the navigation problem with the navigation data(F,V)must be a Kropina metric.

关 键 词：Kropina度量 RANDERS度量 导航术问题 向量场 芬斯勒度量 

分 类 号：O186.13[理学—数学]