The navigation problems on a class of conic Finsler manifolds

Based on our deep understanding on the essential relationships between the Zermelo navigation problems and the geometries of indicatrix on Finsler manifolds, we study navigation problems on conic Kropina manifolds. For a conic Kropina metric F(x,y) and a vector field V with F(x,−Vx)≤1 on an n-dimens...

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Bibliographic Details
Published inDifferential geometry and its applications Vol. 74; p. 101709
Main Authors Cheng, Xinyue, Qu, Qiuhong, Xu, Suiyun
Format Journal Article
LanguageEnglish
Published Elsevier B.V 01.02.2021
Subjects
Flag curvature
Kropina metric
Randers metric
Ricci curvature
S-curvature
Zermelo navigation problem
Kropina metric
53B40
Flag curvature
Ricci curvature
S-curvature
53C60
Randers metric
Zermelo navigation problem
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Summary:Based on our deep understanding on the essential relationships between the Zermelo navigation problems and the geometries of indicatrix on Finsler manifolds, we study navigation problems on conic Kropina manifolds. For a conic Kropina metric F(x,y) and a vector field V with F(x,−Vx)≤1 on an n-dimensional manifold M, let F˜=F˜(x,y) be the solution of the navigation problem with navigation data (F,V). We prove that F˜ must be either a Randers metric or a Kropina metric. Then we establish the relationships between some curvature properties of F and the corresponding properties of F˜ when V is a conformal vector field on (M,F), which involve S-curvature, flag curvature and Ricci curvature.
ISSN:0926-2245
1872-6984
DOI:10.1016/j.difgeo.2020.101709