Studies On The Navigation Problems And The Related Curvature Properties On Finsler Manifolds

We investigate the navigation problems on Finsler manifolds,which involves the important relationships between navigation problems and the geometry of indicatrix on Finsler manifolds,the navigation problems on conic Kropina manifolds and the navigation problems on Randers manifolds.Firstly,we reveal the important relationships between navigation problems and the geometry of indicatrix on Finsler manifolds.Secondly,we study the navigation problems on conic Kropina manifold.For a conic Kropina metric F=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 date(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 conic Kropina manifolds(M,F),which involve S-curvature,flag curvature,Ricci curvature.Finally,we study the navigation problems on Randers manifolds.When F is a Randers metric and the vector field V satisfies F(x,-Vx)=1,we prove that the solution F of navigation problem with the navigation date(F,V) must be a Kropina metric.