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Abstract: In this paper, we study Ricci-flat -metrics which are defined by a Riemann metric α and a 1-form β on a manifold M. We prove that an -metric of Randers type is Ricci-flat Douglas metric if and only if it is a Berwald metric and α is Ricci-flat. Further, we characterize completely Ricci-flat Douglas -metrics of non-Randers type on M when the dimension . [Copyright &y& Elsevier]

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 , − V x) ≤ 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. [ABSTRACT FROM AUTHOR]