关于Kropina度量的数量曲率

On the Scalar Curvature of Kropina Metrics

为了研究和刻画Kropina度量的数量曲率,利用Kropina度量的导航术技巧及数量曲率的定义展开讨论。令F=α^(2)/β是一个由导航数据(h, V)确定的Kropina度量。在F关于Busemann-Hausdorff体积形式的S-曲率是迷向的条件下,讨论了F的数量曲率r(x)和h的数量曲率■(x)的关系。特别地,当F是一个Einstein-Kropina度量时,证明了r(x)=■(x)。

The main purpose is to study and characterize the scalar curvature of Kropina metrics.The discussions are carried out by using navigation technique of Kropina metrics and the definition of scalar curvature.Let F=α^(2)/β be a Kropina metric expressed by navigation data(h,V).If F is of isotropic S-curvature with respect to the Busemann-Hausdorff volume form,the relationship between the scalar curvature r(x)of F and the scalar curvature■(r)of h is discussed.The relationship between the scalar curvature r(x)of F and the scalar curvature■(r)of h is obtained.In particular,when F is an Einstein-Kropina metric,it is proved that r(x)=■(x).