On the Riemann curvature operators in Randers spaces (Q2855845)

For a given Finsler space \((M,F)\), the principal curvatures at a point \(x\) are the eigenvalues of the Riemann curvature operator at \(x\). They are real functions \(k\) on the slit tangent manifold \(TM_0=TM-\{\text{zero\,\,section}\}\). A principal curvature \(k(x,y)\) is said to be isotropic (respectively, quadratic) if \(k(x,y)/F(x,y)\) is a function which depends only on \(x\) (respectively, \(k(x,y)\) is quadratic with respect to \(y\)). On the other hand, one of the most important non-Riemannian quantities in Finsler geometry is the \(S\)-curvature.NEWLINENEWLINEIn the present paper, the author proves that if a Randers metric admits an isotropic principal curvature, then \(F\) is of isotropic \(S\)-curvature. The same result is also proved for the case when \(F\) admits a quadratic principal curvature.