ON THE RIEMANN CURVATURE OPERATORS IN RANDERS SPACES

The Riemann curvature in Riemann-Finsler geometry can be regarded as a collection of linear operators on the tangent spaces. The algebraic properties of these operators may be linked to the geometry and the topology of the underlying space. The principal curvatures of a Finsler space (M, F) at a point x are the eigenvalues of the Riemann curvature operator at x. They are real functions. on the slit tangent manifold TM0. A principal curvature kappa(x, y) is said to be isotropic (respectively, quadratic) if kappa(x, y)/F(x, y) is a function of x only (respectively, kappa(x, y) is quadratic with respect to y). On the other hand, the Randers metrics are the most popular and prominent metrics in pure and applied disciplines. Here, it is proved that if a Randers metric admits an isotropic principal curvature, then F is of isotropic S-curvature. The same result is also established for F to admit a quadratic principal curvature. These results extend Shen's verbal results about Randers metrics of scalar flag curvature K = K(x) as well as those Randers metrics with quadratic Riemann curvature operator. The Riemann curvature R-k(i) may be broken into two operators R-k(i) and J(k)(i). The isotropic and quadratic principal curvature are characterized in terms of the eigenvalues of R and J.