Schur’s Lemma and curvature’s in Riemannian Finsler manifolds

We conclude by discussing the curvatures R and P of two important classes of Finsler spaces. This is the follow- up of the discussion we gave, at the end of their chern connections. Therefore, we now illustrate the usefulness of some of ourVianchi identities in the study of the flag curvature of Riemannian geometry. Further more, the flag curvature is insensitive to whether one is using the chern, cartan, Berwald, orHashiguchi- connection. Then, we calculate the flag curvature for a family of examples that are valid in all dimensions. These examples are interesting because their flag curvatures do not depend on the transverse edges V at all. Moreover, the flag curvatures in question have a simple nonconstant dependence on the basepoint x and the flagpole y. in this paper, we make precise the relation in the flag curvature. InRiemannian geometry, the full cuvature tensor is expressible as a sum of dectional curvatures, each multiplied by the area of the corresponding parallelogram or flag. How ever, for this mater, the Berwald connection is particulary suited for the study of Finsler spaces constant flag curvature.

Keywords:

Flag curvature , chern connections , Schur’s lemma , conventions , Cartan tensor