A NEW QUANTITY IN RIEMANN-FINSLER GEOMETRY

In this note, we study a new Finslerian quantity (C) over cap defined by the Riemannian curvature. We prove that the newFinslerian quantity is a non-Riemannian quantity for a Finsler manifold with dimension n = 3. Then we study Finsler metrics of scalar curvature. We find that the (C) over cap -curvature is closely related to the flag curvature and the H-curvature. We show that (C) over cap -curvature gives, a measure of the failure of a Finsler metric to be of weakly isotropic flag curvature. We also give a simple proof of the Najafi-Shen-Tayebi' theorem.