THE MAIN INVARIANTS OF A COMPLEX FINSLER SPACE

In this paper we extend the results obtained in [3], where are investigated the general settings of the two-dimensional complex Finsler manifolds, with respect to a local complex Berwahl frame. The geometry of such manifolds is controlled by three real invari- ants which live on T＇M： two horizontal curvature invariants K and W and one vertical curvature invariant I. By means of these invariants are defined both the horizontal and the vertical holomorphic sectional curvatures. The complex Landsberg and Berwald spaces are of particular into, rest. Complex Berwald spaces coincide with K/ihler spaces, in the two - dimensional case, We establish the necessary and sufficient condition under which K is a constant and we obtain a characterization for the Kghler purely Hermitian spaces by the fact K = W=constant and I = 0. For the class of complex Berwald spaces we have K =W = 0. Finally, a classitication of two-dimensional complex Finsler spaces for which the horizontal curvature satisfies a special property is obtained.