Bulletin of the
Korean Mathematical Society BKMS

In this paper, we investigate the topology of complex Finsler manifolds. For a complex Finsler manifold $(M,F),$ we introduce a certain condition on the Finsler metric $F$ on $M.$ This is a generalization of the K\"ahler condition for the Hermitian metric. Under this condition, we can produce a K\"ahler metric on $M.$ This enables us to use the usual techniques in the K\"ahler and Riemannian geometry. We show that if the holomorphic sectional curvature of $M$ is $\ge c^2 > 0$ for some $c > 0,$ then $diam (M) \le \tfrac{\pi}{c}$ and hence $M$ is compact. This is a generalization of the Bonnet's theorem in the Riemannian geometry.

Keywords: complex Finsler manifold, holomorphic sectional curvature, Bonnet's theorem

MSC numbers: Primary 53C60 ; Secondary 58B20