Bulletin of the
Korean Mathematical Society BKMS

In \cite{dyw1}, we developed a theory of complex Finsler manifolds to investigate the global geometry of complex Finsler manifolds. There we proved a version of Bonnet-Myers' theorem for complex Finsler manifolds with a certain condition on the Finsler metric which is a generalization of the K\"ahler condition for the Hermitian metric. In this paper, we show that if the holomorphic sectional curvature of $M$ is $\ge c^2 > 0\,,$ then $M\,$ is simply connected. This is a generalization of the Synge's theorem in the Riemannian geometry and the Tsukamoto's theorem for K\"ahler manifolds. The main point of the proof lies in how we can circumvent the convex neighborhood theorem in the Riemannian geometry. A second variation formula of arc length for complex Finsler manifolds is also derived.

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

MSC numbers: Primary 53C60; Secondary 58B20