On the Synge's theorem for complex Finsler manifolds

Abstract

In cite{dyw1}, we developed a theory of complex Finsler manifoldsto investigate the global geometry of complex Finsler manifolds.There we proved a version of Bonnet-Myers' theorem for complexFinsler manifolds with a certain condition on the Finsler metricwhich is a generalization of the K"ahler condition for theHermitian metric.In this paper, we show that if the holomorphic sectional curvatureof M is ge c^2 > 0,, then M, is simply connected. This isa generalization of the Synge's theorem in the Riemannian geometryand the Tsukamoto's theorem for K"ahler manifolds.The main point of the proof lies in how we can circumvent theconvex neighborhood theorem in the Riemannian geometry. A secondvariation formula of arc length for complex Finsler manifolds isalso derived.