K\"ahler Finsler manifolds with curvatures bounded from below

Abstract

We obtain a partial parallelism of the complex structure on K\"ahler Finsler manifolds. As applications, we prove Synge-Tsukamoto theorem and Bonnet-Myers theorem for positively curved K\"ahler Finsler manifolds. Moreover, we generalize a comparison theorem due to Ni-Zheng by introducing the notion of orthogonal Ricci curvature to K\"ahler Finsler geometry.