K\"ahler Finsler manifolds with curvatures bounded from below

TL;DR

This paper explores curvature properties of Kähler Finsler manifolds, establishing fundamental theorems like Synge-Tsukamoto and Bonnet-Myers, and generalizes comparison theorems through orthogonal Ricci curvature.

Contribution

It introduces new curvature bounds and extends classical theorems to the setting of Kähler Finsler geometry, including a novel comparison theorem.

Findings

Proves Synge-Tsukamoto theorem for positively curved Kähler Finsler manifolds

Establishes Bonnet-Myers theorem in this setting

Generalizes Ni-Zheng comparison theorem using orthogonal Ricci curvature

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.