On projective invariants of the complex Finsler spaces

TL;DR

This paper develops projective curvature invariants for complex Finsler spaces, introduces the concept of complex Douglas spaces, and characterizes their properties and special cases such as Berwald and locally projectively flat metrics.

Contribution

It defines projective invariants for complex Finsler spaces and establishes conditions for these spaces to be Douglas or Berwald, including new characterizations and invariants.

Findings

Weakly Kähler Douglas spaces are complex Berwald spaces

A Weyl-type invariant characterizes complex Berwald spaces

Locally projectively flat complex Finsler metrics are classified

Abstract

In this paper the projective curvature invariants of a complex Finsler space are obtained. By means of these invariants the notion of complex Douglas space is then defined. A special approach is devoted to obtain the equivalence conditions that a complex Finsler space should be Douglas. It is shown that any weakly K\"{a}hler Douglas space is a complex Berwald space. A projective curvature invariant of Weyl type characterizes the complex Berwald spaces. They must be either purely Hermitian of constant holomorphic curvature or non purely Hermitian of vanish holomorphic curvature. The locally projectively flat complex Finsler metrics are also studied