# Finsler structures on holomorphic Lie algebroids

## TL;DR

This paper extends the concept of Finsler structures from holomorphic vector bundles to holomorphic Lie algebroids, introducing connections and studying their properties in this more general setting.

## Contribution

It introduces Finsler structures, partial and Chern-Finsler connections on holomorphic Lie algebroids, expanding the geometric framework beyond vector bundles.

## Key findings

- Defined complex Finsler structures on holomorphic Lie algebroids
- Studied nonlinear and linear connections and their relations
- Introduced the analogue of the Chern-Finsler connection for Lie algebroids

## Abstract

Complex Finsler vector bundles have been studied mainly by T. Aikou, who defined complex Finsler structures on holomorphic vector bundles. In this paper, we consider the more general case of a holomorphic Lie algebroid E and we introduce Finsler structures, partial and Chern-Finsler connections on it. First, we recall some basic notions on holomorphic Lie algebroids. Then, using an idea from E. Martinez, we introduce the concept of complexified prolongation of such an algebroid. Also, we study nonlinear and linear connections on the tangent bundle of E and on the prolongation of E and we investigate the relation between their coefficients. The analogue of the classical Chern-Finsler connection is defined and studied in the paper for the case of the holomorphic Lie algebroid.