Laplace operators on holomorphic Lie algebroids

TL;DR

This paper develops Laplace-type operators for functions and forms on Finsler Lie algebroids, utilizing the Chern-Finsler connection to extend classical Laplacian concepts to this geometric setting.

Contribution

It introduces new Laplace operators on Finsler Lie algebroids and provides their local expressions via the Chern-Finsler connection, advancing the geometric analysis in this area.

Findings

Defined Laplace operators on the tangent space of Finsler Lie algebroids

Constructed a horizontal Laplace operator for forms

Expressed operators locally using the Chern-Finsler connection

Abstract

The paper introduces Laplace-type operators for functions defined on the tangent space of a Finsler Lie algebroid, using a volume form on the prolongation of the algebroid. It also presents the construction of a horizontal Laplace operator for forms defined on the prolongation of the algebroid. All of the Laplace operators considered in the paper are also locally expressed using the Chern-Finsler connection of the algebroid.