Tangent Lie algebras to the holonomy group of a Finsler manifold

TL;DR

This paper investigates the structure of the holonomy group in Finsler geometry by defining and analyzing associated Lie algebras, including the curvature, infinitesimal, and holonomy algebras, to understand their tangent properties.

Contribution

It introduces the notion of the holonomy algebra of a Finsler manifold and proves its tangent property to the holonomy group, extending classical concepts in Finsler geometry.

Findings

Holonomy algebra is tangent to the holonomy group.

Defined curvature and infinitesimal holonomy algebras for Finsler manifolds.

Established the holonomy algebra via conjugates and parallel translations.

Abstract

Our goal in this paper is to make an attempt to find the largest Lie algebra of vector fields on the indicatrix such that all its elements are tangent to the holonomy group of a Finsler manifold. First, we introduce the notion of the curvature algebra, generated by curvature vector fields, then we define the infinitesimal holonomy algebra by the smallest Lie algebra of vector fields on an indicatrix, containing the curvature vector fields and their horizontal covariant derivatives with respect to the Berwald connection. At the end we introduce the notion of the holonomy algebra of a Finsler manifold by all conjugates of infinitesimal holonomy algebras by parallel translations with respect to the Berwald connection. We prove that this holonomy algebra is tangent to the holonomy group.