Holonomy of a class of bundles with fibre metrics

TL;DR

This paper investigates the holonomy of certain geometric spaces, including Landsberg spaces in Finsler geometry, using Lie groupoid methods and extends classical theorems to this context.

Contribution

It introduces a Lie groupoid approach to holonomy in Landsberg and related spaces, extending the Ambrose-Singer theorem to these settings.

Findings

Proves a version of the Ambrose-Singer theorem for these spaces

Demonstrates the applicability of Lie groupoid methods in Finsler geometry

Discusses potential extensions to more general Finsler and nonlinear connection spaces

Abstract

This paper is concerned with the holonomy of a class of spaces which includes Landsberg spaces of Finsler geometry. The methods used are those of Lie groupoids and algebroids as developed by Mackenzie. We prove a version of the Ambrose-Singer Theorem for such spaces. The paper ends with a discussion of how the results may be extended to Finsler spaces and homogeneous nonlinear connections in general.