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Average structures associated with a Finsler space

Ricardo Gallego Torromé

TL;DR

The paper introduces an averaging scheme that converts Finsler geometric data into affine and Riemannian representatives on the base manifold by averaging over indicatrices. It develops a general framework for averaging tensors and linear connections, and shows that in Berwald spaces the averaged connection coincides with the Levi-Civita connection of the averaged metric $h$, while the averaged curvature aligns with the average of the original curvature. This approach yields new affine, isometric invariants and enables Riemannian-style results, such as a Gauss–Bonnet type theorem for Berwald surfaces and a metrizable holonomy in Berwald spaces. It also clarifies the relationship between Finsler isometries and the isometries of the averaged metric, including symmetry properties and Randers examples within the averaged setting.

Abstract

Given a Finsler space (M,F) on a manifold M, the averaging method associates to Finslerian geometric objects affine geometric objects} living on $M$. In particular, a Riemannian metric is associated to the fundamental tensor $g$ and an affine, torsion free connection is associated to the Chern-Rund connection. As an illustration of the technique, a generalization of the Gauss-Bonnet theorem to Berwald surfaces using the average metric is presented. The parallel transport and curvature endomorphisms of the average connection are obtained. The holonomy group for a Berwald space is discussed. New affine, local isometric invariants of the original Finsler metric. The heredity of the property of symmetric space from the Finsler space to the average Riemannian metric is proved.

Average structures associated with a Finsler space

TL;DR

The paper introduces an averaging scheme that converts Finsler geometric data into affine and Riemannian representatives on the base manifold by averaging over indicatrices. It develops a general framework for averaging tensors and linear connections, and shows that in Berwald spaces the averaged connection coincides with the Levi-Civita connection of the averaged metric , while the averaged curvature aligns with the average of the original curvature. This approach yields new affine, isometric invariants and enables Riemannian-style results, such as a Gauss–Bonnet type theorem for Berwald surfaces and a metrizable holonomy in Berwald spaces. It also clarifies the relationship between Finsler isometries and the isometries of the averaged metric, including symmetry properties and Randers examples within the averaged setting.

Abstract

Given a Finsler space (M,F) on a manifold M, the averaging method associates to Finslerian geometric objects affine geometric objects} living on . In particular, a Riemannian metric is associated to the fundamental tensor and an affine, torsion free connection is associated to the Chern-Rund connection. As an illustration of the technique, a generalization of the Gauss-Bonnet theorem to Berwald surfaces using the average metric is presented. The parallel transport and curvature endomorphisms of the average connection are obtained. The holonomy group for a Berwald space is discussed. New affine, local isometric invariants of the original Finsler metric. The heredity of the property of symmetric space from the Finsler space to the average Riemannian metric is proved.

Paper Structure

This paper contains 17 sections, 21 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Basic notions of Finsler geometry
  3. Definition of a non-linear connection of $N$
  4. The Chern-Rund connection
  5. Riemannian average metrics from a Finsler space
  6. Definition of the averaging procedure for Finsler metrics
  7. A coordinate-free formula for $h$
  8. General form of the average operation and applications
  9. Average of a family of tensor automorphisms
  10. The average of linear connections on $\hat{\pi}^*TM$
  11. General properties of the averaged connection
  12. Average connection of a Berwald space
  13. The parallel transport of the average connection
  14. Curvature of the average connection
  15. Holonomy of a Berwald space
  16. ...and 2 more sections

Key Result

Theorem 2.6

Let $(M,F)$ be a Finsler space. The vector bundle $\hat{\pi}^{*}TM$ admits a unique linear connection characterized by the collection of connection 1-forms $\{ {\,^{ch}\omega}^i _j,\,\, i,j=1,...,n \}$ such that the following structure equations hold:

Theorems & Definitions (49)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Corollary 2.7
  • proof
  • Corollary 2.8
  • proof
  • ...and 39 more