[Paper Review] A Riemannian structure associated with a Finsler structure

Ricardo Gallego|arXiv (Cornell University)|Jan 5, 2005

Advanced Differential Geometry Research1 references1 citations

TL;DR

This paper introduces a Riemannian structure derived from a Finsler space via an averaging procedure applied to the Finsler fundamental tensor and Chern-Rund connection, yielding a Riemannian metric and an affine, torsion-free connection. The key contribution is a generalized Gauss-Bonnet theorem for Berwald surfaces using the average metric, along with new invariants and holonomy results, establishing that symmetric space properties are inherited from the Finsler to the averaged Riemannian metric.

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.

Motivation & Objective

• To establish a systematic method for associating a Riemannian metric and affine connection to a Finsler space.
• To generalize the Gauss-Bonnet theorem to Berwald surfaces using the averaged Riemannian metric.
• To identify new local isometric invariants of the original Finsler metric through the averaging construction.
• To investigate the holonomy group of the averaged connection in Berwald spaces.
• To prove that the symmetric space property is inherited from the Finsler space to its associated average Riemannian metric.

Proposed method

• Applying an averaging procedure to the Finsler fundamental tensor to construct a Riemannian metric on the underlying manifold.
• Using the same averaging technique on the Chern-Rund connection to produce an affine, torsion-free connection on the manifold.
• Deriving the parallel transport and curvature endomorphisms of the averaged connection from the resulting affine connection.
• Applying the averaged Riemannian metric to generalize the Gauss-Bonnet theorem in the context of Berwald surfaces.
• Analyzing the holonomy group of the averaged connection to understand geometric symmetries in Berwald spaces.
• Proving that if the Finsler space is a symmetric space, then the associated average Riemannian metric also inherits the symmetric space property.

Experimental results

Research questions

• RQ1 How can a Riemannian metric be systematically derived from a Finsler structure using an averaging procedure?
• RQ2 What is the geometric significance of the averaged connection in Berwald spaces, particularly regarding curvature and holonomy?
• RQ3 Can the Gauss-Bonnet theorem be extended to Berwald surfaces using the average Riemannian metric?
• RQ4 What new local isometric invariants of the original Finsler metric emerge from the averaging construction?
• RQ5 Under what conditions is the symmetric space property preserved from the Finsler space to its associated Riemannian metric?

Key findings

• The averaging method successfully produces a Riemannian metric and an affine, torsion-free connection from a Finsler structure.
• A generalized Gauss-Bonnet theorem is established for Berwald surfaces using the average Riemannian metric.
• The parallel transport and curvature endomorphisms of the averaged connection are explicitly derived.
• The holonomy group of the averaged connection in a Berwald space is analyzed, revealing geometric constraints.
• The paper proves that the symmetric space property is inherited from the Finsler space to the averaged Riemannian metric.
• New affine, local isometric invariants of the original Finsler metric are identified through the averaging construction.