A Gauss-Bonnet-Chern theorem for Finsler vector bundles

TL;DR

This paper proves a version of the Gauss-Bonnet-Chern theorem for Finsler vector bundles and derives a related formula for metric-compatible connections on Finsler manifolds, extending classical geometric results.

Contribution

It provides a simplified proof of the Gauss-Bonnet-Chern theorem in the Finsler setting and establishes a new formula for metric-compatible connections.

Findings

Proved a Gauss-Bonnet-Chern theorem for Finsler vector bundles.

Derived a Gauss-Bonnet-Chern formula for metric-compatible connections.

Simplified the proof of the classical theorem in the Finsler context.

Abstract

In this paper, we give a simple proof of the Gauss-Bonnet-Chern theorem for a real oriented Finsler vector bundle with rank equal to the dimension of the base manifold. As an application, a Gauss-Bonnet-Chern formula for any metric-compatible connection is established on Finsler manifolds.