A simple proof of the Gauss-Bonnet-Chern formula for Finsler manifolds
Wei Zhao
TL;DR
This paper presents a straightforward proof of the Gauss-Bonnet-Chern formula for Finsler manifolds using the Cartan connection, extending it to metric-compatible connections and deriving related formulas.
Contribution
It provides a simplified proof of the Gauss-Bonnet-Chern formula for Finsler manifolds and generalizes it to metric-compatible connections.
Findings
Proof of Gauss-Bonnet-Chern formula for Finsler manifolds
Extension to metric-compatible connections
Derivation of Lackey's Gauss-Bonnet-Chern formula
Abstract
From the point of view of index theory, we give a simple proof of a Gauss-Bonnet-Chern formula for all Finsler manifolds by the Cartan connection. Based on this, we establish a Gauss-Bonnet-Chern formula for any metric-compatible connection and also derive the Gauss-Bonnet-Chern formula of Lackey.
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Taxonomy
TopicsAdvanced Differential Geometry Research