Identifying Berwald Finsler Geometries

TL;DR

This paper derives a simple first order PDE condition to identify Berwald Finsler geometries, providing new examples and generalizing previous results in Finsler and relativistic geometries.

Contribution

It establishes a necessary and sufficient PDE condition for Finsler Lagrangians to be Berwald, especially for $(eta)$-spaces and spacetimes, with novel geometric examples.

Findings

Derived a PDE condition for Berwald Finsler geometries.

Provided new examples of $(eta)$-Berwald geometries.

Generalized earlier results in Finsler and relativistic geometries.

Abstract

Berwald geometries are Finsler geometries close to (pseudo)-Riemannian geometries. We establish a simple first order partial differential equation as necessary and sufficient condition, which a given Finsler Lagrangian has to satisfy to be of Berwald type. Applied to $(\alpha,\beta)$-Finsler spaces, respectively $(A,B)$-Finsler spacetimes, this reduces to a necessary and sufficient condition for the Levi-Civita covariant derivative of the defining $1$-form. We illustrate our results with novel examples of $(\alpha,\beta)$-Berwald geometries which represent Finslerian versions of Kundt (constant scalar invariant) spacetimes. The results generalize earlier findings by Tavakol and van den Bergh, as well as the Berwald conditions for Randers and m-Kropina resp. very special/general relativity geometries.