A characterisation for Finsler metrics of constant curvature and a Finslerian version of Beltrami Theorem

TL;DR

This paper introduces a Weyl-type curvature tensor to characterize Finsler metrics with constant flag curvature and establishes a Finslerian analogue of Beltrami's theorem, highlighting the role of projective deformations.

Contribution

It defines a new Weyl-type curvature tensor for Finsler metrics and proves a Finslerian version of Beltrami's theorem relating projective deformations and constant curvature.

Findings

The Weyl-type curvature tensor reduces to the classical tensor in Riemannian cases.

Projective deformation preserves constant flag curvature iff the projective factor is a Hamel function.

The paper establishes a relation between Weyl-type curvature tensors of projectively related Finsler metrics.

Abstract

We define a Weyl-type curvature tensor that provides a characterisation for Finsler metrics of constant flag curvature. When the Finsler metric reduces to a Riemannian metric, the Weyl-type curvature tensor reduces to the classic projective Weyl tensor. In the general case, the Weyl-type curvature tensor differs from the Weyl projective curvature, it is not a projective invariant, and hence Beltrami Theorem does not work in Finsler geometry. We provide the relation between the Weyl-type curvature tensors of two projectively related Finsler metrics. Using this formula we show that a projective deformation preserves the property of having constant flag curvature if and only if the projective factor is a Hamel function. This way we provide a Finslerian version of Beltrami Theorem.