A Characterisation for Finsler Metrics of Constant Curvature and a Finslerian Version of Beltrami Theorem

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 classical 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.

New classes of projectively related Finsler metrics of constant flag curvature

“…We study what happens with the projective Weyl-curvature tensor W 1 if we make a projective deformation of the initial spray. Similar to the tensor introduced in [3] we obtain that W 1 is invariant only to projective factors that are Hamel functions. The main aim of this paper is to construct some families of projectively related Finsler metrics that preserve the Weyl-type curvature tensor W 1 .…”

“…For the direct implication we will assume that F is projectively flat and hence δ S0 F = 0. According to Proposition 2.2 from [3] this is equivalent with…”

An Algebraic Characterisation for Finsler Metrics of Constant Flag Curvature

“…In Finsler geometry there are various characterisations for metrics of constant flag curvature using some Weyl-type curvature tensors [1,2,11]. In this paper we will prove the following algebraic characterisation for Finsler metrics of constant flag curvature.…”

“…Such algebraic identity appears as an obstruction to the formal integrability of some operators in Finsler geometry, [4,7]. This algebraic characterisation for Finsler metrics of constant flag curvature allows to provide yet another proof for the Finslerian version of Beltrami's Theorem, [2,3].…”

Invariant volume forms and first integrals for geodesically equivalent Finsler metrics

“…Finsler geometry has been characterised by Chern as being "just Riemannian geometry without the quadratic restriction" [4]. However, many important results from Riemannian geometry cannot be easily extended to the Finslerian framework without using techniques and tools that are specific to Finsler geometry, [2].…”