Invariant metrizability and projective metrizability on Lie groups and homogeneous spaces

TL;DR

This paper investigates invariant and projective metrizability of geodesic sprays on Lie groups, establishing that projective Finsler metrizability coincides with Riemann metrizability for canonical sprays, indicating structural rigidity.

Contribution

It proves that for Lie groups, the canonical spray is projectively Finsler metrizable if and only if it is Riemann metrizable, extending results to geodesic orbit spaces.

Findings

Projective Finsler metrizability equals Riemann metrizability for canonical sprays.

Canonical sprays on Lie groups are rigid in their metrizability properties.

Results extend to geodesic orbit spaces.

Abstract

In this paper we study the invariant metrizability and projective metrizability problems for the special case of the geodesic spray associated to the canonical connection of a Lie group. We prove that such canonical spray is projectively Finsler metrizable if and only if it is Riemann metrizable. This result means that this structure is rigid in the sense that considering left-invariant metrics, the potentially much larger class of projective Finsler metrizable canonical sprays, corresponding to Lie groups, coincides with the class of Riemann metrizable canonical sprays. Generalisation of these results for geodesic orbit spaces are given.