Invariant metrizability and projective metrizability on Lie groups and homogeneous spaces

Ioan Bucataru, Tamás Milkovszki, Zoltán Muzsnay

Published 2015-09-15Version 1

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