PROJECTIVE AND FINSLER METRIZABILITY: PARAMETERIZATION-RIGIDITY OF THE GEODESICS

In this work we show that for the geodesic spray S of a Finsler function F, the most natural projective deformation e (S) over tilde = S - 2 lambda FC leads to a non-Finsler metrizable spray, for almost every value of lambda is an element of R. This result shows how rigid is the metrizablility property with respect to certain reparameterizations of the geodesics. As a consequence, we obtain that the projective class of an arbitrary spray contains infinitely many sprays that are not Finsler metrizable.