Projective and Finsler metrizability: parameterization-rigidity of the geodesics

Abstract

In this work we show that for the geodesic spray $S$ of a Finsler function $F$ the most natural projective deformation $\tilde{S}=S-2\lambda F\mathbb{C}$ leads to a non-Finsler metrizable spray, for almost every value of $\lambda \in \mathbb{R}$. 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.

References in corpus (2)

• Semi-basic 1-forms and Helmholtz conditions for the inverse problem of the calculus of variations
• A geometric setting for systems of ordinary differential equations

Cited by in corpus (10)

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• Generalized Helmholtz conditions for non-conservative Lagrangian systems
• A setting for higher order differential equations fields and higher order Lagrange and Finsler spaces
• A characterisation for Finsler metrics of constant curvature and a Finslerian version of Beltrami Theorem
• Funk functions and projective deformations of sprays and Finsler spaces of scalar flag curvature
• Invariant metrizability and projective metrizability on Lie groups and homogeneous spaces
• An Algebraic Characterisation for Finsler Metrics of Constant Flag Curvature
• Jacobi fields and conjugate points for a projective class of sprays