Invariant volume forms and first integrals for geodesically equivalent Finsler metrics

Abstract

Two geodesically equivalent Finsler metrics determine a set of invariant volume forms on the projective sphere bundle. The proportionality factors, between these volume forms, are geodesically invariant functions and hence they are first integrals. These first integrals can be easily computed as the coefficients of the characteristic polynomial of a (1, 1)-type tensor field derived from the angular metric.