Funk Functions and Projective Deformations of Sprays and Finsler Spaces of Scalar Flag Curvature

In 2001, Zhongmin Shen asked if it is possible for two projectively related Finsler metrics to have the same Riemann curvature (Shen in Differential Geometry of Spray and Finsler Spaces, p. 184, 2001 ). In this paper we provide an answer to this question within the class of Finsler metrics of scalar flag curvature. In Theorem 3.1 , we show that the answer is negative, for non-vanishing scalar flag curvature (SFC). The answer is known to be positive when the SFC vanishes (Grifone and Muzsnay in Variational Principles for Second-Order Differential Equations, 2000 ; Shen in Differential Geometry of Spray and Finsler Spaces, p. 184, 2001 ), and this positive answer is related to the existence of many solutions to Hilbert’s Fourth Problem. As a generalization of this problem, we can ask if it is possible for a given spray, with non-vanishing SFC, to represent, after reparameterization, the geodesic spray of a Finsler metric. In Proposition 3.3 , we show how to construct sprays whose projective class does not contain any Finsler metrizable spray with the same Riemann curvature.