Non-existence of Funk functions for Finsler spaces of non-vanishing scalar flag curvature

In his book Differential Geometry of Spray and Finsler spaces, page 177, Zhongmin Shen asks whether or not there always exist non-trivial Funk functions on a spray space. In this note, we will prove that the answer is negative for the geodesic spray of a Finslerian function of non- vanishing scalar flag curvature. (C) 2016 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.