On Spherically Symmetric Sprays

This paper studies spherically symmetric sprays, i.e., sprays that are invariant under orthogonal transformations. We first establish a canonical form for such sprays, showing that their geodesic coefficients can be expressed as $G^i = |y|\alpha(r,s) y^i + |y|^2\beta(r,s) x^i$, where $r = |x|^2$ and $s = \langle x,y\rangle/|y|$. For projectively flat spherically symmetric sprays -- which are directly related to Hilbert's fourth problem on characterizing metrics whose geodesics are straight lines -- we derive a complete classification of those with isotropic curvature, and in particular, we obtain the explicit form of those with zero curvature. Furthermore, we characterize sprays of weakly isotropic curvature in this class by a system of partial differential equations. These results may provide a unified framework for understanding symmetry and curvature in spray geometry and could offer new insights into the metrizability problem in Finsler geometry.