On Spherically Symmetric Sprays

TL;DR

This paper characterizes spherically symmetric sprays, providing a canonical form, classifying projectively flat cases with isotropic curvature, and exploring their curvature properties to advance understanding in spray and Finsler geometry.

Contribution

It establishes a canonical form for spherically symmetric sprays and classifies projectively flat cases with isotropic curvature, including explicit zero-curvature solutions.

Findings

Derived a canonical form for spherically symmetric sprays.

Classified projectively flat spherically symmetric sprays with isotropic curvature.

Obtained explicit forms for sprays with zero curvature.

Abstract

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.