On Geodesics of Sprays and Projective Completeness

Abstract: Geodesics, which play an important role in spray-Finsler geometry, are integral curves of a spray vector field on a manifold. Some comparison theorems and rigidity issues are established on the completeness of geodesics of a spray or a Finsler metric. In this paper, projectively flat sprays with weak Ricci constant (eps. constant curvature) are classified at the level of geodesics. Further, a geodesic method is introduced to determine an $n$-dimensional spray based on a family of curves with $2(n-1)$ free constant parameters as geodesics. Finally, it shows that a spray is projectively complete under certain condition satisfied by the domain of geodesic parameter of all geodesics.