On Geodesics of Sprays and Projective Completeness

TL;DR

This paper classifies projectively flat sprays with weak Ricci constant in spray-Finsler geometry, introduces a geodesic-based method to identify sprays, and discusses conditions for projective completeness of sprays.

Contribution

It provides a classification of projectively flat sprays with weak Ricci constant and introduces a new geodesic method for identifying sprays based on parameterized curves.

Findings

Classification of projectively flat sprays with weak Ricci constant.

Introduction of a geodesic method to determine sprays.

Conditions under which a spray is projectively complete.

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.