arXiv:2610.31896v1 Announce Type: new Abstract: In this paper we give an overview of several developments in spray geometry and provide some further interpretations of the geometric quantities that arise naturally in this subject. We discuss, in particular, the $S$-curvature, the $\chi$-curvature, and several notions of Ricci curvature for sprays. We rederive a number of identities relating these quantities and explain their applications to locally projectively flat sprays and to projectively related sprays. We also discuss applications to Finsler geometry, with emphasis on the $\chi$-curvature of a Finsler metric and on the inverse problem of determining whether a given spray is induced by a Finsler metric.
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