Sprays Metrizability by a Finsler Metric of Scalar Curvature

This paper establishes necessary and sufficient conditions for the Finsler metrization of sprays with non-vanishing scalar curvature, by studying symplectic forms, the Berwald connection, and symmetry-based criteria. We derive global and local metrizability conditions for sprays with non-vanishing scalar flag curvature, extending to isotropic and constant curvature cases. Examples on manifolds including S^2 , ℝ^2 × S^1 , ℝ× T^2 , and warped product spaces illustrate these conditions, highlighting the interplay of curvature with non-trivial topologies. These results refine and generalize existing frameworks, advancing applications in Finsler geometry, differential geometry, and mathematical physics.