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Finsler metrics with relatively non-negative (non-positive, respectively),constant and isotropic stretch curvatures are investigated in this paper. Inparticular, it is proved that every non-Riemannian $(\alpha, \beta)$-metricwith a nonzero constant flag curvature and a non-zero relatively isotropicstretch curvature over a manifold of dimension $n\geq 3$ is of a characteristicscalar constant over the Finsler geodesics. It is also shown that every compactFinsler manifold with a relatively non-negative (non-positive, respectively)stretch curvature is a Landsberg metric. Finsler manifolds with $2$-dimensionalrelative stretch curvature are also investigated.

On Some Properties of Finsler Manifolds of Stretch Curvature