Let (M,F) be a Finsler manifold, that is, M is a smooth manifold endowed with a Finsler metric F. In this paper, we introduce on the slit tangent bundle TM˜ a Riemannian metric G˜ which is naturally induced by F, and a family of framed f-structures which are parameterized by a real parameter c≠0. We prove that (i) the parameterized framed f-structure reduces to an almost contact structure on IM; (ii) the almost contact structure on IM is a Sasakian structure iff (M,F) is of constant flag curvature K=c; (iii) if S=yiδi is the geodesic spray of F and R(⋅,⋅) the curvature operator of the Sasaki–Finsler metric which is induced by F, then R(⋅,⋅)S=0 iff (M,F) is a locally flat Riemannian manifold.
