Foliations on the tangent bundle of Finsler manifolds

Let M be a smooth manifold with Finsler metric F, and let $widetilde{TM}$ be the slit tangent bundle of M with a generalized Riemannian metric G, which is induced by F. In this paper, we prove that (i) (M, F) is a Landsberg manifold if and only if the vertical foliation F _V is totally geodesic in ( $widetilde{TM}$ ,G); (ii) letting a:= a(??) be a positive function of ?? = F ² and k, c be two positive numbers such that $c = sqrt {tfrac{2} {{k(1 + a)}}} $ , then (M,F) is of constant curvature k if and only if the restriction of G on the c-indicatrix bundle IM(c) is bundle-like for the horizontal Liouville foliation on IM(c), if and only if the horizontal Liouville vector field is a Killing vector field on (IM(c),G), if and only if the curvature-angular form ?? of (M,F) satisfies $Lambda = tfrac{{1 - a}} {2}R $ on IM(c).