identification of riemannian foliations on the tangent bundle via sode structure

the geometry of a system of second order differential equations is the geometry of a semispray, which is a globally defined vector field on tm. the metrizability of a given semispray is of special importance. in this paper, the metric associated with the semispray s is applied in order to study some types of foliations on the tangent bundle which are compatible with sode structure. indeed, sufficient conditions for the metric associated with the semispray s are obtained to extend to a bundle-like metric for the lifted foliation on tm. thus, the lifted foliation converts to a riemanian foliation on the tangent space which is adapted to the sode structure. particularly, the metrizability property of the semispray s is applied in order to induce sode structure on transversals. finally, some equivalent conditions are presented for the transversals to be totally geodesic.