In this paper we study the horizontal and vertical distributions of a class of g-natural metrics on the tangent bundle of Finsler manifolds. Then we characterize the Riemannian manifolds among Finsler manifolds from the viewpoint of the geometry of slit tangent bundle and obtain some results on the Riemannian curvature of these metrics. Finally we prove that if the slit tangent bundle is locally symmetric, then the base manifold is locally Euclidean.
