Abstract
The cotangent bundle of a smooth manifold, as a particular submersion, carries a natural foliation called vertical defined by the kernel of the differential of the projection of the cotangent bundle on its base manifold. The vertical foliation is a La-grangian one with respect to the natural symplectic structure of the cotangent bundle. It has new properties if the cotangent bundle has additional geometrical structures, for instance those induced by a non-degenerate homogeneous Hamiltonian. A Cartan space is a manifold whose cotangent bundle is endowed with a smooth non-degenerate Hamiltonian K^2 which is positively homogeneous of degree 2 in momenta. Then the vertical foliation becomes a semi Riemannian foliation whose transversal distri-bution is completely determined by K and is orthogonal on the vertical distribution with respect to a semi Riemannian metric of Sasaki type. In this framework various linear connections will be associated to and some properties of the vertical foliation will be pointed out.