MIHAI ANASTASIEI 1 AND MANUELA GÎRŢU 2 1. Faculty of Mathematics, “Alexandru Ioan Cuza” University of Iaşi
700506, Iaşi, ROMANIA
and
Mathematics Institute "O.Mayer"
Romanian Academy Iaşi Branch
700506, Iaşi, ROMANIA
e-mail : anastas@uaic.ro
2. Department of Mathematics, Informatics and Education Sciences,
Faculty of Sciences, "Vasile Alecsandri" University of Bacău, 157 Calea Mărăşeşti, 600115 Bacău, ROMANIA,
e-mail: girtum@yahoo.com
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.
