A vertical Liouville subfoliation on the cotangent bundle of a Cartan space and some related structures

In this paper we study some problems related to a vertical Liouville distribution (called vertical Liouville-Hamilton distribution) on the cotangent bundle of a Cartan space. We study the existence of some linear connections of Vr\u{a}nceanu type on Cartan spaces related to some foliated structures. Also, we identify a certain $(n,2n-1)$–codimensional subfoliation $({\mathcal{F}}_V,{\mathcal{F}}_{C^{\ast }})$ on $T^{\ast }{M}_0$ given by vertical foliation ${\mathcal{F}}_V$ and the line foliation ${\mathcal{F}}_{C^{\ast }}$ spanned by the vertical Liouville-Hamilton vector field $C^{\ast }$ and we give a triplet of basic connections adapted to this subfoliation. Finally, using the vertical Liouville foliation ${\mathcal{F}}_{V_{C^{\ast }}}$ and the natural almost complex structure on $T^{\ast }{M}_0$ we study some aspects concerning the cohomology of $c$–indicatrix cotangent bundle.