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 $(\mathcal{F}_V,\mathcal{F}_{C^*})$ on $T^*M_0$ given by vertical foliation $\mathcal{F}_V$ and the line foliation $\mathcal{F}_{C^*}$ spanned by the vertical Liouville-Hamilton vector field $C^*$ and we give a triplet of basic connections adapted to this subfoliation. Finally, using the vertical Liouville foliation $\mathcal{F}_{V_{C^*}}$ and the natural almost complex structure on $T^*M_0$ we study some aspects concerning the cohomology of $c$--indicatrix cotangent bundle.