  On the slit tangent manifold $TM^0$ of a Finsler space $(M,F)$ there are
given some natural foliations as vertical foliation and some other fundamental
foliations produced by the vertical and horizontal Liouville vector fields, see
[A. Bejancu, H. R. Farran, Finsler Geometry and Natural Foliations on the
Tangent Bundle, Rep. Math. Physics 58, No. 1 (2006), 131-146]. In this paper we
consider a $(n,2n-1)$-codimensional subfoliation
$(\mathcal{F}_V,\mathcal{F}_{\Gamma})$ on $TM^0$ given by vertical foliation
$\mathcal{F}_V$ and the line foliation spanned by vertical Liouville vector
field $\Gamma$ and we give a triplet of basic connections adapted to this
subfoliation.
