  The theory of connections in Finsler geometry is not satisfactorily
established as in Riemannian geometry. Many trials have been carried out to
build up an adequate theory. One of the most important in this direction is
that of Grifone ([3] and [4]). His approach to the theory of nonlinear
connections was accomplished in [3], in which his new definition of a nonlinear
connection is easly handled from the algebraic point of view. Grifone's
approach is based essentially on the natural almost-tangent structure $J$ on
the tangent bundle $T(M)$ of a differentiable manifold $M$. This structure was
introduced and investigated by Klein and Voutier [5]. Anona in [1] generalized
the natural almost-tangent structure by considering a vector 1-form $L$ on a
manifold $M$ (not on $T(M)$) satisfying certain conditions. He investigated the
$d_L$-cohomology induced on $M$ by $L$ and generalized some of Grifone's
results. \par In this paper, we adopt the point of view of Anona [1] to
generalize Grifone's theory of nonlinear connections [3]: We consider a vector
1-form $L$ on $M$ of constant rank such that $[L,L]=0$ and that
$Im(L_z)=Ker(L_z)$; $z\in M$. We found that $L$ has properties similar to those
of $J$, which enables us to generalize systematically the most important
results of Grifone's theory. \par The theory of Grifone is retrieved, as a
special case of our work, by letting $M$ be the tangent bundle of a
differentiable manifold and $L$ the natural almost-tangent structure $J$.
