L-connections and Associated Tensors

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 dL-cohomology induced on M by L and generalized some of Grifone’s results. 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(Lz) = Ker(Lz); z ∈ 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. 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 .