L-regular linear connections

Abstract

The aim of this paper is to generalize the theory of nonlinear connections of Grifone ([3] and [4]). We adopt the point of view of Anona [1] and continue developing the approach established by the first author in [10]. The first part of the work is devoted to the problem of associating to each $L$-regular linear connection on $M$ a nonlinear $L$-connection on $M$. The route we have followed is significantly different from that of Grifone. We introduce an almost-complex and an almost-product structures on $M$ by means of a given $L$-regular linear connection on $M$. The product of these two structures defines a nonlinear $L$-connection on $M$, which generalizes Grifone's nonlinear connection. The seconed part is devoted to the converse problem: associating to each nonlinear $L$-connection \G on $M$ an $L$-regular linear connection on $M$; called the $L$-lift of \G. The existence of this lift is established and the fundamental tensors associated with it are studied. In the third part, we investigate the $L$-lift of a homogeneous $L$-connection \G, called the Berwald $L$-lift of \G. Then we particularize our study to the $L$-lift of a conservative $L$-connection. This $L$-lift enjoys some interesting properties. We finally deduce various identities concerning the curvature tensors of such a lift. Grifone's theory can be retrieved by letting $M$ be the tangent bundle of a differentiable manifold and $L$ be the natural almost-tangent structure $J$ on $M$.