L-regular linear connections

Details L-regular linear connections L-regular linear connections

2006-08-13

arxiv.org/abs/math/0608314v2

Tensor, N. S., Vol. 60 (1998), 144-154

Mathematics

Differential Geometry

12 pages, LaTeX file, Minor change (concerning reference No. 10)

Scientific paper

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 $M$.

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