Abstract. A study of webs on differentiable manifolds is, roughly speaking, investigating of local invariants (under the group of diffeomorphisms) of a set of $d$ foliations in general position. A $d$-web of codimension $n$ on an $m$-dimensional manifold is usually introduced as a $d$-tuple of foliations such that their leaves are submanifolds of codimension $n$, and at any point, the corresponding tangent spaces are in general position. If $m$ is a $k$-multiple of $n$, we speak about a $(d,k,n)$-web. Here we will be concerned by the case $d=3$, $k=2$, $m=2n$. Tangent spaces to the foliations are usually described by systems of differential forms (satisfying suitable integrability conditions). In the papers of Chern as well as in works of Akivis - Goldberg school, the main and fruitfull tool is the use of Cartan methods. We will try to present here a dual approach (occuring also in [Ng 2]) making use of distributions, tensor fields, and projectors. In the first two parts, a definition of a three-web of codimension $n$ on a (real) $2n$-dimensional differentiable manifold is given in terms of distributions, and a (local) equivalence of webs is treated. Regarding a 3-web ${\cal W}$ on $M_{2n}$ as a triple of $n$-dimensional involutive distributions which are pairwise complementary, we associate with a web a set of (six) projectors $P_{\alpha }^ {\beta }$, and a triple $B_1$, $B_2$, $B_3$ of associated $(1,1)$- tensor fields. The original distributions can be regarded either as kernels (or images) of projectors, or as invariant subspaces under $B_{\gamma }$. No distribution is preferred (no one is chosen as ``horizontal'' or ``vertical'') which is an advantage of this new approach. In the set of associated projectors, as well as in the set of associated fields, all informations about the web are involved, and ${\cal W}$ can be fully described by a suitably chosen couple of them. Using this description, all linear connections are found with respect to which the distributions of the web are parallel. All objects under consideration are supposed to be smooth (of the class $C^{\infty }$).

AMS Subject Classification (1991): 53C05

Keyword(s): Distribution, projector, manifold, connection, web

Received December 21, 1992 and in revised form May 25, 1994. (Registered under 5613/2009.)