Three-webs $W(r,r,2)$

Local differential-geometric properties of three-webs $W(r,r,2)$ formed on a $2r$-dimensional manifold by foliations of codimension $r$, $r$ and $2$, respectively, are considered. In particular, three-webs defined by complex analytic functions of $r$ complex arguments belong to this class of webs. The structure equations of a three-web $W(r,r,2)$ in an adapted co-frame (in particular, in a natural co-frame) are deduced; the canonical connection $\Gamma$ on the manifold of a web $W(r,r,2)$ is introduced; formulae are obtained to calculate (in a natural co-basis) the components of the first structure tensor of a three-web $W(r,r,2)$ in terms of the derivatives of the function of this web. Three special classes of three-webs $W(r,r,2)$ are considered in detail: regular and group three-webs and also three-webs $W(r,r,2)$ generated by holomorphic functions.

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