Cartan–Laptev method in the theory of multidimensional three-webs

We show how the Cartan–Laptev method which generalizes Elie Cartan's method of external forms and moving frames is supplied to the study of closed $G$-structures defined by multidimensional three-webs formed on a $C^s$-smooth manifold of dimension $2r$, $r\ge1$, $s\ge3$, by a triple of foliations of codimension $r$. We say that a tensor $T$ belonging to a differential-geometric object of order $s$ of three-web $W$ is closed if it can be expressed in terms of components of objects of lower order $s$. We find all closed tensors of a three-web and the geometric sense of one of relations connecting three-web tensors. We also point out some sufficient conditions for the web to have a closed $G$-structure. It follows from our results that the $G$-structure associated with a hexagonal three-web $W$ is a closed $G$-structure of class 4. It is proved that basic tensors of a three-web $W$ belonging to a differential-geometric object of order $s$ of the web can be expressed in terms of $s$-jet of the canonical expansion of its coordinate loop, and conversely. This implies that the canonical expansion of every coordinate loop of a three-web $W$ with closed $G$-structure of class $s$ is completely defined by an $s$-jet of this expansion. We also consider webs with one-digit identities of $k$th order in their coordinate loops and find the conditions for these webs to have the closed $G$-structure.