| Abstract: | This survey is a study of the geometry of a multidimensional three-web defined by its higher-order differential-geometric objects. Relations are obtained that connect the fundamental tensors of akth-order differential neighborhood. Using these relations we solve the problem of determining whether theg-structure defined by a multidimensional hexagonal three-web is closed. The latter turns out to be a closedg-structure of class 4. This result is then generalized. To be specific, analytic conditions are found for ag-structure of arbitrary order to be closed. These, as shown in the article, can be interpreted algebraically using a certain number of identities in one variable which are satisfied in the coordinate loops of the web. A classification of identities of this type is given.Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 19, pp. 101–154, 1987. |
