Geometry of the Fourth Order Differential Neighborhood of a Multidimensional Three-Web

Abstract

The success in the study of web geometry on the level of the third order differential neighborhood has been achieved due to the two circumstances. First of all, each of the classes of webs listed in Table 2.2 (p. 83) can be characterized by a special structure of the torsion and curvature tensors a and b. Secondly, these tensors have a clear geometric meaning: they determine the principal part of the “deviation” from commutativity and associativity in the coordinate loop l_p of a web W, respectively (see Section 2.5). As a result, each of these classes has three characterizations: the closure condition, the algebraic identity and the tensor equation. For example, for the group three-webs these three characterizations are: a) the figures R are closed on them; b) their coordinate loops are associative; and c) the curvature tensor is zero. The same kind of characterizations can be given for other classes of webs.