Classification of multidimensional three-webs according to closure conditions.

*(Russian)*Zbl 0705.53014The author considers three approaches to classification of smooth multidimensional three-webs according to closure conditions, i.e., according to identities which are satisfied in the coordinate loops of these three-webs. First, he considers three-webs for which an identity of Bol type (i.e. a weighted identity of the length four with three variables) is satisfied in their coordinate loops. Such identities were described by F. Fenyves [Publ. Math. 16, No.1-4, 187-192 (1969; Zbl 0221.20097)]. It is proved that three-webs of this kind are always Bol or Moufang three-webs. The second approach is connected with the identities of order k, \(k=3,4,...\), in one variable which generalize the identity of monoassociativity (for the latter \(k=2)\). The author refers to his early paper [J. Sov. Math. 44, No.2, 153-190 (1989); translation from Itogi Nauki Tekh., Ser. Probl. Geom. 19, 101-154 (1987)], where he gave a classification of such identities.

The third approach is most interesting. In the coordinate loops of a three-web the author considers a family of diffeomorphisms of some special kind and evaluates their derivation from an automorphism. In particular, the internal diffeomorphisms of a coordinate loop Q() are considered: \[ \ell_{x,y}=L^{-1}_{x\cdot y}\circ L_ x\circ L_ y,\quad r_{x,y}=R^{-1}_{x\cdot y}\circ R_ y\circ R_ x,\quad m_{x,y}=L_ x^{-1}\circ_ y^{-1}\circ L_ x\circ R_ y, \] (L\({}_ x\) and \(R_ x\) are translations on Q). Their deviation from an automorphism in Q() is determined by the functions \({\mathcal L}_{x,y}(u,v)\), \({\mathcal R}_{x,y}(u,v)\), \({\mathcal M}_{x,y}(u,v)\), where, for example, \[ (1)\quad {\mathcal L}_{x,y}(u,v)=^{- 1}\ell_{x,y}(u,v)\cdot (\ell_{x,y}(u)\cdot (\ell_{x,y}(u)\cdot (\ell_{x,y}(v)),\quad x,y,u,v\in Q. \] It is proved that the principal parts of the functions \({\mathcal L}\), \({\mathcal R}\) and \({\mathcal M}\) are tensors of type (1,4) and they are expressed in terms of the covariant derivatives (in Chern’s canonical connection associated with any three- web) of the torsion tensor of the three-web. The equation (1) and equations for \({\mathcal R}\) and \({\mathcal M}\) which are similar to (1), allow to describe adequately in pure algebraic terms the geometry of the differential neighborhood of order 4 of three-webs and to extend their well-known classification. New special classes of three-webs are distinguished by the identities of the form \({\mathcal L}_{x,y}(u,v)=e\) (e is the unit of Q), where some variables can be identified. This approach allows the author to solve some known open problems of the theory of loops.

Using rather complicated calculations, the author extends the results outlined above, to differential neighborhoods of arbitrary order. He finds wide special classes of three-webs and proves that the G-structures associated with them are closed G-structures. In the last section the author discusses the relation between the existence of non-trivial autotopies on a web and the closure of its G-structure. This leads to the notion of a parametric closed G-structure which generalizes the notion of a closed G-structure introduced by M. A. Akivis [Itogi Nauki Tekh., Ser. Probl. Geom. 7, 69-79 (1975; Zbl 0549.53032)].

The third approach is most interesting. In the coordinate loops of a three-web the author considers a family of diffeomorphisms of some special kind and evaluates their derivation from an automorphism. In particular, the internal diffeomorphisms of a coordinate loop Q() are considered: \[ \ell_{x,y}=L^{-1}_{x\cdot y}\circ L_ x\circ L_ y,\quad r_{x,y}=R^{-1}_{x\cdot y}\circ R_ y\circ R_ x,\quad m_{x,y}=L_ x^{-1}\circ_ y^{-1}\circ L_ x\circ R_ y, \] (L\({}_ x\) and \(R_ x\) are translations on Q). Their deviation from an automorphism in Q() is determined by the functions \({\mathcal L}_{x,y}(u,v)\), \({\mathcal R}_{x,y}(u,v)\), \({\mathcal M}_{x,y}(u,v)\), where, for example, \[ (1)\quad {\mathcal L}_{x,y}(u,v)=^{- 1}\ell_{x,y}(u,v)\cdot (\ell_{x,y}(u)\cdot (\ell_{x,y}(u)\cdot (\ell_{x,y}(v)),\quad x,y,u,v\in Q. \] It is proved that the principal parts of the functions \({\mathcal L}\), \({\mathcal R}\) and \({\mathcal M}\) are tensors of type (1,4) and they are expressed in terms of the covariant derivatives (in Chern’s canonical connection associated with any three- web) of the torsion tensor of the three-web. The equation (1) and equations for \({\mathcal R}\) and \({\mathcal M}\) which are similar to (1), allow to describe adequately in pure algebraic terms the geometry of the differential neighborhood of order 4 of three-webs and to extend their well-known classification. New special classes of three-webs are distinguished by the identities of the form \({\mathcal L}_{x,y}(u,v)=e\) (e is the unit of Q), where some variables can be identified. This approach allows the author to solve some known open problems of the theory of loops.

Using rather complicated calculations, the author extends the results outlined above, to differential neighborhoods of arbitrary order. He finds wide special classes of three-webs and proves that the G-structures associated with them are closed G-structures. In the last section the author discusses the relation between the existence of non-trivial autotopies on a web and the closure of its G-structure. This leads to the notion of a parametric closed G-structure which generalizes the notion of a closed G-structure introduced by M. A. Akivis [Itogi Nauki Tekh., Ser. Probl. Geom. 7, 69-79 (1975; Zbl 0549.53032)].

Reviewer: V.V.Goldberg