Geometry and algebra of multidimensional three-webs. Translated from the Russian by Vladislav V. Goldberg.

*(English)*Zbl 0771.53001
Mathematics and Its Applications. Soviet Series. 82. Dordrecht etc.: Kluwer Academic Publishers. xvii, 358 p (1992).

The book under review contains a detailed exposition of the theory of three-webs formed on a differentiable manifold \(X\) of dimension \(2r\) by three smooth foliations of dimension \(r\) which are in general position. The web theory is closely connected with the theory of binary quasigroups, and these two theories are jointly studied in the book under review. In addition to the theory of quasigroups and other nonassociative algebraic systems, the web theory is related to other branches of mathematics: the theory of Grassmann and almost Grassmann structures, algebraic geometry, the foundations of geometry, nomography etc. All these relations are traced in the book.

The book summarizes investigations in the theory of multidimensional three-webs which were originated in the papers of S. S. Chern and G. Bol in the middle of the 1930’s and after a rather long break were continued in the papers which started to appear at the end of 1960’s and continue to appear at present. An extensive and valuable contribution to these investigations has been made by the geometers of Moscow. The authors present not only the results which were already published but also some new constructions and conclusions.

The book consists of eight chapters. In Chapter 1 the authors consider some geometric structures associated by a three-web \(W\). In the tangent subspace \(T_ x\) tangent to the manifold \(X\) at a point \(x\), there are three \(r\)-dimensional subspaces \(T_ \alpha\), \(\alpha = 1,2,3\), which are tangent to the web leaves passing through the point \(x\). These subspaces are invariant under transformations of an \(r\)-parameter linear group GL\((r)\) which is a subgroup of the group GL\((2r)\) of all admissible transformations of the subspace \(T_ x\). This is the reason that a web \(W\) defines on \(X\) a \(G\)-structure with the structural group \(G = \text{GL}(r)\). The structure equations of a web are invariant with respect to this group. The structure tensors of this group, defined in the second and third order neighborhoods, are called the torsion and curvature tensors of a three-web \(W\). A few \(G\)-connections are defined on \(W\). One of them which is called the canonical Chern connection is distinguished. Actually this connection was introduced by Chern in his paper of 1936. The authors consider the simplest classes of three-webs, parallelizable and group three-webs, and analytically characterize these two classes of webs by means of the torsion and curvature tensors.

In Chapter 2 the algebraic structures associated with three-webs are under investigation. The authors review the basic notions of the algebraic theory of quasi-groups and loops and clarify their relation with the theory of abstract three-webs. Next the authors introduce the notion of a local differentiable loop and a local \(W\)-algebra for a three-web defined on a differentiable manifold \(X\). These \(W\)-algebras (which were later named the Akivis algebras by K. H. Hofmann and K. Strambach [see their paper in Mathematika 33, No. 1, 87-95 (1986; Zbl 0601.22002)]) have dimension \(r\), a binary operation and a ternary operation which satisfy an identity generalizing the Jacobi identity. The binary and ternary operations in this algebra are connected with the commutation and the association in a local loop. The tensors defining these operations are not essentially different from the torsion and curvature tensors introduced in Chapter 1.

In the tangent subspace \(T_ x(X)\), the subspaces \(T_ \alpha\) indicated above define an \(r\)-parameter family of isoclinic subspaces (the subspaces \(T_ \alpha\) are isoclinic themselves) and an \((r-1)\)- parameter family of two-dimensional transversal subspaces which intersect all isoclinic subspaces along straight lines. These two families of subspaces are two families of plane generators of the Segre cone \(C_ x(2,r)\) belonging to the subspace \(T_ x\). The fibration of the Segre cones on the submanifold \(X\) defines on \(X\) an almost Grassmann structure associated with a three-web \(W\). This structure is studied in Chapter 3. A three-web for which this structure is 2-semi-integrable or \(r\)-semi- integrable is called transversally geodesic or isoclinic respectively. If this structure is integrable, then a web is Grassmannizable, i.e. it can be mapped onto a three-web formed on the Grassmannian \(G(1,r+1)\) of straight lines of a projective space \(P^{r+1}\) by three foliations whose bases are three hypersurfaces \(X_ \alpha\), \(\alpha = 1,2,3\), of the space \(P^{r+1}\) and whose leaves are the bundles of straight lines with centers on \(X_ \alpha\). Analytic conditions for a three-web to be one of these types are found.

The Moufang and Bol webs are the subjects of Chapter 4. It is proved that all local algebras of a Moufang web and a Bol web are Mal’cev algebras and Bol’s algebras respectively. The relationship between the Bol webs and the theory of symmetric spaces is established, and some special Bol webs are studied. In Chapter 5 the authors introduce the notion of a closed \(G\)-structure: such a \(G\)-structure is a \(G\)-structure whose structural tensors of order exceeding some number \(p\) are algebraically expressed in terms of the tensors of order not exceeding \(p\). It is proved that a closed \(G\)-structure is defined by a completely integrable system of Pfaffian equations, and thus this structure depends on a finite number of parameters. It is also proved that all multidimensional three- webs, on which one of the classical closure conditions (Thomson, Reidemeister, Moufang, Bol and hexagonality) holds, define on the manifold \(X\) closed \(G\)-structures. The local algebra of each of these five types of webs given at a point of a submanifold \(X\), at a neighborhood of this point, defines a three-web on which some closure condition holds. Thus, the third S. Lie theorem on determination of a Lie group by its Lie algebra is generalized to a wide class of three-webs and local differentiable quasigroups associated with these webs.

In Chapter 6 webs admitting smooth families of automorphisms are studied. In particular, this leads to the investigation of \(G\)-webs admitting a transitive group of automorphisms. In Chapter 7 properties related to the 4th differential neighborhood of webs are investigated, and the corresponding tangent algebra is defined. The theory of internal automorphisms of coordinate loops of a three-web is con nected with this algebra.

Finally, Chapter 8 of the book is devoted to the study of \(d\)-webs of codimension \(r\) given on a differentiable manifold of dimension \(nr\) where \(d \geq n+1\). Such webs were investigated in detail in the book of the reviewer [Theory of multi-codimensional \((n+1)\)-webs (1988; Zbl 0668.53001)]. In the book under review only a brief outline of this theory is given. Special attention is given to the connections of the theory of \(d\)-webs with the theory of Grassmann and almost Grassmann structures, the problems of algebraizability and linearizability of these \(d\)-webs and the rank problems for these \(d\)-webs.

There is an appendix to the book written by E. V. Ferapontov. In this appendix the author considers some applications of the web theory to the investigation of systems of partial differential equations of hydrodynamic type and the application of the notion of rank to finding invariants of certain wave systems.

The material presented in the book may constitute the content of some selective courses for graduate students majoring in geometry. Each chapter of the book is concluded by numerous problems which supplement the text and by notes with references to source material. The book contains an extensive bibliography up to 1992, the list of symbols and the index.

The book summarizes investigations in the theory of multidimensional three-webs which were originated in the papers of S. S. Chern and G. Bol in the middle of the 1930’s and after a rather long break were continued in the papers which started to appear at the end of 1960’s and continue to appear at present. An extensive and valuable contribution to these investigations has been made by the geometers of Moscow. The authors present not only the results which were already published but also some new constructions and conclusions.

The book consists of eight chapters. In Chapter 1 the authors consider some geometric structures associated by a three-web \(W\). In the tangent subspace \(T_ x\) tangent to the manifold \(X\) at a point \(x\), there are three \(r\)-dimensional subspaces \(T_ \alpha\), \(\alpha = 1,2,3\), which are tangent to the web leaves passing through the point \(x\). These subspaces are invariant under transformations of an \(r\)-parameter linear group GL\((r)\) which is a subgroup of the group GL\((2r)\) of all admissible transformations of the subspace \(T_ x\). This is the reason that a web \(W\) defines on \(X\) a \(G\)-structure with the structural group \(G = \text{GL}(r)\). The structure equations of a web are invariant with respect to this group. The structure tensors of this group, defined in the second and third order neighborhoods, are called the torsion and curvature tensors of a three-web \(W\). A few \(G\)-connections are defined on \(W\). One of them which is called the canonical Chern connection is distinguished. Actually this connection was introduced by Chern in his paper of 1936. The authors consider the simplest classes of three-webs, parallelizable and group three-webs, and analytically characterize these two classes of webs by means of the torsion and curvature tensors.

In Chapter 2 the algebraic structures associated with three-webs are under investigation. The authors review the basic notions of the algebraic theory of quasi-groups and loops and clarify their relation with the theory of abstract three-webs. Next the authors introduce the notion of a local differentiable loop and a local \(W\)-algebra for a three-web defined on a differentiable manifold \(X\). These \(W\)-algebras (which were later named the Akivis algebras by K. H. Hofmann and K. Strambach [see their paper in Mathematika 33, No. 1, 87-95 (1986; Zbl 0601.22002)]) have dimension \(r\), a binary operation and a ternary operation which satisfy an identity generalizing the Jacobi identity. The binary and ternary operations in this algebra are connected with the commutation and the association in a local loop. The tensors defining these operations are not essentially different from the torsion and curvature tensors introduced in Chapter 1.

In the tangent subspace \(T_ x(X)\), the subspaces \(T_ \alpha\) indicated above define an \(r\)-parameter family of isoclinic subspaces (the subspaces \(T_ \alpha\) are isoclinic themselves) and an \((r-1)\)- parameter family of two-dimensional transversal subspaces which intersect all isoclinic subspaces along straight lines. These two families of subspaces are two families of plane generators of the Segre cone \(C_ x(2,r)\) belonging to the subspace \(T_ x\). The fibration of the Segre cones on the submanifold \(X\) defines on \(X\) an almost Grassmann structure associated with a three-web \(W\). This structure is studied in Chapter 3. A three-web for which this structure is 2-semi-integrable or \(r\)-semi- integrable is called transversally geodesic or isoclinic respectively. If this structure is integrable, then a web is Grassmannizable, i.e. it can be mapped onto a three-web formed on the Grassmannian \(G(1,r+1)\) of straight lines of a projective space \(P^{r+1}\) by three foliations whose bases are three hypersurfaces \(X_ \alpha\), \(\alpha = 1,2,3\), of the space \(P^{r+1}\) and whose leaves are the bundles of straight lines with centers on \(X_ \alpha\). Analytic conditions for a three-web to be one of these types are found.

The Moufang and Bol webs are the subjects of Chapter 4. It is proved that all local algebras of a Moufang web and a Bol web are Mal’cev algebras and Bol’s algebras respectively. The relationship between the Bol webs and the theory of symmetric spaces is established, and some special Bol webs are studied. In Chapter 5 the authors introduce the notion of a closed \(G\)-structure: such a \(G\)-structure is a \(G\)-structure whose structural tensors of order exceeding some number \(p\) are algebraically expressed in terms of the tensors of order not exceeding \(p\). It is proved that a closed \(G\)-structure is defined by a completely integrable system of Pfaffian equations, and thus this structure depends on a finite number of parameters. It is also proved that all multidimensional three- webs, on which one of the classical closure conditions (Thomson, Reidemeister, Moufang, Bol and hexagonality) holds, define on the manifold \(X\) closed \(G\)-structures. The local algebra of each of these five types of webs given at a point of a submanifold \(X\), at a neighborhood of this point, defines a three-web on which some closure condition holds. Thus, the third S. Lie theorem on determination of a Lie group by its Lie algebra is generalized to a wide class of three-webs and local differentiable quasigroups associated with these webs.

In Chapter 6 webs admitting smooth families of automorphisms are studied. In particular, this leads to the investigation of \(G\)-webs admitting a transitive group of automorphisms. In Chapter 7 properties related to the 4th differential neighborhood of webs are investigated, and the corresponding tangent algebra is defined. The theory of internal automorphisms of coordinate loops of a three-web is con nected with this algebra.

Finally, Chapter 8 of the book is devoted to the study of \(d\)-webs of codimension \(r\) given on a differentiable manifold of dimension \(nr\) where \(d \geq n+1\). Such webs were investigated in detail in the book of the reviewer [Theory of multi-codimensional \((n+1)\)-webs (1988; Zbl 0668.53001)]. In the book under review only a brief outline of this theory is given. Special attention is given to the connections of the theory of \(d\)-webs with the theory of Grassmann and almost Grassmann structures, the problems of algebraizability and linearizability of these \(d\)-webs and the rank problems for these \(d\)-webs.

There is an appendix to the book written by E. V. Ferapontov. In this appendix the author considers some applications of the web theory to the investigation of systems of partial differential equations of hydrodynamic type and the application of the notion of rank to finding invariants of certain wave systems.

The material presented in the book may constitute the content of some selective courses for graduate students majoring in geometry. Each chapter of the book is concluded by numerous problems which supplement the text and by notes with references to source material. The book contains an extensive bibliography up to 1992, the list of symbols and the index.

Reviewer: V.V.Goldberg (Newark / New Jersey)

##### MSC:

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

53A60 | Differential geometry of webs |

20N05 | Loops, quasigroups |

53C10 | \(G\)-structures |