Theory of multicodimensional \((n+1)\)-webs.

*(English)*Zbl 0668.53001The book contains a detailed presentation of the theory of d-webs W(d,n,r) of codimension r on a differentiable manifold X of dimension nr. These webs are given by a set of d differentiable foliations of codimension r which are in general position on X, \(d\geq n+1\). In this case the G-structure on X defined by the web W(d,n,r) is locally non- trivial. The local differential geometry of this structure and its numerous connections with the other differential-geometric and algebraic structures are studied in the book.

The foundations of web theory were given by W. Blaschke and his co- workers [see W. Blaschke and G. Bol, Geometrie der Gewebe (Berlin 1938; Zbl 0020.06701), Geometrie der Waben (Basel und Stuttgart 1955; Zbl 0068.365)]. These works were devoted to webs W(d,n,1), mostly for \(n=2,3\). Then webs W(3,2,r) were studied by S. S. Chern in 1936 and the reviewer with co-workers since 1969.

The author of this monograph mostly studies the webs \(W(n+1,n,r)\), W(4,2,r) and the problem of a rank for webs W(d,2,r). They are the main topic of the monograph. Related results by other authors are given, too. The book consists of eight chapters. In Chapter 1 the structure equations of a web \(W(n+1,n,r)\) are derived and its fundamental tensors, namely the torsion and curvature tensors connected with the neighbourhood of the second and the third orders respectively, are introduced. The affine connection naturally associated with a web \(W(n+1,n,r)\) and connections induced by it on the leaves of a web and its three-subwebs are considered. Special classes of webs defined by specific properties of their torsion and curvature tensors, in particular isoclinic and transversally geodesic webs, are studied.

In Chapter 2 the author proves that a web \(W(n+1,n,r)\) defines an almost Grassmann structure \(AG(n-1,r+n-1)\) which becomes a Grassmann one if and only if the web is isoclinic and transversally geodesic simultaneously. In this case the web \(W(n+1,n,r)\) is Grassmannisable, i.e. permits a mapping on a Grassmann manifold \(G(n-1,r+n-1)\) of (n-1)-dimensional subspaces of a projective space \(P^{r+n-1}\). The conditions of Grassmannisability for these webs and for webs W(d,n,r) where \(d>n+1\), \(n\geq 2\), \(r\geq 2\) are found. Then the conditions of algebraisability for webs W(d,n,r) for \(d=n+1\) and \(d>n+1\) are derived. A realisation of these latter conditions leads to a web realized in \(P^{r+n-1}\) by an algebraic manifold of dimension r and degree d. These results are connected with work by the reviewer [Sib. Math. J. 23, No.6, 763-770 (1982); translation from Sib. Mat. Zh. 23, No.6, 6-15 (1982; Zbl 0505.53004); Sov. Math., Dokl. 28, 507-509 (1983); translation from Dokl. Akad. Nauk SSSR 272, 1289-1291 (1983; Zbl 0547.53006)] and J. A. Wood [Duke Math. J. 51, 235-237 (1984; Zbl 0584.14021)].

In Chapter 3 a link between webs \(W(n+1,n,r)\) and local differentiable n- quasigroups is studied. Torsion and curvature tensors of the web are expressed in terms of partial derivatives of second and third orders of functions defining an operation in its coordinate n-quasigroup, and canonical expansions of these functions are found. Then a condition for the existence of a one-parameter n-quasigroup for a web \(W(n+1,n,r)\) is established. Following J. D. H. Smith [Arch. Math. 51, No.2, 169- 177 (1988; Zbl 0627.22003)] a local algebra for an analytic r-dimensional ternary loop is constructed and a theorem analogous to Lie’s third fundamental theorem is proved. This construction is generalized to the case of an analytic n-loop.

In Chapter 4 the author considers some more special classes of webs \(W(n+1,n,r):\) reducible and completely reducible ones, group webs, \((2n+2)\)-hedral and \((n+1)\)-Bol webs. The latter class of webs was studied earlier by S. A. Gerasimenko [Problems of the theory of webs and quasi-groups, Collect. sci. Works, Kalinin 1985, 148-152 (1985; Zbl 0572.53020); Multidimensional Bol \((n+1)\)-webs. (Russian), Preprint, 76 pp., Mosc. Gos. Pedag. Inst., Moscow (1985)]. In Chapter 5 realisations of Grassmann webs \(W(n+1,n,r)\) in projective space \(P^{r+n+1}\) and some of their subclasses are studied. Furthermore in this chapter a web W(4,3,r) formed by four r-pencils of 2r-planes with (2r-1)-dimensional vertices in \(P^{3r}\) is considered.

Chapter 6 is devoted to applications of web theory to a geometry of point correspondences of \(n+1\) projective lines and \(n+1\) projective spaces as well as to a theory of holomorphic mappings between polyhedral domains of complex space \({\mathbb{C}}^ n\). Here V. S. Bolodurin’s [Sov. Math. 28, No.12, 20-26 (1984); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1984, No.12, 17-23 (1984; Zbl 0564.53005); Sov. Math. 29, No.5, 13- 21 (1985); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1985, No.5(276), 15-22 (1985; Zbl 0592.53008)] and J. Baumann’s [Schriftenr. Math. Inst. Univ. Münster, 2. Ser. 24, 152 pp. (1982; Zbl 0545.32001)] results are presented. Chapter 7 deals with four-webs W(4,2,r) on a manifold X of a dimension 2r. Fundamental equations of these webs are derived and their basis affinor is found. Special classes of webs are considered. They are parallelisable, group, almost Grassmannisable and Grassmannisable webs. A link between four-webs and pairs of orthogonal differentiable quasigroups is established. The Desargues and triangle closure conditions on a four-web are considered and their algebraic interpretation in terms of a pair of orthogonal quasigroups is given.

The final Chapter of the monograph is devoted to a rank problem for webs. This chapter is closely connected with S. S. Chern’s and P. A. Griffiths’ investigations [Proc. int. Symp. on algebraic geometry, Kyoto 1977, 85-91 (1977; Zbl 0406.14003); Jahresber. Dtsch. Math.-Ver. 80, 13- 110 (1978; Zbl 0386.14002); Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 5, 539-557 (1978; Zbl 0402.57001)]. In the book a rank problem is considered for webs W(d,2,r), \(d\geq 3\). For these webs the concepts of almost Grassmannisability and almost algebraisability are introduced. The upper bounds for the 1-rank and r-rank of almost Grassmannisable webs W(d,2,r) are established. It is proved that an almost Grassmannisable web W(d,2,r) of maximum 1-rank is always parallelisable and a web W(d,2,2) of maximum 2-rank is always algebraisable for \(d>4\). For \(d=4\) the existence of non-algebraisable webs of maximum rank is proved.

The monograph contains an extended bibliography reporting on all related work up to 1988 and a detailed index.

The foundations of web theory were given by W. Blaschke and his co- workers [see W. Blaschke and G. Bol, Geometrie der Gewebe (Berlin 1938; Zbl 0020.06701), Geometrie der Waben (Basel und Stuttgart 1955; Zbl 0068.365)]. These works were devoted to webs W(d,n,1), mostly for \(n=2,3\). Then webs W(3,2,r) were studied by S. S. Chern in 1936 and the reviewer with co-workers since 1969.

The author of this monograph mostly studies the webs \(W(n+1,n,r)\), W(4,2,r) and the problem of a rank for webs W(d,2,r). They are the main topic of the monograph. Related results by other authors are given, too. The book consists of eight chapters. In Chapter 1 the structure equations of a web \(W(n+1,n,r)\) are derived and its fundamental tensors, namely the torsion and curvature tensors connected with the neighbourhood of the second and the third orders respectively, are introduced. The affine connection naturally associated with a web \(W(n+1,n,r)\) and connections induced by it on the leaves of a web and its three-subwebs are considered. Special classes of webs defined by specific properties of their torsion and curvature tensors, in particular isoclinic and transversally geodesic webs, are studied.

In Chapter 2 the author proves that a web \(W(n+1,n,r)\) defines an almost Grassmann structure \(AG(n-1,r+n-1)\) which becomes a Grassmann one if and only if the web is isoclinic and transversally geodesic simultaneously. In this case the web \(W(n+1,n,r)\) is Grassmannisable, i.e. permits a mapping on a Grassmann manifold \(G(n-1,r+n-1)\) of (n-1)-dimensional subspaces of a projective space \(P^{r+n-1}\). The conditions of Grassmannisability for these webs and for webs W(d,n,r) where \(d>n+1\), \(n\geq 2\), \(r\geq 2\) are found. Then the conditions of algebraisability for webs W(d,n,r) for \(d=n+1\) and \(d>n+1\) are derived. A realisation of these latter conditions leads to a web realized in \(P^{r+n-1}\) by an algebraic manifold of dimension r and degree d. These results are connected with work by the reviewer [Sib. Math. J. 23, No.6, 763-770 (1982); translation from Sib. Mat. Zh. 23, No.6, 6-15 (1982; Zbl 0505.53004); Sov. Math., Dokl. 28, 507-509 (1983); translation from Dokl. Akad. Nauk SSSR 272, 1289-1291 (1983; Zbl 0547.53006)] and J. A. Wood [Duke Math. J. 51, 235-237 (1984; Zbl 0584.14021)].

In Chapter 3 a link between webs \(W(n+1,n,r)\) and local differentiable n- quasigroups is studied. Torsion and curvature tensors of the web are expressed in terms of partial derivatives of second and third orders of functions defining an operation in its coordinate n-quasigroup, and canonical expansions of these functions are found. Then a condition for the existence of a one-parameter n-quasigroup for a web \(W(n+1,n,r)\) is established. Following J. D. H. Smith [Arch. Math. 51, No.2, 169- 177 (1988; Zbl 0627.22003)] a local algebra for an analytic r-dimensional ternary loop is constructed and a theorem analogous to Lie’s third fundamental theorem is proved. This construction is generalized to the case of an analytic n-loop.

In Chapter 4 the author considers some more special classes of webs \(W(n+1,n,r):\) reducible and completely reducible ones, group webs, \((2n+2)\)-hedral and \((n+1)\)-Bol webs. The latter class of webs was studied earlier by S. A. Gerasimenko [Problems of the theory of webs and quasi-groups, Collect. sci. Works, Kalinin 1985, 148-152 (1985; Zbl 0572.53020); Multidimensional Bol \((n+1)\)-webs. (Russian), Preprint, 76 pp., Mosc. Gos. Pedag. Inst., Moscow (1985)]. In Chapter 5 realisations of Grassmann webs \(W(n+1,n,r)\) in projective space \(P^{r+n+1}\) and some of their subclasses are studied. Furthermore in this chapter a web W(4,3,r) formed by four r-pencils of 2r-planes with (2r-1)-dimensional vertices in \(P^{3r}\) is considered.

Chapter 6 is devoted to applications of web theory to a geometry of point correspondences of \(n+1\) projective lines and \(n+1\) projective spaces as well as to a theory of holomorphic mappings between polyhedral domains of complex space \({\mathbb{C}}^ n\). Here V. S. Bolodurin’s [Sov. Math. 28, No.12, 20-26 (1984); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1984, No.12, 17-23 (1984; Zbl 0564.53005); Sov. Math. 29, No.5, 13- 21 (1985); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1985, No.5(276), 15-22 (1985; Zbl 0592.53008)] and J. Baumann’s [Schriftenr. Math. Inst. Univ. Münster, 2. Ser. 24, 152 pp. (1982; Zbl 0545.32001)] results are presented. Chapter 7 deals with four-webs W(4,2,r) on a manifold X of a dimension 2r. Fundamental equations of these webs are derived and their basis affinor is found. Special classes of webs are considered. They are parallelisable, group, almost Grassmannisable and Grassmannisable webs. A link between four-webs and pairs of orthogonal differentiable quasigroups is established. The Desargues and triangle closure conditions on a four-web are considered and their algebraic interpretation in terms of a pair of orthogonal quasigroups is given.

The final Chapter of the monograph is devoted to a rank problem for webs. This chapter is closely connected with S. S. Chern’s and P. A. Griffiths’ investigations [Proc. int. Symp. on algebraic geometry, Kyoto 1977, 85-91 (1977; Zbl 0406.14003); Jahresber. Dtsch. Math.-Ver. 80, 13- 110 (1978; Zbl 0386.14002); Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 5, 539-557 (1978; Zbl 0402.57001)]. In the book a rank problem is considered for webs W(d,2,r), \(d\geq 3\). For these webs the concepts of almost Grassmannisability and almost algebraisability are introduced. The upper bounds for the 1-rank and r-rank of almost Grassmannisable webs W(d,2,r) are established. It is proved that an almost Grassmannisable web W(d,2,r) of maximum 1-rank is always parallelisable and a web W(d,2,2) of maximum 2-rank is always algebraisable for \(d>4\). For \(d=4\) the existence of non-algebraisable webs of maximum rank is proved.

The monograph contains an extended bibliography reporting on all related work up to 1988 and a detailed index.

Reviewer: M.A.Akivis

##### MSC:

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

53A60 | Differential geometry of webs |

20N05 | Loops, quasigroups |

14C21 | Pencils, nets, webs in algebraic geometry |