Theory of Multicodimensional (n+1)-Webs

• 1 Differential Geometry of Multicodimensional (n + 1)-Webs.- 1.1 Fibrations, Foliations, and d-Webs W(d, n, r) of Codimension r on a Differentiable Manifold Xnr.- 1.2 The Structure Equations and Fundamental Tensors of a Web W(n + 1, n, r).- 1.3 Invariant Affine Connections Associated with a Web W(n + 1, n, r).- 1.4 Webs W(n + 1, n, r) with Vanishing Curvature.- 1.5 Parallelisable (n + 1)-Webs.- 1.6 (n + 1)-Webs with Paratactical 3-Subwebs.- 1.7 (n + 1)-Webs with Integrable Diagonal Distributions of 4-Subwebs.- 1.8 (n + 1)-Webs with Integrable Diagonal Distributions.- 1.9 Transversally Geodesic (n + 1)-Webs.- 1.10 Hexagonal (n + 1)-Webs.- 1.11 Isoclinic (n + 1)-Webs.- Notes.- 2 Almost Grassmann Structures Associated with Webs W(n + 1, n, r).- 2.1 Almost Grassmann Structures on a Differentiable Manifold.- 2.2 Structure Equations and Torsion Tensor of an Almost Grassmann Manifold.- 2.3 An Almost Grassmann Structure Associated with a Web W(n + 1, n, r).- 2.4 Semiintegrable Almost Grassmann Structures and Transversally Geodesic and Isoclinic (n + 1)-Webs.- 2.5 Double Webs.- 2.6 Problems of Grassmannisation and Algebraisation and Their Solution for Webs W(d, n, r), d ? n + 1.- Notes.- 3 Local Differentiable n-Quasigroups Associated with a Web W(n + 1, n, r).- 3.1 Local Differentiable n-Quasigroups of a Web W(n + 1, n, r).- 3.2 Structure of a Web W(n + 1, n, r) and Its Coordinate n-Quasigroups in a Neighbourhood of a Point.- 3.3 Computation of the Components of the Torsion and Curvature Tensors of a Web W(n + 1, n, r) in Terms of Its Closed Form Equations.- 3.4 The Relations between the Torsion Tensors and Alternators of Parastrophic Coordinate n-Quasigroups.- 3.5 Canonical Expansions of the Equations of a Local Analytic n-Quasigroup.- 3.6 The One-Parameter n-Subquasigroupsof a Local Differentiable n-Quasigroup.- 3.7 Comtrans Algebras.- Notes.- 4 Special Classes of Multicodimensional (n + 1)-Webs.- 4.1 Reducible (n + 1)-Webs.- 4.2 Multiple Reducible and Completely Reducible (n + 1)-Webs.- 4.3 Group (n + 1)-Webs.- 4.4 (2n + 2)-Hedral (n + 1)-Webs.- 4.5 Bol (n + 1)-Webs.- 5 Realisations of Multicodimensional (n + 1)-Webs.- 5.1 Grassmann (n + 1)-Webs.- 5.2 The Grassmannisation Theorem for Multicodimensional (n + 1)-Webs.- 5.3 Reducible Grassmann (n + 1)-Webs.- 5.4 Algebraic, Bol Algebraic, and Reducible Algebraic (n + 1)-Webs.- 5.5 Moufang Algebraic (n + 1)-Webs.- 5.6 (2n + 2)-Hedral Grassmann (n + 1)-Webs.- 5.7 The Fundamental Equations of a Diagonal 4-Web Formed by Four Pencils of (2r)-Planes in P3r.- 5.8 The Geometry of Diagonal 4-Webs in P3r.- Notes.- 6 Applications of the Theory of (n + 1)-Webs.- 6.1 The Application of the Theory of (n + 1)-Webs to the Theory of Point Correspondences of n + 1 Projective Lines.- 6.2 The Application of the Theory of (n + 1)-Webs to the Theory of Point Correspondences of n + 1 Projective Spaces.- 6.3 Application of the Theory of (n + 1)-Webs to the Theory of Holomorphic Mappings between Polyhedral Domains.- Notes.- 7 The Theory of Four-Webs W(4, 2, r).- 7.1 Differential geometry of Four-Webs W(4, 2, r).- 7.2 Special Classes of Webs W(4, 2, r).- 7.3 The Canonical Expansions of the Equations of a Pair of Orthogonal Quasigroups Associated with a Web W(4, 2, r).- 7.4 Webs W(4, 2, r) Satisfying the Desargues and Triangle Closure Conditions.- 7.5 A Classification of Group Webs W(4, 2, 3).- 7.6 Grassmann Webs GW(4, 2, r).- 7.7 Grassmann Webs GW(4, 2, r) with Algebraic 3-Subwebs.- 7.8 Algebraic Webs AW(4, 2, r).- Notes.- 8 Rank Problems for Webs W(d, 2, r).- 8.1 Almost Grassmannisable and Almost Algebraisable Webs W(d, 2, r).- 8.2 1-Rank Problems for Almost Grassmannisable Webs AGW(d, 2, r).- 8.3 r-Rank Problems for Webs W(d, 2, r).- 8.4 Examples of Webs W(4, 2, 2) of Maximum 2-Rank.- 8.5 The Geometry of The Exceptional Webs W(4, 2, 2) of Maximum 2-Rank.- Notes.- Symbols Frequently Used.

'This book is a basic reference of web geometry and gives a new impulse to the further development of this theory and of the related fields: non-associative algebra, topological, combinatorial and algebraic geometry, theory of foliations and their applications. It is highly recommended to all mathematicians interested in the interrelations of analytical theory, geometry, algebra and topology.' Acta scientiarum mathematicarum, 54:3/4, 1990