Table of Contents
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.
