Theory of Multicodimensional (N+1)-Webs

Problem for a Web W(d, n, r), d> n + 1.- 2.6.3 The Algebraisation Problem for a Web W(3, 2, r).- 2.6.4 The Algebraisation Problem for a Web W(n + 1, n, r).- 2.6.5 The Algebraisation Problem 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-Subquasigroups of a Local Differentiable n-Quasigroup.- 3.7 Comtrans Algebras.- 3.7.1 Preliminaries.- 3.7.2 Comtrans Structures.- 3.7.3 Masking.- 3.7.4 Lie's Third Fundamental Theorem for Analytic 3-Loops.- 3.7.5 General Case of Analytic n-Loops.- 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.- 4.5.1 Definition and Properties of Bol and Moufang (n + 1)-Webs.- 4.5.2 The Bol Closure Conditions.- 4.5.3 A Geometric Characteristic of Bol (n + 1)-Webs.- 4.5.4 An Analytic Characteristic of the Bol Closure Condition (Bn + 1n + 1).- Notes.- 5 Realisations of Multicodimensional (n + 1)-Webs.- 5.1 Grassmann (n + 1)-Webs.- 5.1.1 Basic Definitions.- 5.1.2 The Structure Equations of Projective Space.- 5.1.3 Specialisation of Moving Frames.- 5.1.4 The Structure Equations and the Fundamental Tensors of a Grassmann (n + 1)-Web.- 5.1.5 Transversally Geodesic and Isoclinic Surfaces of a Grassmann (n + 1)-Web.- 5.1.6 The Hexagonality Tensor of a Grassmann (n + 1)-Web and the 2nd Fundamental Forms of Surfaces U?.- 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.- 5.4.1 General Algebraic (n + 1)-Webs.- 5.4.2 Bol Algebraic (n + 1)-Webs.- 5.4.3 Reducible Algebraic (n + 1)-Webs.- 5.4.4 Multiple Reducible Algebraic (n + 1)-Webs.- 5.4.5 Reducible Algebraic Four-Webs.- 5.4.6 Completely 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.- 6.1.1 The Fundamental Equations.- 6.1.2 Correspondences among n + 1 Projective Lines and One-Codimensional (n + 1)-Webs.- 6.1.3 Parallelisable Correpondences.- 6.1.4 Hexagonal Correspondences.- 6.1.5 The Godeaux Homography.- 6.1.6 Parallelisable Godeaux Homographies.- 6.2 The Application of the Theory of (n + 1)-Webs to the Theory of Point Correspondences of n + 1 Projective Spaces.- 6.2.1 The Fundamental Equations.- 6.2.2 Correspondences among n + 1 Projective Lines and Multicodimensional (n + 1)-Webs.- 6.2.3 Parallelisable Correpondences.- 6.2.4 Godeaux Homographies.- 6.2.5 Parallelisable Godeaux Homographies.- 6.2.6 Paratactical Correspondences.- 6.3 Application of the Theory of (n + 1)-Webs to the Theory of Holomorphic Mappings between Polyhedral Domains.- 6.3.1 Introductory Note.- 6.3.2 Analytical Polyhedral Domains in Cn, n> 1.- 6.3.3 Meromorphic Webs in Domains of Cn, n> 1.- 6.3.4 Partition Webs Generated by Analytical Polyhedral Domains.- 6.3.5 Partition Webs with Parallelisable Foliations.- 6.3.6 Partition Webs with Invariant Functions.- Notes.- 7 The Theory of Four-Webs W(4, 2, r).- 7.1 Differential geometry of Four-Webs W(4, 2, r).- 7.1.1 Basic Notions and Equations.- 7.1.2 The Geometrical Meaning of the Basis Affinor.- 7.1.3 Transversal Bivectors Associated with a 4-Web.- 7.1.4 Permutability of Transformations [?, ?].- 7.1.5 Fundamental Equations of a Web W(4, 2, r).- 7.1.6 The Affine Connections Associated with a Web W(4, 2, r).- 7.1.7 Conditions of Geodesicity of Some Lines on the Leaves of a 4-Web in the Canonical Affine Connection ?123.- 7.2 Special Classes of Webs W(4, 2, r).- 7.2.1 Parallelisable Webs W(4, 2, r).- 7.2.2 Webs W(4, 2, r) with Special 3-Subwebs.- 7.2.3 Group Webs W(4, 2, r).- 7.2.4 Parallelisable Webs W(4, 2, r) (Continuation).- 7.2.5 Webs W(4, 2, r) with a Group 3-Subweb.- 7.3 The Canonical Expansions of the Equations of a Pair of Orthogonal Quasigroups Associated with a Web W(4, 2, r).- 7.3.1 A Pair of Orthogonal Quasigroups Associated with a Web W(4, 2, r).- 7.3.2 The Canonical Expansions of the Equations of the Quasigroups A and B.- 7.4 Webs W(4, 2, r) Satisfying the Desargues and Triangle Closure Conditions.- 7.4.1 Webs W(4, 2, r) Satisfying the Desargues Closure Condition D1.- 7.4.2 Group Webs and Webs W(4, 2, r) Satisfying Two Desargues Closure Conditions, D1 and D2.- 7.4.3 Webs W(4, 2, r) Satisfying the Desargues Closure Condition D12.- 7.4.4 Webs W(4, 2, r) Satisfying the Triangle Closure Conditions.- 7.4.5 Properties of Orthogonal Quasigroups Associated with 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).- 7.6.1 Basic Notions.- 7.6.2 Specialisation of Moving Frames.- 7.6.3 The Structure Equations, the Fundamental Tensors, and the Basis Affinor of a Grassmann Web GW(4, 2, r).- 7.6.4 The Connection Forms and the Fundamental Tensors of 3-Subwebs of a Grassmann Web 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).- 8.1.1 Basic Notions and Equations for a Web W(d, 2, r), d ? 3.- 8.1.2 Almost Grassmannisable Webs AGW(d,2,r), d > 3.- 8.1.3 Isoclinic Almost Grassmannisable Webs AGW(d, 2, r).- 8.1.4 Almost Algebraisable Webs AAW(d, 2, r).- 8.1.5 Non-Isoclinic Almost Grassmannisable Webs AGW(4, 2, 2).- 8.1.6 Examples of Non-Extendable Non-Isoclinic Webs W(3, 2, 2).- 8.2 1-Rank Problems for Almost Grassmannisable Webs AGW(d, 2, r).- 8.2.1 Basic Equations for a Web W(d, 2, r) of Non-Zero 1-Rank.- 8.2.2 The Upper Bound for the 1-Rank of an Almost Grassmannisable Web AGW(d, 2, r), r > 1.- 8.2.3 Description of the Webs AGW(d, 2, r), d ? 4,r ? 1, of Maximum 1-Rank.- 8.2.4 Explicit Expressions of the Functions ?? and Description of Their Level Sets.- 8.3 r-Rank Problems for Webs W(d, 2, r).- 8.3.1 The r-Rank of Webs W(d, n, r).- 8.3.2 Almost Grassmannisable Webs AGW(d, 2, 2) of Maximum 2-Rank.- 8.3.3 Webs W(d, 2, 2), d > 4, of Maximum 2-Rank.- 8.3.4 Four-Webs W(4, 2, 2) of Maximum 2-Rank.- 8.4 Examples of Webs W(4, 2, 2) of Maximum 2-Rank.- 8.4.1 The Isoclinic Case.- 8.4.2 The Non-Isoclinic Case.- 8.5 The Geometry of The Exceptional Webs W(4, 2, 2) of Maximum 2-Rank.- 8.5.1 Double Fibrations and Webs.- 8.5.2 Interior Products Associated with an Exceptional Four-Web.- 8.5.3 Exterior 3-Forms Associated with an Exceptional Four-Web.- 8.5.4 Infinitesimal Automorphisms of Exterior Cubic Forms Associated with an Exceptional Four-Web.- 8.5.5 Infinitesimal Conformai Transformations of Exterior Cubic Forms Associated with an Exceptional Four-Web.- Notes.- Symbols Frequently Used.