Theory of Multicodimensional (n+1)-Webs

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Editorial Reviews

From the Publisher
'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
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Product Details

  • ISBN-13: 9789401078542
  • Publisher: Springer Netherlands
  • Publication date: 12/31/2013
  • Series: Mathematics and Its Applications (closed) Series , #44
  • Edition description: Softcover reprint of the original 1st ed. 1988
  • Pages: 466
  • Product dimensions: 6.14 (w) x 9.21 (h) x 0.99 (d)

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.- 1.1.1 Definitions and Examples.- 1.1.2 Closed Form Equations of a Web W(n + 1, n, r) and Further Examples.- 1.2 The Structure Equations and Fundamental Tensors of a Web W(n + 1, n, r).- 1.2.1 Moving Frames Associated with a Web W(n + 1, n, r).- 1.2.2 The Structure Equations and Fundamental Tensors of a Web W(n + 1, n, r).- 1.2.3 The Structure Equations and Fundamental Tensors of a Web W(3, 2, r).- 1.2.4 Special Classes of 3-Webs W(3, 2, r).- 1.3 Invariant Affine Connections Associated with a Web W(n + 1, n, r).- 1.3.1 The Geometrical Meaning of the Forms Wji(?).- 1.3.2 Affine Connections Associated with an (n + 1)-Web.- 1.3.3 The Affine Connections Induced by the Connnection—n+ 1 on Leaves.- 1.3.4 Affine Connections Associated with 3-Subwebs of an (n + 1)-Web.- 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.- 2.1.1 The Segre Variety and the Segre Cone.- 2.1.2 Grassmann and Almost Grassmann Structures.- 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.- 2.6.1 The Grassmannisation Problem for a Web W(n + l, n, r).- 2.6.2 The Grassmannisation 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.

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