Differential Geometry of Varieties With Degenerate Gauss Maps
by Akivis, Maks A.; Goldberg, Vladislav V.Format: Hardcover
Pub. Date: 2003-12-01
Publisher(s): Springer Verlag
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Summary
In this book the authors study the differential geometry of varieties with degenerate Gauss maps. They use the main methods of differential geometry, namely, the methods of moving frames and exterior differential forms as well as tensor methods. By means of these methods, the authors discover the structure of varieties with degenerate Gauss maps, determine the singular points and singular varieties, find focal images and construct a classification of the varieties with degenerate Gauss maps.The authors introduce the above mentioned methods and apply them to a series of concrete problems arising in the theory of varieties with degenerate Gauss maps. What makes this book unique is the authors' use of a systematic application of methods of projective differential geometry along with methods of the classical algebraic geometry for studying varieties with degenerate Gauss maps.This book is intended for researchers and graduate students interested in projective differential geometry and algebraic geometry and their applications. It can be used as a text for advanced undergraduate and graduate students.Each author has published over 100 papers and they have each written a number of books, including Conformal Differential Geometry and Its Generalizations (Wiley 1996), Projective Differential Geometry of Submanifolds (North-Holland 1993), and Introductory Linear Algebra (Prentice-Hall 1972), which were written by them jointly.
Table of Contents
| Preface | p. xi |
| Foundational Material | p. 1 |
| Vector Space | p. 1 |
| The General Linear Group | p. 1 |
| Vectors and Tensors | p. 3 |
| Differentiable Manifolds | p. 5 |
| The Tangent Space, the Frame Bundle, and Tensor Fields | p. 5 |
| Mappings of Differentiable Manifolds | p. 7 |
| Exterior Algebra, Pfaffian Forms, and the Cartan Lemma | p. 9 |
| The Structure Equations of the General Linear Group | p. 12 |
| The Frobenius Theorem | p. 12 |
| The Cartan Test | p. 13 |
| The Structure Equations of a Differentiable Manifold | p. 15 |
| Affine Connections on a Differentiable Manifold | p. 18 |
| Projective Space | p. 19 |
| Projective Transformations, Projective Frames, and the Structure Equations of a Projective Space | p. 19 |
| The Duality Principle | p. 22 |
| Projectivization | p. 24 |
| Classical Homogeneous Spaces (Affine, Euclidean, Non-Euclidean) and Their Transformations | p. 25 |
| Specializations of Moving Frames | p. 28 |
| The First Specialization | p. 28 |
| Power Series Expansion of an Equation of a Curve | p. 30 |
| The Osculating Conic to a Curve | p. 32 |
| The Second and Third Specializations and Their Geometric Meaning | p. 33 |
| The Osculating Cubic to a Curve | p. 35 |
| Two More Specializations and Their Geometric Meaning | p. 37 |
| Conclusions | p. 39 |
| Some Algebraic Manifolds | p. 41 |
| Grassmannians | p. 41 |
| Determinant Submanifolds | p. 44 |
| Notes | p. 46 |
| Varieties in Projective Spaces and Their Gauss Maps | p. 49 |
| Varieties in a Projective Space | p. 49 |
| Equations of a Variety | p. 49 |
| The Bundle of First-Order Frames Associated with a Variety | p. 51 |
| The Prolongation of Basic Equations | p. 53 |
| The Second Fundamental Tensor and the Second Fundamental Form | p. 54 |
| The Second Fundamental Tensor, the Second Fundamental Form, and the Osculating Subspace of a Variety | p. 54 |
| Further Specialization of Moving Frames and Reduced Normal Subspaces | p. 56 |
| Asymptotic Lines and Asymptotic Cone | p. 58 |
| The Osculating Subspace, the Second Fundamental Form, and the Asymptotic Cone of the Grassmannian | p. 59 |
| Varieties with One-Dimensional Normal Subspaces | p. 61 |
| Rank and Defect of Varieties with Degenerate Gauss Maps | p. 63 |
| Examples of Varieties with Degenerate Gauss Maps | p. 65 |
| Application of the Duality Principle | p. 70 |
| Dual Variety | p. 70 |
| The Main Theorem | p. 72 |
| Cubic Symmetroid | p. 76 |
| Singular Points of the Cubic Symmetroid | p. 78 |
| Correlative Transformations | p. 80 |
| Hypersurface with a Degenerate Gauss Map Associated with a Veronese Variety | p. 81 |
| Veronese Varieties and Varieties with Degenerate Gauss Maps | p. 81 |
| Singular Points | p. 85 |
| Notes | p. 86 |
| Basic Equations of Varieties with Degenerate Gauss Maps | p. 91 |
| The Monge-Ampère Foliation | p. 91 |
| The Monge-Ampère Foliation Associated with a Variety with a Degenerate Gauss Map | p. 91 |
| Basic Equations of Varieties with Degenerate Gauss Maps | p. 92 |
| The Structure of Leaves of the Monge-Ampère Foliation | p. 95 |
| The Generalized Griffiths-Harris Theorem | p. 96 |
| Focal Images | p. 99 |
| The Focus Hypersurfaces | p. 99 |
| The Focus Hypercones | p. 101 |
| Examples | p. 102 |
| The Case n = 2 | p. 103 |
| The Case n = 3 | p. 104 |
| Some Algebraic Hypersurfaces with Degenerate Gauss Maps in P4 | p. 105 |
| The Sacksteder-Bourgain Hypersurface | p. 116 |
| The Sacksteder Hypersurface | p. 116 |
| The Bourgain Hypersurface | p. 118 |
| Local Equivalence of Sacksteder's and Bourgain's Hypersurfaces | p. 123 |
| Computation of the Matrices Ci and B¿ for Sacksteder-Bourgain Hypersurfaces | p. 125 |
| Complete Varieties with Degenerate Gauss Maps in Real Projective and Non-Euclidean Spaces | p. 126 |
| Parabolic Varieties | p. 126 |
| Examples | p. 128 |
| Notes | p. 132 |
| Main Structure Theorems | p. 135 |
| Torsal Varieties | p. 135 |
| Hypersurfaces with Degenerate Gauss Maps | p. 141 |
| Sufficient Condition for a Variety with a Degenerate Gauss Map to be a Hypersurface in a Subspace of PN | p. 141 |
| Focal Images of a Hypersurface with a Degenerate Gauss Map | p. 144 |
| Examples of Hypersurfaces with Degenerate Gauss Maps | p. 145 |
| Cones and Affine Analogue of the Hartman-Nirenberg Cylinder Theorem | p. 146 |
| Structure of Focus Hypersurfaces of Cones | p. 146 |
| Affine Analogue of the Hartman-Nirenberg Cylinder Theorem | p. 149 |
| Varieties with Degenerate Gauss Maps with Multiple Foci and Twisted Cones | p. 151 |
| Basic Equations of a Hypersurface of Rank r with r-Multiple Focus Hyperplanes | p. 151 |
| Hypersurfaces with Degenerate Gauss Maps of Rank r with a One-Dimensional Monge-Ampere Foliation and r-Multiple Foci | p. 152 |
| Hypersurfaces with Degenerate Gauss Maps with Double Foci on Their Rectilinear Generators in the Space P4 | p. 154 |
| The Case n = 3 (Continuation) | p. 164 |
| Reducible Varieties with Degenerate Gauss Maps | p. 165 |
| Some Definitions | p. 165 |
| The Structure of Focal Images of Reducible Varieties with Degenerate Gauss Maps | p. 165 |
| The Structure Theorems for Reducible Varieties with Degenerate Gauss Maps | p. 166 |
| Embedding Theorems for Varieties with Degenerate Gauss Maps | p. 169 |
| The Embedding Theorem | p. 169 |
| A Sufficient Condition for a Variety with a Degenerate Gauss Map to be a Cone | p. 172 |
| Notes | p. 172 |
| Further Examples and Applications of the Theory of Varieties with Degenerate Gauss Maps | p. 175 |
| Lightlike Hypersurfaces in the de Sitter Space and Their Focal Properties | p. 176 |
| Lightlike Hypersurfaces and Physics | p. 176 |
| The de Sitter Space | p. 177 |
| Lightlike Hypersurfaces in the de Sitter Space | p. 181 |
| Singular Points of Lightlike Hypersurfaces in the de Sitter Space | p. 184 |
| Lightlike Hypersurfaces of Reduced Rank in the de Sitter Space | p. 192 |
| Induced Connections on Submanifolds | p. 195 |
| Congruences and Pseudocongruences in a Projective Space | p. 195 |
| Normalized Varieties in a Multidimensional Projective Space | p. 198 |
| Normalization of Varieties of Affine and Euclidean Spaces | p. 203 |
| Varieties with Degenerate Gauss Maps Associated with Smooth Lines on Projective Planes over Two-Dimensional Algebras | p. 207 |
| Two-Dimensional Algebras and Their Representations | p. 207 |
| The Projective Planes over the Algebras <$>{\op C}<$>, <$>{\op C}^ | |
| <$>, <$>{\op C}^0<$>, and <$>{\op M}<$> | p. 208 |
| Equation of a Straight Line | p. 209 |
| Moving Frames in Projective Planes over Algebras | p. 210 |
| Focal Properties of the Congruences K, K1, and K0 | p. 211 |
| Smooth Lines in Projective Planes | p. 214 |
| Singular Points of Varieties Corresponding to Smooth Lines in the Projective Spaces over Two-Dimensional Algebras | p. 215 |
| Curvature of Smooth Lines over Algebras | p. 217 |
| Notes | p. 218 |
| Bibliography | p. 221 |
| Symbols Frequently Used | p. 237 |
| Author Index | p. 239 |
| Subject Index | p. 241 |
| Table of Contents provided by Publisher. All Rights Reserved. |
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